%------------------------------------------------------------------------------
% File     : ITP293^3 : TPTP v8.2.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Example 00027_000692
% 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    : 0097_VEBT_Example_00027_000692 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 11020 (6167 unt; 776 typ;   0 def)
%            Number of atoms       : 25326 (11172 equ;   0 cnn)
%            Maximal formula atoms :   28 (   2 avg)
%            Number of connectives : 94481 (2186   ~; 499   |;1294   &;82634   @)
%                                         (   0 <=>;7868  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   40 (   5 avg)
%            Number of types       :   48 (  47 usr)
%            Number of type conns  : 2283 (2283   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  732 ( 729 usr;  56 con; 0-4 aty)
%            Number of variables   : 21985 (1811   ^;19743   !; 431   ?;21985   :)
% 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 23:43:25.825
%------------------------------------------------------------------------------
% Could-be-implicit typings (47)
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thf(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
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thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
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thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
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thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
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thf(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
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thf(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
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thf(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
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thf(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
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thf(sy_c_Assertions_Opure__assn,type,
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thf(sy_c_Binomial_Obinomial,type,
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thf(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
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thf(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
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thf(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
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thf(sy_c_Bit__Comprehension_Obit__comprehension__class_Oset__bits_001t__Int__Oint,type,
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thf(sy_c_Bit__Comprehension_Obit__comprehension__class_Oset__bits_001t__Uint32__Ouint32,type,
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thf(sy_c_Bit__Comprehension_Owf__set__bits__int,type,
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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__Int__Oint,type,
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thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Uint32__Ouint32,type,
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thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Word__Oword_It__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_J,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Code____Numeral__Ointeger,type,
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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_Oand_001t__Uint32__Ouint32,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_Odrop__bit_001t__Uint32__Ouint32,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Word__Oword_It__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_J,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Ointeger,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_Oflip__bit_001t__Uint32__Ouint32,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Word__Oword_It__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_J,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__Code____Numeral__Ointeger,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Uint32__Ouint32,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Uint32__Ouint32,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Word__Oword_It__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_J,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Uint32__Ouint32,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Word__Oword_It__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_J,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Word__Oword_It__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_J,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Uint32__Ouint32,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Word__Oword_It__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_J,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_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__Operations_Osemiring__bits__class_Obit_001t__Uint32__Ouint32,type,
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thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Word__Oword_It__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_J,type,
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thf(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftl_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Word__Oword_It__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_J,type,
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thf(sy_c_Groups_Osgn__class_Osgn_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups_Osgn__class_Osgn_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Osgn__class_Osgn_001t__Int__Oint,type,
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thf(sy_c_Groups_Osgn__class_Osgn_001t__Rat__Orat,type,
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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,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
    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,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Uint32__Ouint32,type,
    times_times_uint32: uint32 > uint32 > uint32 ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Word__Oword_It__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_J,type,
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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,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Real__Oreal_M_Eo_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
    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,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Uint32__Ouint32,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Word__Oword_It__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_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Rat__Orat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Uint32__Ouint32,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Word__Oword_It__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_J,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
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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,
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thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001_Eo_001t__Int__Oint,type,
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thf(sy_c_HOL_OThe_001t__Int__Oint,type,
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thf(sy_c_HOL_OThe_001t__Real__Oreal,type,
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thf(sy_c_HOL_Oundefined_001_062_I_062_It__Code____Numeral__Ointeger_Mt__Uint32__Ouint32_J_M_062_It__Code____Numeral__Ointeger_Mt__Uint32__Ouint32_J_J,type,
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thf(sy_c_HOL_Oundefined_001_062_I_062_It__Nat__Onat_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_M_062_It__Code____Numeral__Ointeger_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J,type,
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thf(sy_c_HOL_Oundefined_001_062_I_062_It__Nat__Onat_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_M_062_It__Uint32__Ouint32_M_062_It__Code____Numeral__Ointeger_Mt__Uint32__Ouint32_J_J_J,type,
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thf(sy_c_HOL_Oundefined_001_062_I_062_It__Uint32__Ouint32_M_062_It__Nat__Onat_M_062_I_Eo_Mt__Uint32__Ouint32_J_J_J_M_062_It__Uint32__Ouint32_M_062_It__Code____Numeral__Ointeger_M_062_I_Eo_Mt__Uint32__Ouint32_J_J_J_J,type,
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thf(sy_c_HOL_Oundefined_001_062_I_062_It__Uint32__Ouint32_M_062_It__Nat__Onat_M_Eo_J_J_M_062_It__Uint32__Ouint32_M_062_It__Code____Numeral__Ointeger_M_Eo_J_J_J,type,
    undefi6981832269580975664eger_o: ( uint32 > nat > $o ) > uint32 > code_integer > $o ).

thf(sy_c_HOL_Oundefined_001_062_I_062_It__Uint32__Ouint32_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_M_062_It__Uint32__Ouint32_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J,type,
    undefi332904144742839227uint32: ( uint32 > uint32 > uint32 ) > uint32 > uint32 > uint32 ).

thf(sy_c_HOL_Oundefined_001_062_I_062_It__Uint32__Ouint32_Mt__Code____Numeral__Ointeger_J_M_062_It__Uint32__Ouint32_Mt__Code____Numeral__Ointeger_J_J,type,
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thf(sy_c_HOL_Oundefined_001_062_It__Code____Numeral__Ointeger_M_062_It__Code____Numeral__Ointeger_M_062_I_Eo_Mt__Code____Numeral__Ointeger_J_J_J,type,
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thf(sy_c_HOL_Oundefined_001_062_It__Code____Numeral__Ointeger_M_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
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thf(sy_c_Hoare__Triple_Ohoare__triple_001_Eo,type,
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thf(sy_c_Hoare__Triple_Ohoare__triple_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_If_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_If_001t__Complex__Ocomplex,type,
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thf(sy_c_If_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
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thf(sy_c_If_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
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thf(sy_c_If_001t__Int__Oint,type,
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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,
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thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
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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,
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thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_If_001t__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J,type,
    if_Pro1135515155860407935uint32: $o > produc827990862158126777uint32 > produc827990862158126777uint32 > produc827990862158126777uint32 ).

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__Uint32__Ouint32,type,
    if_uint32: $o > uint32 > uint32 > uint32 ).

thf(sy_c_If_001t__Word__Oword_It__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_J,type,
    if_wor5778924947035936048l_num1: $o > word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

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_Int_Oring__1__class_Oof__int_001t__Uint32__Ouint32,type,
    ring_1_of_int_uint32: int > uint32 ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Word__Oword_It__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_J,type,
    ring_17408606157368542149l_num1: int > word_N3645301735248828278l_num1 ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Complex__Ocomplex_M_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Int__Oint_M_Eo_J,type,
    sup_sup_int_o: ( int > $o ) > ( int > $o ) > int > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Real__Oreal_M_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Assertions__Oassn,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Extended____Nat__Oenat,type,
    sup_su3973961784419623482d_enat: extended_enat > extended_enat > extended_enat ).

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

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

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

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

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Least__significant__bit_Olsb__class_Olsb_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Least__significant__bit_Olsb__class_Olsb_001t__Int__Oint,type,
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thf(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal,type,
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thf(sy_c_Limits_Oat__infinity_001t__Real__Oreal,type,
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thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
    cons_int: int > list_int > list_int ).

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

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

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

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

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

thf(sy_c_List_Ounion_001t__Complex__Ocomplex,type,
    union_complex: list_complex > list_complex > list_complex ).

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

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

thf(sy_c_List_Ounion_001t__Real__Oreal,type,
    union_real: list_real > list_real > list_real ).

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

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

thf(sy_c_Most__significant__bit_Omsb__class_Omsb_001t__Uint32__Ouint32,type,
    most_s9063628576841037300uint32: uint32 > $o ).

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

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

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

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

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

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

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

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Uint32__Ouint32,type,
    semiri2565882477558803405uint32: nat > uint32 ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Word__Oword_It__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_J,type,
    semiri8819519690708144855l_num1: nat > word_N3645301735248828278l_num1 ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Code____Numeral__Ointeger,type,
    semiri4055485073559036834nteger: ( code_integer > code_integer ) > nat > code_integer > code_integer ).

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

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

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

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
    semiri7787848453975740701ux_rat: ( rat > rat ) > nat > rat > rat ).

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

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Word__Oword_It__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_J,type,
    semiri2846968517960172219l_num1: ( word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ) > nat > word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

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

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

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

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__Uint32__Ouint32,type,
    size_size_uint32: uint32 > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Word__Oword_It__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_J,type,
    size_s8261804613246490634l_num1: word_N3645301735248828278l_num1 > nat ).

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Uint32__Ouint32,type,
    neg_nu5314729912787363643uint32: uint32 > uint32 ).

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Word__Oword_It__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_J,type,
    neg_nu7865238048354675525l_num1: word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

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

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

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

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

thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Uint32__Ouint32,type,
    neg_nu965353292909893953uint32: uint32 > uint32 ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Word__Oword_It__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_J,type,
    neg_nu93272222329896523l_num1: word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

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

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

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

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

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Uint32__Ouint32,type,
    neg_nu4269007558841261821uint32: uint32 > uint32 ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Word__Oword_It__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_J,type,
    neg_nu8115118780965096967l_num1: word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

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

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

thf(sy_c_Num_Onum_OOne,type,
    one: num ).

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

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

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

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

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

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

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

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

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Uint32__Ouint32,type,
    numera9087168376688890119uint32: num > uint32 ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Word__Oword_It__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_J,type,
    numera7442385471795722001l_num1: num > word_N3645301735248828278l_num1 ).

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

thf(sy_c_Num_Opred__numeral,type,
    pred_numeral: num > 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__Extended____Nat__Oenat,type,
    bot_bo4199563552545308370d_enat: extended_enat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    bot_bot_filter_nat: filter_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
    bot_bot_set_o: set_o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    bot_bot_set_complex: set_complex ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__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__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__Real__Oreal_J,type,
    bot_bot_set_real: set_real ).

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_001t__Assertions__Oassn,type,
    ord_less_assn: assn > assn > $o ).

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

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

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

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__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__Real__Oreal_J,type,
    ord_less_set_real: set_real > set_real > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Uint32__Ouint32_J,type,
    ord_less_set_uint32: set_uint32 > set_uint32 > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Word__Oword_It__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_J_J,type,
    ord_le6726900395242856064l_num1: set_wo3913738467083021356l_num1 > set_wo3913738467083021356l_num1 > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Uint32__Ouint32,type,
    ord_less_uint32: uint32 > uint32 > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Word__Oword_It__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_J,type,
    ord_le750835935415966154l_num1: word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 > $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_001t__Assertions__Oassn,type,
    ord_less_eq_assn: assn > assn > $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__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__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).

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thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,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_Oof__int,type,
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thf(sy_c_Rat_Oquotient__of,type,
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thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
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thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,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__Real__Oreal,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__Real__Oreal,type,
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    the_elem_real: set_real > real ).

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Code____Numeral__Ointeger,type,
    set_fo1084959871951514735nteger: ( nat > code_integer > code_integer ) > nat > nat > code_integer > code_integer ).

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

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

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

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

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

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Word__Oword_It__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_J,type,
    set_fo4709898541180519304l_num1: ( nat > word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ) > nat > nat > word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Code____Numeral__Ointeger,type,
    set_or189985376899183464nteger: code_integer > code_integer > set_Code_integer ).

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

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

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

thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Code____Numeral__Ointeger,type,
    set_or8404916559141939852nteger: code_integer > code_integer > set_Code_integer ).

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

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

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

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

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

thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Code____Numeral__Ointeger,type,
    set_or2715278749043346189nteger: code_integer > code_integer > set_Code_integer ).

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_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__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__divide_001t__Word__Oword_It__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_J,type,
    signed6753297604338940182l_num1: word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

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_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__VEBT____BuildupMemImp__OVEBTi,type,
    time_htt_VEBT_VEBTi: assn > heap_T8145700208782473153_VEBTi > ( vEBT_VEBTi > assn ) > nat > $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Transcendental_Odiffs_001t__Code____Numeral__Ointeger,type,
    diffs_Code_integer: ( nat > code_integer ) > nat > code_integer ).

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

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

thf(sy_c_Transcendental_Odiffs_001t__Rat__Orat,type,
    diffs_rat: ( nat > rat ) > nat > rat ).

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

thf(sy_c_Transcendental_Odiffs_001t__Uint32__Ouint32,type,
    diffs_uint32: ( nat > uint32 ) > nat > uint32 ).

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

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

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

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

thf(sy_c_Transcendental_Opi,type,
    pi: real ).

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

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

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

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

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

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

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

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

thf(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____1,type,
    type_l31302759751748491nite_1: itself_finite_1 > nat ).

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_Type__Length_Olen0__class_Olen__of_001t__Numeral____Type__Onum1,type,
    type_l4264026598287037465l_num1: itself_Numeral_num1 > nat ).

thf(sy_c_Uint32_ORep__uint32_H,type,
    rep_uint32: uint32 > word_N3645301735248828278l_num1 ).

thf(sy_c_Uint32_OUint32,type,
    uint322: code_integer > uint32 ).

thf(sy_c_Uint32_OUint32__signed,type,
    uint32_signed: code_integer > uint32 ).

thf(sy_c_Uint32_Odiv0__uint32,type,
    div0_uint32: uint32 > uint32 ).

thf(sy_c_Uint32_Ointeger__of__uint32,type,
    integer_of_uint32: uint32 > code_integer ).

thf(sy_c_Uint32_Ointeger__of__uint32__signed,type,
    intege5370686899274169573signed: uint32 > code_integer ).

thf(sy_c_Uint32_Omod0__uint32,type,
    mod0_uint32: uint32 > uint32 ).

thf(sy_c_Uint32_Oset__bits__aux__uint32,type,
    set_bits_aux_uint32: ( nat > $o ) > nat > uint32 > uint32 ).

thf(sy_c_Uint32_Osigned__drop__bit__uint32,type,
    signed489701013188660438uint32: nat > uint32 > uint32 ).

thf(sy_c_Uint32_Ouint32_OAbs__uint32,type,
    abs_uint32: word_N3645301735248828278l_num1 > uint32 ).

thf(sy_c_Uint32_Ouint32_ORep__uint32,type,
    rep_uint322: uint32 > word_N3645301735248828278l_num1 ).

thf(sy_c_Uint32_Ouint32__div,type,
    uint32_div: uint32 > uint32 > uint32 ).

thf(sy_c_Uint32_Ouint32__divmod,type,
    uint32_divmod: uint32 > uint32 > produc827990862158126777uint32 ).

thf(sy_c_Uint32_Ouint32__mod,type,
    uint32_mod: uint32 > uint32 > uint32 ).

thf(sy_c_Uint32_Ouint32__sdiv,type,
    uint32_sdiv: uint32 > uint32 > uint32 ).

thf(sy_c_Uint32_Ouint32__set__bit,type,
    uint32_set_bit: uint32 > code_integer > $o > uint32 ).

thf(sy_c_Uint32_Ouint32__shiftl,type,
    uint32_shiftl: uint32 > code_integer > uint32 ).

thf(sy_c_Uint32_Ouint32__shiftr,type,
    uint32_shiftr: uint32 > code_integer > uint32 ).

thf(sy_c_Uint32_Ouint32__sshiftr,type,
    uint32_sshiftr: uint32 > code_integer > uint32 ).

thf(sy_c_Uint32_Ouint32__test__bit,type,
    uint32_test_bit: uint32 > code_integer > $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__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__memberi,type,
    vEBT_vebt_memberi: vEBT_VEBTi > nat > heap_Time_Heap_o ).

thf(sy_c_VEBT__DelImperative_Ovebt__deletei,type,
    vEBT_vebt_deletei: vEBT_VEBTi > nat > heap_T8145700208782473153_VEBTi ).

thf(sy_c_VEBT__Example__Setup_Omfold_001t__Nat__Onat_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_E6105538542217078229_VEBTi: ( nat > vEBT_VEBTi > heap_T8145700208782473153_VEBTi ) > list_nat > vEBT_VEBTi > heap_T8145700208782473153_VEBTi ).

thf(sy_c_VEBT__Intf__Imperative_Ovebt__assn,type,
    vEBT_Intf_vebt_assn: nat > set_nat > vEBT_VEBTi > assn ).

thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: nat > nat > 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,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_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_Word_Oeven__word_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,
    even_w9054469088133485505l_num1: word_N3645301735248828278l_num1 > $o ).

thf(sy_c_Word_Oring__1__class_Osigned_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_001t__Complex__Ocomplex,type,
    ring_17006344825680464911omplex: word_N3645301735248828278l_num1 > complex ).

thf(sy_c_Word_Oring__1__class_Osigned_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_001t__Int__Oint,type,
    ring_18494264989212010381m1_int: word_N3645301735248828278l_num1 > int ).

thf(sy_c_Word_Oring__1__class_Osigned_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_001t__Rat__Orat,type,
    ring_17861625399634564921m1_rat: word_N3645301735248828278l_num1 > rat ).

thf(sy_c_Word_Oring__1__class_Osigned_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_001t__Real__Oreal,type,
    ring_130596761880696677251_real: word_N3645301735248828278l_num1 > real ).

thf(sy_c_Word_Oring__1__class_Osigned_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_001t__Uint32__Ouint32,type,
    ring_12341578652349764045uint32: word_N3645301735248828278l_num1 > uint32 ).

thf(sy_c_Word_Oring__1__class_Osigned_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_001t__Word__Oword_It__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_J,type,
    ring_14059547012839848151l_num1: word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

thf(sy_c_Word_Osemiring__1__class_Ounsigned_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_001t__Complex__Ocomplex,type,
    semiri7067251934024306614omplex: word_N3645301735248828278l_num1 > complex ).

thf(sy_c_Word_Osemiring__1__class_Ounsigned_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_001t__Int__Oint,type,
    semiri7338730514057886004m1_int: word_N3645301735248828278l_num1 > int ).

thf(sy_c_Word_Osemiring__1__class_Ounsigned_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_001t__Nat__Onat,type,
    semiri7341220984566936280m1_nat: word_N3645301735248828278l_num1 > nat ).

thf(sy_c_Word_Osemiring__1__class_Ounsigned_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_001t__Rat__Orat,type,
    semiri6706090924480440544m1_rat: word_N3645301735248828278l_num1 > rat ).

thf(sy_c_Word_Osemiring__1__class_Ounsigned_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_001t__Real__Oreal,type,
    semiri46416754965307273481_real: word_N3645301735248828278l_num1 > real ).

thf(sy_c_Word_Osemiring__1__class_Ounsigned_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_001t__Word__Oword_It__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_J,type,
    semiri1312839663145358974l_num1: word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

thf(sy_c_Word_Osigned__drop__bit_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,
    signed5000768011106662067l_num1: nat > word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

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

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

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

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

thf(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > set_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__Real__Oreal,type,
    member_real: real > set_real > $o ).

thf(sy_v_n,type,
    n: nat ).

thf(sy_v_s,type,
    s: set_nat ).

thf(sy_v_t,type,
    t: vEBT_VEBTi ).

thf(sy_v_xs,type,
    xs: list_nat ).

% Relevant facts (10204)
thf(fact_0_n__less__equal__power__2,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% n_less_equal_power_2
thf(fact_1_semiring__norm_I85_J,axiom,
    ! [M: num] :
      ( ( bit0 @ M )
     != one ) ).

% semiring_norm(85)
thf(fact_2_semiring__norm_I83_J,axiom,
    ! [N: num] :
      ( one
     != ( bit0 @ N ) ) ).

% semiring_norm(83)
thf(fact_3_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_4_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_5_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_6_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_7_UnCI,axiom,
    ! [C: real,B: set_real,A: set_real] :
      ( ( ~ ( member_real @ C @ B )
       => ( member_real @ C @ A ) )
     => ( member_real @ C @ ( sup_sup_set_real @ A @ B ) ) ) ).

% UnCI
thf(fact_8_UnCI,axiom,
    ! [C: int,B: set_int,A: set_int] :
      ( ( ~ ( member_int @ C @ B )
       => ( member_int @ C @ A ) )
     => ( member_int @ C @ ( sup_sup_set_int @ A @ B ) ) ) ).

% UnCI
thf(fact_9_UnCI,axiom,
    ! [C: complex,B: set_complex,A: set_complex] :
      ( ( ~ ( member_complex @ C @ B )
       => ( member_complex @ C @ A ) )
     => ( member_complex @ C @ ( sup_sup_set_complex @ A @ B ) ) ) ).

% UnCI
thf(fact_10_UnCI,axiom,
    ! [C: nat,B: set_nat,A: set_nat] :
      ( ( ~ ( member_nat @ C @ B )
       => ( member_nat @ C @ A ) )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% UnCI
thf(fact_11_Un__iff,axiom,
    ! [C: real,A: set_real,B: set_real] :
      ( ( member_real @ C @ ( sup_sup_set_real @ A @ B ) )
      = ( ( member_real @ C @ A )
        | ( member_real @ C @ B ) ) ) ).

% Un_iff
thf(fact_12_Un__iff,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( ( member_int @ C @ ( sup_sup_set_int @ A @ B ) )
      = ( ( member_int @ C @ A )
        | ( member_int @ C @ B ) ) ) ).

% Un_iff
thf(fact_13_Un__iff,axiom,
    ! [C: complex,A: set_complex,B: set_complex] :
      ( ( member_complex @ C @ ( sup_sup_set_complex @ A @ B ) )
      = ( ( member_complex @ C @ A )
        | ( member_complex @ C @ B ) ) ) ).

% Un_iff
thf(fact_14_Un__iff,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
      = ( ( member_nat @ C @ A )
        | ( member_nat @ C @ B ) ) ) ).

% Un_iff
thf(fact_15_sup_Oidem,axiom,
    ! [A2: set_complex] :
      ( ( sup_sup_set_complex @ A2 @ A2 )
      = A2 ) ).

% sup.idem
thf(fact_16_sup_Oidem,axiom,
    ! [A2: set_int] :
      ( ( sup_sup_set_int @ A2 @ A2 )
      = A2 ) ).

% sup.idem
thf(fact_17_sup_Oidem,axiom,
    ! [A2: set_real] :
      ( ( sup_sup_set_real @ A2 @ A2 )
      = A2 ) ).

% sup.idem
thf(fact_18_sup_Oidem,axiom,
    ! [A2: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ A2 )
      = A2 ) ).

% sup.idem
thf(fact_19_sup_Oidem,axiom,
    ! [A2: int] :
      ( ( sup_sup_int @ A2 @ A2 )
      = A2 ) ).

% sup.idem
thf(fact_20_sup_Oidem,axiom,
    ! [A2: nat] :
      ( ( sup_sup_nat @ A2 @ A2 )
      = A2 ) ).

% sup.idem
thf(fact_21_sup_Oidem,axiom,
    ! [A2: extended_enat] :
      ( ( sup_su3973961784419623482d_enat @ A2 @ A2 )
      = A2 ) ).

% sup.idem
thf(fact_22_sup_Oidem,axiom,
    ! [A2: assn] :
      ( ( sup_sup_assn @ A2 @ A2 )
      = A2 ) ).

% sup.idem
thf(fact_23_sup__idem,axiom,
    ! [X: set_complex] :
      ( ( sup_sup_set_complex @ X @ X )
      = X ) ).

% sup_idem
thf(fact_24_sup__idem,axiom,
    ! [X: set_int] :
      ( ( sup_sup_set_int @ X @ X )
      = X ) ).

% sup_idem
thf(fact_25_sup__idem,axiom,
    ! [X: set_real] :
      ( ( sup_sup_set_real @ X @ X )
      = X ) ).

% sup_idem
thf(fact_26_sup__idem,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ X )
      = X ) ).

% sup_idem
thf(fact_27_sup__idem,axiom,
    ! [X: int] :
      ( ( sup_sup_int @ X @ X )
      = X ) ).

% sup_idem
thf(fact_28_sup__idem,axiom,
    ! [X: nat] :
      ( ( sup_sup_nat @ X @ X )
      = X ) ).

% sup_idem
thf(fact_29_sup__idem,axiom,
    ! [X: extended_enat] :
      ( ( sup_su3973961784419623482d_enat @ X @ X )
      = X ) ).

% sup_idem
thf(fact_30_sup__idem,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ X )
      = X ) ).

% sup_idem
thf(fact_31_sup_Oleft__idem,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ( sup_sup_set_complex @ A2 @ ( sup_sup_set_complex @ A2 @ B2 ) )
      = ( sup_sup_set_complex @ A2 @ B2 ) ) ).

% sup.left_idem
thf(fact_32_sup_Oleft__idem,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( sup_sup_set_int @ A2 @ ( sup_sup_set_int @ A2 @ B2 ) )
      = ( sup_sup_set_int @ A2 @ B2 ) ) ).

% sup.left_idem
thf(fact_33_sup_Oleft__idem,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( sup_sup_set_real @ A2 @ ( sup_sup_set_real @ A2 @ B2 ) )
      = ( sup_sup_set_real @ A2 @ B2 ) ) ).

% sup.left_idem
thf(fact_34_sup_Oleft__idem,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
      = ( sup_sup_set_nat @ A2 @ B2 ) ) ).

% sup.left_idem
thf(fact_35_sup_Oleft__idem,axiom,
    ! [A2: int,B2: int] :
      ( ( sup_sup_int @ A2 @ ( sup_sup_int @ A2 @ B2 ) )
      = ( sup_sup_int @ A2 @ B2 ) ) ).

% sup.left_idem
thf(fact_36_sup_Oleft__idem,axiom,
    ! [A2: nat,B2: nat] :
      ( ( sup_sup_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) )
      = ( sup_sup_nat @ A2 @ B2 ) ) ).

% sup.left_idem
thf(fact_37_sup_Oleft__idem,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( sup_su3973961784419623482d_enat @ A2 @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) )
      = ( sup_su3973961784419623482d_enat @ A2 @ B2 ) ) ).

% sup.left_idem
thf(fact_38_sup_Oleft__idem,axiom,
    ! [A2: assn,B2: assn] :
      ( ( sup_sup_assn @ A2 @ ( sup_sup_assn @ A2 @ B2 ) )
      = ( sup_sup_assn @ A2 @ B2 ) ) ).

% sup.left_idem
thf(fact_39_sup__left__idem,axiom,
    ! [X: set_complex,Y: set_complex] :
      ( ( sup_sup_set_complex @ X @ ( sup_sup_set_complex @ X @ Y ) )
      = ( sup_sup_set_complex @ X @ Y ) ) ).

% sup_left_idem
thf(fact_40_sup__left__idem,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( sup_sup_set_int @ X @ ( sup_sup_set_int @ X @ Y ) )
      = ( sup_sup_set_int @ X @ Y ) ) ).

% sup_left_idem
thf(fact_41_sup__left__idem,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( sup_sup_set_real @ X @ ( sup_sup_set_real @ X @ Y ) )
      = ( sup_sup_set_real @ X @ Y ) ) ).

% sup_left_idem
thf(fact_42_sup__left__idem,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
      = ( sup_sup_set_nat @ X @ Y ) ) ).

% sup_left_idem
thf(fact_43_sup__left__idem,axiom,
    ! [X: int,Y: int] :
      ( ( sup_sup_int @ X @ ( sup_sup_int @ X @ Y ) )
      = ( sup_sup_int @ X @ Y ) ) ).

% sup_left_idem
thf(fact_44_sup__left__idem,axiom,
    ! [X: nat,Y: nat] :
      ( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
      = ( sup_sup_nat @ X @ Y ) ) ).

% sup_left_idem
thf(fact_45_sup__left__idem,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( sup_su3973961784419623482d_enat @ X @ ( sup_su3973961784419623482d_enat @ X @ Y ) )
      = ( sup_su3973961784419623482d_enat @ X @ Y ) ) ).

% sup_left_idem
thf(fact_46_sup__left__idem,axiom,
    ! [X: assn,Y: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ X @ Y ) )
      = ( sup_sup_assn @ X @ Y ) ) ).

% sup_left_idem
thf(fact_47_sup_Oright__idem,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ B2 )
      = ( sup_sup_set_nat @ A2 @ B2 ) ) ).

% sup.right_idem
thf(fact_48_sup_Oright__idem,axiom,
    ! [A2: int,B2: int] :
      ( ( sup_sup_int @ ( sup_sup_int @ A2 @ B2 ) @ B2 )
      = ( sup_sup_int @ A2 @ B2 ) ) ).

% sup.right_idem
thf(fact_49_sup_Oright__idem,axiom,
    ! [A2: nat,B2: nat] :
      ( ( sup_sup_nat @ ( sup_sup_nat @ A2 @ B2 ) @ B2 )
      = ( sup_sup_nat @ A2 @ B2 ) ) ).

% sup.right_idem
thf(fact_50_sup_Oright__idem,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( sup_su3973961784419623482d_enat @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) @ B2 )
      = ( sup_su3973961784419623482d_enat @ A2 @ B2 ) ) ).

% sup.right_idem
thf(fact_51_sup_Oright__idem,axiom,
    ! [A2: assn,B2: assn] :
      ( ( sup_sup_assn @ ( sup_sup_assn @ A2 @ B2 ) @ B2 )
      = ( sup_sup_assn @ A2 @ B2 ) ) ).

% sup.right_idem
thf(fact_52_sup_Oright__idem,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ( sup_sup_set_complex @ ( sup_sup_set_complex @ A2 @ B2 ) @ B2 )
      = ( sup_sup_set_complex @ A2 @ B2 ) ) ).

% sup.right_idem
thf(fact_53_sup_Oright__idem,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( sup_sup_set_int @ ( sup_sup_set_int @ A2 @ B2 ) @ B2 )
      = ( sup_sup_set_int @ A2 @ B2 ) ) ).

% sup.right_idem
thf(fact_54_sup_Oright__idem,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( sup_sup_set_real @ ( sup_sup_set_real @ A2 @ B2 ) @ B2 )
      = ( sup_sup_set_real @ A2 @ B2 ) ) ).

% sup.right_idem
thf(fact_55_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_real @ M )
        = ( numeral_numeral_real @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_56_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_rat @ M )
        = ( numeral_numeral_rat @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_57_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_nat @ M )
        = ( numeral_numeral_nat @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_58_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_int @ M )
        = ( numeral_numeral_int @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_59_semiring__norm_I87_J,axiom,
    ! [M: num,N: num] :
      ( ( ( bit0 @ M )
        = ( bit0 @ N ) )
      = ( M = N ) ) ).

% semiring_norm(87)
thf(fact_60_semiring__norm_I78_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(78)
thf(fact_61_semiring__norm_I75_J,axiom,
    ! [M: num] :
      ~ ( ord_less_num @ M @ one ) ).

% semiring_norm(75)
thf(fact_62_semiring__norm_I76_J,axiom,
    ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).

% semiring_norm(76)
thf(fact_63_sup__set__def,axiom,
    ( sup_su6024340866399070445nt_int
    = ( ^ [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ( sup_su8463660629351352368_int_o
            @ ^ [X2: product_prod_int_int] : ( member5262025264175285858nt_int @ X2 @ A3 )
            @ ^ [X2: product_prod_int_int] : ( member5262025264175285858nt_int @ X2 @ B3 ) ) ) ) ) ).

% sup_set_def
thf(fact_64_sup__set__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( collect_nat
          @ ( sup_sup_nat_o
            @ ^ [X2: nat] : ( member_nat @ X2 @ A3 )
            @ ^ [X2: nat] : ( member_nat @ X2 @ B3 ) ) ) ) ) ).

% sup_set_def
thf(fact_65_sup__set__def,axiom,
    ( sup_sup_set_complex
    = ( ^ [A3: set_complex,B3: set_complex] :
          ( collect_complex
          @ ( sup_sup_complex_o
            @ ^ [X2: complex] : ( member_complex @ X2 @ A3 )
            @ ^ [X2: complex] : ( member_complex @ X2 @ B3 ) ) ) ) ) ).

% sup_set_def
thf(fact_66_sup__set__def,axiom,
    ( sup_sup_set_int
    = ( ^ [A3: set_int,B3: set_int] :
          ( collect_int
          @ ( sup_sup_int_o
            @ ^ [X2: int] : ( member_int @ X2 @ A3 )
            @ ^ [X2: int] : ( member_int @ X2 @ B3 ) ) ) ) ) ).

% sup_set_def
thf(fact_67_sup__set__def,axiom,
    ( sup_sup_set_real
    = ( ^ [A3: set_real,B3: set_real] :
          ( collect_real
          @ ( sup_sup_real_o
            @ ^ [X2: real] : ( member_real @ X2 @ A3 )
            @ ^ [X2: real] : ( member_real @ X2 @ B3 ) ) ) ) ) ).

% sup_set_def
thf(fact_68_sup__left__commute,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
      = ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_69_sup__left__commute,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z ) )
      = ( sup_sup_int @ Y @ ( sup_sup_int @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_70_sup__left__commute,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
      = ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_71_sup__left__commute,axiom,
    ! [X: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( ( sup_su3973961784419623482d_enat @ X @ ( sup_su3973961784419623482d_enat @ Y @ Z ) )
      = ( sup_su3973961784419623482d_enat @ Y @ ( sup_su3973961784419623482d_enat @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_72_sup__left__commute,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ Y @ Z ) )
      = ( sup_sup_assn @ Y @ ( sup_sup_assn @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_73_sup__left__commute,axiom,
    ! [X: set_complex,Y: set_complex,Z: set_complex] :
      ( ( sup_sup_set_complex @ X @ ( sup_sup_set_complex @ Y @ Z ) )
      = ( sup_sup_set_complex @ Y @ ( sup_sup_set_complex @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_74_sup__left__commute,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( sup_sup_set_int @ X @ ( sup_sup_set_int @ Y @ Z ) )
      = ( sup_sup_set_int @ Y @ ( sup_sup_set_int @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_75_sup__left__commute,axiom,
    ! [X: set_real,Y: set_real,Z: set_real] :
      ( ( sup_sup_set_real @ X @ ( sup_sup_set_real @ Y @ Z ) )
      = ( sup_sup_set_real @ Y @ ( sup_sup_set_real @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_76_sup_Oleft__commute,axiom,
    ! [B2: set_nat,A2: set_nat,C: set_nat] :
      ( ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ C ) )
      = ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).

% sup.left_commute
thf(fact_77_sup_Oleft__commute,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( sup_sup_int @ B2 @ ( sup_sup_int @ A2 @ C ) )
      = ( sup_sup_int @ A2 @ ( sup_sup_int @ B2 @ C ) ) ) ).

% sup.left_commute
thf(fact_78_sup_Oleft__commute,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( sup_sup_nat @ B2 @ ( sup_sup_nat @ A2 @ C ) )
      = ( sup_sup_nat @ A2 @ ( sup_sup_nat @ B2 @ C ) ) ) ).

% sup.left_commute
thf(fact_79_sup_Oleft__commute,axiom,
    ! [B2: extended_enat,A2: extended_enat,C: extended_enat] :
      ( ( sup_su3973961784419623482d_enat @ B2 @ ( sup_su3973961784419623482d_enat @ A2 @ C ) )
      = ( sup_su3973961784419623482d_enat @ A2 @ ( sup_su3973961784419623482d_enat @ B2 @ C ) ) ) ).

% sup.left_commute
thf(fact_80_sup_Oleft__commute,axiom,
    ! [B2: assn,A2: assn,C: assn] :
      ( ( sup_sup_assn @ B2 @ ( sup_sup_assn @ A2 @ C ) )
      = ( sup_sup_assn @ A2 @ ( sup_sup_assn @ B2 @ C ) ) ) ).

% sup.left_commute
thf(fact_81_sup_Oleft__commute,axiom,
    ! [B2: set_complex,A2: set_complex,C: set_complex] :
      ( ( sup_sup_set_complex @ B2 @ ( sup_sup_set_complex @ A2 @ C ) )
      = ( sup_sup_set_complex @ A2 @ ( sup_sup_set_complex @ B2 @ C ) ) ) ).

% sup.left_commute
thf(fact_82_sup_Oleft__commute,axiom,
    ! [B2: set_int,A2: set_int,C: set_int] :
      ( ( sup_sup_set_int @ B2 @ ( sup_sup_set_int @ A2 @ C ) )
      = ( sup_sup_set_int @ A2 @ ( sup_sup_set_int @ B2 @ C ) ) ) ).

% sup.left_commute
thf(fact_83_sup_Oleft__commute,axiom,
    ! [B2: set_real,A2: set_real,C: set_real] :
      ( ( sup_sup_set_real @ B2 @ ( sup_sup_set_real @ A2 @ C ) )
      = ( sup_sup_set_real @ A2 @ ( sup_sup_set_real @ B2 @ C ) ) ) ).

% sup.left_commute
thf(fact_84_sup__commute,axiom,
    ( sup_sup_set_nat
    = ( ^ [X2: set_nat,Y2: set_nat] : ( sup_sup_set_nat @ Y2 @ X2 ) ) ) ).

% sup_commute
thf(fact_85_sup__commute,axiom,
    ( sup_sup_int
    = ( ^ [X2: int,Y2: int] : ( sup_sup_int @ Y2 @ X2 ) ) ) ).

% sup_commute
thf(fact_86_sup__commute,axiom,
    ( sup_sup_nat
    = ( ^ [X2: nat,Y2: nat] : ( sup_sup_nat @ Y2 @ X2 ) ) ) ).

% sup_commute
thf(fact_87_sup__commute,axiom,
    ( sup_su3973961784419623482d_enat
    = ( ^ [X2: extended_enat,Y2: extended_enat] : ( sup_su3973961784419623482d_enat @ Y2 @ X2 ) ) ) ).

% sup_commute
thf(fact_88_sup__commute,axiom,
    ( sup_sup_assn
    = ( ^ [X2: assn,Y2: assn] : ( sup_sup_assn @ Y2 @ X2 ) ) ) ).

% sup_commute
thf(fact_89_sup__commute,axiom,
    ( sup_sup_set_complex
    = ( ^ [X2: set_complex,Y2: set_complex] : ( sup_sup_set_complex @ Y2 @ X2 ) ) ) ).

% sup_commute
thf(fact_90_sup__commute,axiom,
    ( sup_sup_set_int
    = ( ^ [X2: set_int,Y2: set_int] : ( sup_sup_set_int @ Y2 @ X2 ) ) ) ).

% sup_commute
thf(fact_91_sup__commute,axiom,
    ( sup_sup_set_real
    = ( ^ [X2: set_real,Y2: set_real] : ( sup_sup_set_real @ Y2 @ X2 ) ) ) ).

% sup_commute
thf(fact_92_sup_Ocommute,axiom,
    ( sup_sup_set_nat
    = ( ^ [A4: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A4 ) ) ) ).

% sup.commute
thf(fact_93_sup_Ocommute,axiom,
    ( sup_sup_int
    = ( ^ [A4: int,B4: int] : ( sup_sup_int @ B4 @ A4 ) ) ) ).

% sup.commute
thf(fact_94_sup_Ocommute,axiom,
    ( sup_sup_nat
    = ( ^ [A4: nat,B4: nat] : ( sup_sup_nat @ B4 @ A4 ) ) ) ).

% sup.commute
thf(fact_95_sup_Ocommute,axiom,
    ( sup_su3973961784419623482d_enat
    = ( ^ [A4: extended_enat,B4: extended_enat] : ( sup_su3973961784419623482d_enat @ B4 @ A4 ) ) ) ).

% sup.commute
thf(fact_96_sup_Ocommute,axiom,
    ( sup_sup_assn
    = ( ^ [A4: assn,B4: assn] : ( sup_sup_assn @ B4 @ A4 ) ) ) ).

% sup.commute
thf(fact_97_sup_Ocommute,axiom,
    ( sup_sup_set_complex
    = ( ^ [A4: set_complex,B4: set_complex] : ( sup_sup_set_complex @ B4 @ A4 ) ) ) ).

% sup.commute
thf(fact_98_sup_Ocommute,axiom,
    ( sup_sup_set_int
    = ( ^ [A4: set_int,B4: set_int] : ( sup_sup_set_int @ B4 @ A4 ) ) ) ).

% sup.commute
thf(fact_99_sup_Ocommute,axiom,
    ( sup_sup_set_real
    = ( ^ [A4: set_real,B4: set_real] : ( sup_sup_set_real @ B4 @ A4 ) ) ) ).

% sup.commute
thf(fact_100_sup__assoc,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
      = ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).

% sup_assoc
thf(fact_101_sup__assoc,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( sup_sup_int @ ( sup_sup_int @ X @ Y ) @ Z )
      = ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z ) ) ) ).

% sup_assoc
thf(fact_102_sup__assoc,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
      = ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).

% sup_assoc
thf(fact_103_sup__assoc,axiom,
    ! [X: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( ( sup_su3973961784419623482d_enat @ ( sup_su3973961784419623482d_enat @ X @ Y ) @ Z )
      = ( sup_su3973961784419623482d_enat @ X @ ( sup_su3973961784419623482d_enat @ Y @ Z ) ) ) ).

% sup_assoc
thf(fact_104_sup__assoc,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ ( sup_sup_assn @ X @ Y ) @ Z )
      = ( sup_sup_assn @ X @ ( sup_sup_assn @ Y @ Z ) ) ) ).

% sup_assoc
thf(fact_105_sup__assoc,axiom,
    ! [X: set_complex,Y: set_complex,Z: set_complex] :
      ( ( sup_sup_set_complex @ ( sup_sup_set_complex @ X @ Y ) @ Z )
      = ( sup_sup_set_complex @ X @ ( sup_sup_set_complex @ Y @ Z ) ) ) ).

% sup_assoc
thf(fact_106_sup__assoc,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( sup_sup_set_int @ ( sup_sup_set_int @ X @ Y ) @ Z )
      = ( sup_sup_set_int @ X @ ( sup_sup_set_int @ Y @ Z ) ) ) ).

% sup_assoc
thf(fact_107_sup__assoc,axiom,
    ! [X: set_real,Y: set_real,Z: set_real] :
      ( ( sup_sup_set_real @ ( sup_sup_set_real @ X @ Y ) @ Z )
      = ( sup_sup_set_real @ X @ ( sup_sup_set_real @ Y @ Z ) ) ) ).

% sup_assoc
thf(fact_108_sup_Oassoc,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C )
      = ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).

% sup.assoc
thf(fact_109_sup_Oassoc,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( sup_sup_int @ ( sup_sup_int @ A2 @ B2 ) @ C )
      = ( sup_sup_int @ A2 @ ( sup_sup_int @ B2 @ C ) ) ) ).

% sup.assoc
thf(fact_110_sup_Oassoc,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( sup_sup_nat @ ( sup_sup_nat @ A2 @ B2 ) @ C )
      = ( sup_sup_nat @ A2 @ ( sup_sup_nat @ B2 @ C ) ) ) ).

% sup.assoc
thf(fact_111_sup_Oassoc,axiom,
    ! [A2: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( sup_su3973961784419623482d_enat @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) @ C )
      = ( sup_su3973961784419623482d_enat @ A2 @ ( sup_su3973961784419623482d_enat @ B2 @ C ) ) ) ).

% sup.assoc
thf(fact_112_sup_Oassoc,axiom,
    ! [A2: assn,B2: assn,C: assn] :
      ( ( sup_sup_assn @ ( sup_sup_assn @ A2 @ B2 ) @ C )
      = ( sup_sup_assn @ A2 @ ( sup_sup_assn @ B2 @ C ) ) ) ).

% sup.assoc
thf(fact_113_sup_Oassoc,axiom,
    ! [A2: set_complex,B2: set_complex,C: set_complex] :
      ( ( sup_sup_set_complex @ ( sup_sup_set_complex @ A2 @ B2 ) @ C )
      = ( sup_sup_set_complex @ A2 @ ( sup_sup_set_complex @ B2 @ C ) ) ) ).

% sup.assoc
thf(fact_114_sup_Oassoc,axiom,
    ! [A2: set_int,B2: set_int,C: set_int] :
      ( ( sup_sup_set_int @ ( sup_sup_set_int @ A2 @ B2 ) @ C )
      = ( sup_sup_set_int @ A2 @ ( sup_sup_set_int @ B2 @ C ) ) ) ).

% sup.assoc
thf(fact_115_sup_Oassoc,axiom,
    ! [A2: set_real,B2: set_real,C: set_real] :
      ( ( sup_sup_set_real @ ( sup_sup_set_real @ A2 @ B2 ) @ C )
      = ( sup_sup_set_real @ A2 @ ( sup_sup_set_real @ B2 @ C ) ) ) ).

% sup.assoc
thf(fact_116_inf__sup__aci_I5_J,axiom,
    ( sup_sup_set_nat
    = ( ^ [X2: set_nat,Y2: set_nat] : ( sup_sup_set_nat @ Y2 @ X2 ) ) ) ).

% inf_sup_aci(5)
thf(fact_117_inf__sup__aci_I5_J,axiom,
    ( sup_sup_int
    = ( ^ [X2: int,Y2: int] : ( sup_sup_int @ Y2 @ X2 ) ) ) ).

% inf_sup_aci(5)
thf(fact_118_inf__sup__aci_I5_J,axiom,
    ( sup_sup_nat
    = ( ^ [X2: nat,Y2: nat] : ( sup_sup_nat @ Y2 @ X2 ) ) ) ).

% inf_sup_aci(5)
thf(fact_119_inf__sup__aci_I5_J,axiom,
    ( sup_su3973961784419623482d_enat
    = ( ^ [X2: extended_enat,Y2: extended_enat] : ( sup_su3973961784419623482d_enat @ Y2 @ X2 ) ) ) ).

% inf_sup_aci(5)
thf(fact_120_inf__sup__aci_I5_J,axiom,
    ( sup_sup_assn
    = ( ^ [X2: assn,Y2: assn] : ( sup_sup_assn @ Y2 @ X2 ) ) ) ).

% inf_sup_aci(5)
thf(fact_121_inf__sup__aci_I5_J,axiom,
    ( sup_sup_set_complex
    = ( ^ [X2: set_complex,Y2: set_complex] : ( sup_sup_set_complex @ Y2 @ X2 ) ) ) ).

% inf_sup_aci(5)
thf(fact_122_inf__sup__aci_I5_J,axiom,
    ( sup_sup_set_int
    = ( ^ [X2: set_int,Y2: set_int] : ( sup_sup_set_int @ Y2 @ X2 ) ) ) ).

% inf_sup_aci(5)
thf(fact_123_inf__sup__aci_I5_J,axiom,
    ( sup_sup_set_real
    = ( ^ [X2: set_real,Y2: set_real] : ( sup_sup_set_real @ Y2 @ X2 ) ) ) ).

% inf_sup_aci(5)
thf(fact_124_inf__sup__aci_I6_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
      = ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_125_inf__sup__aci_I6_J,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( sup_sup_int @ ( sup_sup_int @ X @ Y ) @ Z )
      = ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_126_inf__sup__aci_I6_J,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
      = ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_127_inf__sup__aci_I6_J,axiom,
    ! [X: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( ( sup_su3973961784419623482d_enat @ ( sup_su3973961784419623482d_enat @ X @ Y ) @ Z )
      = ( sup_su3973961784419623482d_enat @ X @ ( sup_su3973961784419623482d_enat @ Y @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_128_inf__sup__aci_I6_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ ( sup_sup_assn @ X @ Y ) @ Z )
      = ( sup_sup_assn @ X @ ( sup_sup_assn @ Y @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_129_inf__sup__aci_I6_J,axiom,
    ! [X: set_complex,Y: set_complex,Z: set_complex] :
      ( ( sup_sup_set_complex @ ( sup_sup_set_complex @ X @ Y ) @ Z )
      = ( sup_sup_set_complex @ X @ ( sup_sup_set_complex @ Y @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_130_inf__sup__aci_I6_J,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( sup_sup_set_int @ ( sup_sup_set_int @ X @ Y ) @ Z )
      = ( sup_sup_set_int @ X @ ( sup_sup_set_int @ Y @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_131_inf__sup__aci_I6_J,axiom,
    ! [X: set_real,Y: set_real,Z: set_real] :
      ( ( sup_sup_set_real @ ( sup_sup_set_real @ X @ Y ) @ Z )
      = ( sup_sup_set_real @ X @ ( sup_sup_set_real @ Y @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_132_inf__sup__aci_I7_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
      = ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_133_inf__sup__aci_I7_J,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z ) )
      = ( sup_sup_int @ Y @ ( sup_sup_int @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_134_inf__sup__aci_I7_J,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
      = ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_135_inf__sup__aci_I7_J,axiom,
    ! [X: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( ( sup_su3973961784419623482d_enat @ X @ ( sup_su3973961784419623482d_enat @ Y @ Z ) )
      = ( sup_su3973961784419623482d_enat @ Y @ ( sup_su3973961784419623482d_enat @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_136_inf__sup__aci_I7_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ Y @ Z ) )
      = ( sup_sup_assn @ Y @ ( sup_sup_assn @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_137_inf__sup__aci_I7_J,axiom,
    ! [X: set_complex,Y: set_complex,Z: set_complex] :
      ( ( sup_sup_set_complex @ X @ ( sup_sup_set_complex @ Y @ Z ) )
      = ( sup_sup_set_complex @ Y @ ( sup_sup_set_complex @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_138_inf__sup__aci_I7_J,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( sup_sup_set_int @ X @ ( sup_sup_set_int @ Y @ Z ) )
      = ( sup_sup_set_int @ Y @ ( sup_sup_set_int @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_139_inf__sup__aci_I7_J,axiom,
    ! [X: set_real,Y: set_real,Z: set_real] :
      ( ( sup_sup_set_real @ X @ ( sup_sup_set_real @ Y @ Z ) )
      = ( sup_sup_set_real @ Y @ ( sup_sup_set_real @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_140_inf__sup__aci_I8_J,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
      = ( sup_sup_set_nat @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_141_inf__sup__aci_I8_J,axiom,
    ! [X: int,Y: int] :
      ( ( sup_sup_int @ X @ ( sup_sup_int @ X @ Y ) )
      = ( sup_sup_int @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_142_inf__sup__aci_I8_J,axiom,
    ! [X: nat,Y: nat] :
      ( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
      = ( sup_sup_nat @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_143_inf__sup__aci_I8_J,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( sup_su3973961784419623482d_enat @ X @ ( sup_su3973961784419623482d_enat @ X @ Y ) )
      = ( sup_su3973961784419623482d_enat @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_144_inf__sup__aci_I8_J,axiom,
    ! [X: assn,Y: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ X @ Y ) )
      = ( sup_sup_assn @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_145_inf__sup__aci_I8_J,axiom,
    ! [X: set_complex,Y: set_complex] :
      ( ( sup_sup_set_complex @ X @ ( sup_sup_set_complex @ X @ Y ) )
      = ( sup_sup_set_complex @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_146_inf__sup__aci_I8_J,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( sup_sup_set_int @ X @ ( sup_sup_set_int @ X @ Y ) )
      = ( sup_sup_set_int @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_147_inf__sup__aci_I8_J,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( sup_sup_set_real @ X @ ( sup_sup_set_real @ X @ Y ) )
      = ( sup_sup_set_real @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_148_Un__left__commute,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) )
      = ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C2 ) ) ) ).

% Un_left_commute
thf(fact_149_Un__left__commute,axiom,
    ! [A: set_complex,B: set_complex,C2: set_complex] :
      ( ( sup_sup_set_complex @ A @ ( sup_sup_set_complex @ B @ C2 ) )
      = ( sup_sup_set_complex @ B @ ( sup_sup_set_complex @ A @ C2 ) ) ) ).

% Un_left_commute
thf(fact_150_Un__left__commute,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( ( sup_sup_set_int @ A @ ( sup_sup_set_int @ B @ C2 ) )
      = ( sup_sup_set_int @ B @ ( sup_sup_set_int @ A @ C2 ) ) ) ).

% Un_left_commute
thf(fact_151_Un__left__commute,axiom,
    ! [A: set_real,B: set_real,C2: set_real] :
      ( ( sup_sup_set_real @ A @ ( sup_sup_set_real @ B @ C2 ) )
      = ( sup_sup_set_real @ B @ ( sup_sup_set_real @ A @ C2 ) ) ) ).

% Un_left_commute
thf(fact_152_Un__left__absorb,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) )
      = ( sup_sup_set_nat @ A @ B ) ) ).

% Un_left_absorb
thf(fact_153_Un__left__absorb,axiom,
    ! [A: set_complex,B: set_complex] :
      ( ( sup_sup_set_complex @ A @ ( sup_sup_set_complex @ A @ B ) )
      = ( sup_sup_set_complex @ A @ B ) ) ).

% Un_left_absorb
thf(fact_154_Un__left__absorb,axiom,
    ! [A: set_int,B: set_int] :
      ( ( sup_sup_set_int @ A @ ( sup_sup_set_int @ A @ B ) )
      = ( sup_sup_set_int @ A @ B ) ) ).

% Un_left_absorb
thf(fact_155_Un__left__absorb,axiom,
    ! [A: set_real,B: set_real] :
      ( ( sup_sup_set_real @ A @ ( sup_sup_set_real @ A @ B ) )
      = ( sup_sup_set_real @ A @ B ) ) ).

% Un_left_absorb
thf(fact_156_Un__commute,axiom,
    ( sup_sup_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).

% Un_commute
thf(fact_157_Un__commute,axiom,
    ( sup_sup_set_complex
    = ( ^ [A3: set_complex,B3: set_complex] : ( sup_sup_set_complex @ B3 @ A3 ) ) ) ).

% Un_commute
thf(fact_158_Un__commute,axiom,
    ( sup_sup_set_int
    = ( ^ [A3: set_int,B3: set_int] : ( sup_sup_set_int @ B3 @ A3 ) ) ) ).

% Un_commute
thf(fact_159_Un__commute,axiom,
    ( sup_sup_set_real
    = ( ^ [A3: set_real,B3: set_real] : ( sup_sup_set_real @ B3 @ A3 ) ) ) ).

% Un_commute
thf(fact_160_Un__absorb,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ A @ A )
      = A ) ).

% Un_absorb
thf(fact_161_Un__absorb,axiom,
    ! [A: set_complex] :
      ( ( sup_sup_set_complex @ A @ A )
      = A ) ).

% Un_absorb
thf(fact_162_Un__absorb,axiom,
    ! [A: set_int] :
      ( ( sup_sup_set_int @ A @ A )
      = A ) ).

% Un_absorb
thf(fact_163_Un__absorb,axiom,
    ! [A: set_real] :
      ( ( sup_sup_set_real @ A @ A )
      = A ) ).

% Un_absorb
thf(fact_164_Un__assoc,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
      = ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).

% Un_assoc
thf(fact_165_Un__assoc,axiom,
    ! [A: set_complex,B: set_complex,C2: set_complex] :
      ( ( sup_sup_set_complex @ ( sup_sup_set_complex @ A @ B ) @ C2 )
      = ( sup_sup_set_complex @ A @ ( sup_sup_set_complex @ B @ C2 ) ) ) ).

% Un_assoc
thf(fact_166_Un__assoc,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( ( sup_sup_set_int @ ( sup_sup_set_int @ A @ B ) @ C2 )
      = ( sup_sup_set_int @ A @ ( sup_sup_set_int @ B @ C2 ) ) ) ).

% Un_assoc
thf(fact_167_Un__assoc,axiom,
    ! [A: set_real,B: set_real,C2: set_real] :
      ( ( sup_sup_set_real @ ( sup_sup_set_real @ A @ B ) @ C2 )
      = ( sup_sup_set_real @ A @ ( sup_sup_set_real @ B @ C2 ) ) ) ).

% Un_assoc
thf(fact_168_ball__Un,axiom,
    ! [A: set_nat,B: set_nat,P: nat > $o] :
      ( ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) )
           => ( P @ X2 ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ A )
           => ( P @ X2 ) )
        & ! [X2: nat] :
            ( ( member_nat @ X2 @ B )
           => ( P @ X2 ) ) ) ) ).

% ball_Un
thf(fact_169_ball__Un,axiom,
    ! [A: set_complex,B: set_complex,P: complex > $o] :
      ( ( ! [X2: complex] :
            ( ( member_complex @ X2 @ ( sup_sup_set_complex @ A @ B ) )
           => ( P @ X2 ) ) )
      = ( ! [X2: complex] :
            ( ( member_complex @ X2 @ A )
           => ( P @ X2 ) )
        & ! [X2: complex] :
            ( ( member_complex @ X2 @ B )
           => ( P @ X2 ) ) ) ) ).

% ball_Un
thf(fact_170_ball__Un,axiom,
    ! [A: set_int,B: set_int,P: int > $o] :
      ( ( ! [X2: int] :
            ( ( member_int @ X2 @ ( sup_sup_set_int @ A @ B ) )
           => ( P @ X2 ) ) )
      = ( ! [X2: int] :
            ( ( member_int @ X2 @ A )
           => ( P @ X2 ) )
        & ! [X2: int] :
            ( ( member_int @ X2 @ B )
           => ( P @ X2 ) ) ) ) ).

% ball_Un
thf(fact_171_ball__Un,axiom,
    ! [A: set_real,B: set_real,P: real > $o] :
      ( ( ! [X2: real] :
            ( ( member_real @ X2 @ ( sup_sup_set_real @ A @ B ) )
           => ( P @ X2 ) ) )
      = ( ! [X2: real] :
            ( ( member_real @ X2 @ A )
           => ( P @ X2 ) )
        & ! [X2: real] :
            ( ( member_real @ X2 @ B )
           => ( P @ X2 ) ) ) ) ).

% ball_Un
thf(fact_172_bex__Un,axiom,
    ! [A: set_nat,B: set_nat,P: nat > $o] :
      ( ( ? [X2: nat] :
            ( ( member_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) )
            & ( P @ X2 ) ) )
      = ( ? [X2: nat] :
            ( ( member_nat @ X2 @ A )
            & ( P @ X2 ) )
        | ? [X2: nat] :
            ( ( member_nat @ X2 @ B )
            & ( P @ X2 ) ) ) ) ).

% bex_Un
thf(fact_173_bex__Un,axiom,
    ! [A: set_complex,B: set_complex,P: complex > $o] :
      ( ( ? [X2: complex] :
            ( ( member_complex @ X2 @ ( sup_sup_set_complex @ A @ B ) )
            & ( P @ X2 ) ) )
      = ( ? [X2: complex] :
            ( ( member_complex @ X2 @ A )
            & ( P @ X2 ) )
        | ? [X2: complex] :
            ( ( member_complex @ X2 @ B )
            & ( P @ X2 ) ) ) ) ).

% bex_Un
thf(fact_174_bex__Un,axiom,
    ! [A: set_int,B: set_int,P: int > $o] :
      ( ( ? [X2: int] :
            ( ( member_int @ X2 @ ( sup_sup_set_int @ A @ B ) )
            & ( P @ X2 ) ) )
      = ( ? [X2: int] :
            ( ( member_int @ X2 @ A )
            & ( P @ X2 ) )
        | ? [X2: int] :
            ( ( member_int @ X2 @ B )
            & ( P @ X2 ) ) ) ) ).

% bex_Un
thf(fact_175_bex__Un,axiom,
    ! [A: set_real,B: set_real,P: real > $o] :
      ( ( ? [X2: real] :
            ( ( member_real @ X2 @ ( sup_sup_set_real @ A @ B ) )
            & ( P @ X2 ) ) )
      = ( ? [X2: real] :
            ( ( member_real @ X2 @ A )
            & ( P @ X2 ) )
        | ? [X2: real] :
            ( ( member_real @ X2 @ B )
            & ( P @ X2 ) ) ) ) ).

% bex_Un
thf(fact_176_UnI2,axiom,
    ! [C: nat,B: set_nat,A: set_nat] :
      ( ( member_nat @ C @ B )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% UnI2
thf(fact_177_UnI2,axiom,
    ! [C: complex,B: set_complex,A: set_complex] :
      ( ( member_complex @ C @ B )
     => ( member_complex @ C @ ( sup_sup_set_complex @ A @ B ) ) ) ).

% UnI2
thf(fact_178_UnI2,axiom,
    ! [C: int,B: set_int,A: set_int] :
      ( ( member_int @ C @ B )
     => ( member_int @ C @ ( sup_sup_set_int @ A @ B ) ) ) ).

% UnI2
thf(fact_179_UnI2,axiom,
    ! [C: real,B: set_real,A: set_real] :
      ( ( member_real @ C @ B )
     => ( member_real @ C @ ( sup_sup_set_real @ A @ B ) ) ) ).

% UnI2
thf(fact_180_UnI1,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C @ A )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% UnI1
thf(fact_181_UnI1,axiom,
    ! [C: complex,A: set_complex,B: set_complex] :
      ( ( member_complex @ C @ A )
     => ( member_complex @ C @ ( sup_sup_set_complex @ A @ B ) ) ) ).

% UnI1
thf(fact_182_UnI1,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( ( member_int @ C @ A )
     => ( member_int @ C @ ( sup_sup_set_int @ A @ B ) ) ) ).

% UnI1
thf(fact_183_UnI1,axiom,
    ! [C: real,A: set_real,B: set_real] :
      ( ( member_real @ C @ A )
     => ( member_real @ C @ ( sup_sup_set_real @ A @ B ) ) ) ).

% UnI1
thf(fact_184_UnE,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
     => ( ~ ( member_nat @ C @ A )
       => ( member_nat @ C @ B ) ) ) ).

% UnE
thf(fact_185_UnE,axiom,
    ! [C: complex,A: set_complex,B: set_complex] :
      ( ( member_complex @ C @ ( sup_sup_set_complex @ A @ B ) )
     => ( ~ ( member_complex @ C @ A )
       => ( member_complex @ C @ B ) ) ) ).

% UnE
thf(fact_186_UnE,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( ( member_int @ C @ ( sup_sup_set_int @ A @ B ) )
     => ( ~ ( member_int @ C @ A )
       => ( member_int @ C @ B ) ) ) ).

% UnE
thf(fact_187_UnE,axiom,
    ! [C: real,A: set_real,B: set_real] :
      ( ( member_real @ C @ ( sup_sup_set_real @ A @ B ) )
     => ( ~ ( member_real @ C @ A )
       => ( member_real @ C @ B ) ) ) ).

% UnE
thf(fact_188_Collect__disj__eq,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ( collec213857154873943460nt_int
        @ ^ [X2: product_prod_int_int] :
            ( ( P @ X2 )
            | ( Q @ X2 ) ) )
      = ( sup_su6024340866399070445nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_189_Collect__disj__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X2: nat] :
            ( ( P @ X2 )
            | ( Q @ X2 ) ) )
      = ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_190_Collect__disj__eq,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( collect_complex
        @ ^ [X2: complex] :
            ( ( P @ X2 )
            | ( Q @ X2 ) ) )
      = ( sup_sup_set_complex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_191_Collect__disj__eq,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( collect_int
        @ ^ [X2: int] :
            ( ( P @ X2 )
            | ( Q @ X2 ) ) )
      = ( sup_sup_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_192_Collect__disj__eq,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ( collect_real
        @ ^ [X2: real] :
            ( ( P @ X2 )
            | ( Q @ X2 ) ) )
      = ( sup_sup_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_193_Un__def,axiom,
    ( sup_su6024340866399070445nt_int
    = ( ^ [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ^ [X2: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X2 @ A3 )
              | ( member5262025264175285858nt_int @ X2 @ B3 ) ) ) ) ) ).

% Un_def
thf(fact_194_Un__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( collect_nat
          @ ^ [X2: nat] :
              ( ( member_nat @ X2 @ A3 )
              | ( member_nat @ X2 @ B3 ) ) ) ) ) ).

% Un_def
thf(fact_195_Un__def,axiom,
    ( sup_sup_set_complex
    = ( ^ [A3: set_complex,B3: set_complex] :
          ( collect_complex
          @ ^ [X2: complex] :
              ( ( member_complex @ X2 @ A3 )
              | ( member_complex @ X2 @ B3 ) ) ) ) ) ).

% Un_def
thf(fact_196_Un__def,axiom,
    ( sup_sup_set_int
    = ( ^ [A3: set_int,B3: set_int] :
          ( collect_int
          @ ^ [X2: int] :
              ( ( member_int @ X2 @ A3 )
              | ( member_int @ X2 @ B3 ) ) ) ) ) ).

% Un_def
thf(fact_197_Un__def,axiom,
    ( sup_sup_set_real
    = ( ^ [A3: set_real,B3: set_real] :
          ( collect_real
          @ ^ [X2: real] :
              ( ( member_real @ X2 @ A3 )
              | ( member_real @ X2 @ B3 ) ) ) ) ) ).

% Un_def
thf(fact_198_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_nat,B2: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ C @ B2 )
     => ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_199_sup_Ostrict__coboundedI2,axiom,
    ! [C: extended_enat,B2: extended_enat,A2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ C @ B2 )
     => ( ord_le72135733267957522d_enat @ C @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_200_sup_Ostrict__coboundedI2,axiom,
    ! [C: assn,B2: assn,A2: assn] :
      ( ( ord_less_assn @ C @ B2 )
     => ( ord_less_assn @ C @ ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_201_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_complex,B2: set_complex,A2: set_complex] :
      ( ( ord_less_set_complex @ C @ B2 )
     => ( ord_less_set_complex @ C @ ( sup_sup_set_complex @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_202_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_int,B2: set_int,A2: set_int] :
      ( ( ord_less_set_int @ C @ B2 )
     => ( ord_less_set_int @ C @ ( sup_sup_set_int @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_203_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_real,B2: set_real,A2: set_real] :
      ( ( ord_less_set_real @ C @ B2 )
     => ( ord_less_set_real @ C @ ( sup_sup_set_real @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_204_sup_Ostrict__coboundedI2,axiom,
    ! [C: real,B2: real,A2: real] :
      ( ( ord_less_real @ C @ B2 )
     => ( ord_less_real @ C @ ( sup_sup_real @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_205_sup_Ostrict__coboundedI2,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ C @ B2 )
     => ( ord_less_rat @ C @ ( sup_sup_rat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_206_sup_Ostrict__coboundedI2,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( ( ord_less_nat @ C @ B2 )
     => ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_207_sup_Ostrict__coboundedI2,axiom,
    ! [C: int,B2: int,A2: int] :
      ( ( ord_less_int @ C @ B2 )
     => ( ord_less_int @ C @ ( sup_sup_int @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_208_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ C @ A2 )
     => ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_209_sup_Ostrict__coboundedI1,axiom,
    ! [C: extended_enat,A2: extended_enat,B2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ C @ A2 )
     => ( ord_le72135733267957522d_enat @ C @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_210_sup_Ostrict__coboundedI1,axiom,
    ! [C: assn,A2: assn,B2: assn] :
      ( ( ord_less_assn @ C @ A2 )
     => ( ord_less_assn @ C @ ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_211_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_complex,A2: set_complex,B2: set_complex] :
      ( ( ord_less_set_complex @ C @ A2 )
     => ( ord_less_set_complex @ C @ ( sup_sup_set_complex @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_212_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_int,A2: set_int,B2: set_int] :
      ( ( ord_less_set_int @ C @ A2 )
     => ( ord_less_set_int @ C @ ( sup_sup_set_int @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_213_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_real,A2: set_real,B2: set_real] :
      ( ( ord_less_set_real @ C @ A2 )
     => ( ord_less_set_real @ C @ ( sup_sup_set_real @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_214_sup_Ostrict__coboundedI1,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( ord_less_real @ C @ A2 )
     => ( ord_less_real @ C @ ( sup_sup_real @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_215_sup_Ostrict__coboundedI1,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ C @ A2 )
     => ( ord_less_rat @ C @ ( sup_sup_rat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_216_sup_Ostrict__coboundedI1,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ C @ A2 )
     => ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_217_sup_Ostrict__coboundedI1,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_int @ C @ A2 )
     => ( ord_less_int @ C @ ( sup_sup_int @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_218_mem__Collect__eq,axiom,
    ! [A2: real,P: real > $o] :
      ( ( member_real @ A2 @ ( collect_real @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_219_mem__Collect__eq,axiom,
    ! [A2: nat,P: nat > $o] :
      ( ( member_nat @ A2 @ ( collect_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_220_mem__Collect__eq,axiom,
    ! [A2: complex,P: complex > $o] :
      ( ( member_complex @ A2 @ ( collect_complex @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_221_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_222_mem__Collect__eq,axiom,
    ! [A2: int,P: int > $o] :
      ( ( member_int @ A2 @ ( collect_int @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_223_Collect__mem__eq,axiom,
    ! [A: set_real] :
      ( ( collect_real
        @ ^ [X2: real] : ( member_real @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_224_Collect__mem__eq,axiom,
    ! [A: set_nat] :
      ( ( collect_nat
        @ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_225_Collect__mem__eq,axiom,
    ! [A: set_complex] :
      ( ( collect_complex
        @ ^ [X2: complex] : ( member_complex @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_226_Collect__mem__eq,axiom,
    ! [A: set_Pr958786334691620121nt_int] :
      ( ( collec213857154873943460nt_int
        @ ^ [X2: product_prod_int_int] : ( member5262025264175285858nt_int @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_227_Collect__mem__eq,axiom,
    ! [A: set_int] :
      ( ( collect_int
        @ ^ [X2: int] : ( member_int @ X2 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_228_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_229_Collect__cong,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ! [X3: complex] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_complex @ P )
        = ( collect_complex @ Q ) ) ) ).

% Collect_cong
thf(fact_230_Collect__cong,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ! [X3: product_prod_int_int] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec213857154873943460nt_int @ P )
        = ( collec213857154873943460nt_int @ Q ) ) ) ).

% Collect_cong
thf(fact_231_Collect__cong,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_int @ P )
        = ( collect_int @ Q ) ) ) ).

% Collect_cong
thf(fact_232_sup_Ostrict__order__iff,axiom,
    ( ord_less_set_nat
    = ( ^ [B4: set_nat,A4: set_nat] :
          ( ( A4
            = ( sup_sup_set_nat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_233_sup_Ostrict__order__iff,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [B4: extended_enat,A4: extended_enat] :
          ( ( A4
            = ( sup_su3973961784419623482d_enat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_234_sup_Ostrict__order__iff,axiom,
    ( ord_less_assn
    = ( ^ [B4: assn,A4: assn] :
          ( ( A4
            = ( sup_sup_assn @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_235_sup_Ostrict__order__iff,axiom,
    ( ord_less_set_complex
    = ( ^ [B4: set_complex,A4: set_complex] :
          ( ( A4
            = ( sup_sup_set_complex @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_236_sup_Ostrict__order__iff,axiom,
    ( ord_less_set_int
    = ( ^ [B4: set_int,A4: set_int] :
          ( ( A4
            = ( sup_sup_set_int @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_237_sup_Ostrict__order__iff,axiom,
    ( ord_less_set_real
    = ( ^ [B4: set_real,A4: set_real] :
          ( ( A4
            = ( sup_sup_set_real @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_238_sup_Ostrict__order__iff,axiom,
    ( ord_less_real
    = ( ^ [B4: real,A4: real] :
          ( ( A4
            = ( sup_sup_real @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_239_sup_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [B4: rat,A4: rat] :
          ( ( A4
            = ( sup_sup_rat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_240_sup_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( A4
            = ( sup_sup_nat @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_241_sup_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B4: int,A4: int] :
          ( ( A4
            = ( sup_sup_int @ A4 @ B4 ) )
          & ( A4 != B4 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_242_sup_Ostrict__boundedE,axiom,
    ! [B2: set_nat,C: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_set_nat @ B2 @ A2 )
         => ~ ( ord_less_set_nat @ C @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_243_sup_Ostrict__boundedE,axiom,
    ! [B2: extended_enat,C: extended_enat,A2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( sup_su3973961784419623482d_enat @ B2 @ C ) @ A2 )
     => ~ ( ( ord_le72135733267957522d_enat @ B2 @ A2 )
         => ~ ( ord_le72135733267957522d_enat @ C @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_244_sup_Ostrict__boundedE,axiom,
    ! [B2: assn,C: assn,A2: assn] :
      ( ( ord_less_assn @ ( sup_sup_assn @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_assn @ B2 @ A2 )
         => ~ ( ord_less_assn @ C @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_245_sup_Ostrict__boundedE,axiom,
    ! [B2: set_complex,C: set_complex,A2: set_complex] :
      ( ( ord_less_set_complex @ ( sup_sup_set_complex @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_set_complex @ B2 @ A2 )
         => ~ ( ord_less_set_complex @ C @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_246_sup_Ostrict__boundedE,axiom,
    ! [B2: set_int,C: set_int,A2: set_int] :
      ( ( ord_less_set_int @ ( sup_sup_set_int @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_set_int @ B2 @ A2 )
         => ~ ( ord_less_set_int @ C @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_247_sup_Ostrict__boundedE,axiom,
    ! [B2: set_real,C: set_real,A2: set_real] :
      ( ( ord_less_set_real @ ( sup_sup_set_real @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_set_real @ B2 @ A2 )
         => ~ ( ord_less_set_real @ C @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_248_sup_Ostrict__boundedE,axiom,
    ! [B2: real,C: real,A2: real] :
      ( ( ord_less_real @ ( sup_sup_real @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_real @ B2 @ A2 )
         => ~ ( ord_less_real @ C @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_249_sup_Ostrict__boundedE,axiom,
    ! [B2: rat,C: rat,A2: rat] :
      ( ( ord_less_rat @ ( sup_sup_rat @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_rat @ B2 @ A2 )
         => ~ ( ord_less_rat @ C @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_250_sup_Ostrict__boundedE,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( ( ord_less_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_nat @ B2 @ A2 )
         => ~ ( ord_less_nat @ C @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_251_sup_Ostrict__boundedE,axiom,
    ! [B2: int,C: int,A2: int] :
      ( ( ord_less_int @ ( sup_sup_int @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_int @ B2 @ A2 )
         => ~ ( ord_less_int @ C @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_252_sup_Oabsorb4,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ( ( sup_sup_set_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_253_sup_Oabsorb4,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A2 @ B2 )
     => ( ( sup_su3973961784419623482d_enat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_254_sup_Oabsorb4,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ord_less_assn @ A2 @ B2 )
     => ( ( sup_sup_assn @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_255_sup_Oabsorb4,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ( ord_less_set_complex @ A2 @ B2 )
     => ( ( sup_sup_set_complex @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_256_sup_Oabsorb4,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ord_less_set_int @ A2 @ B2 )
     => ( ( sup_sup_set_int @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_257_sup_Oabsorb4,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( ord_less_set_real @ A2 @ B2 )
     => ( ( sup_sup_set_real @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_258_sup_Oabsorb4,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( sup_sup_real @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_259_sup_Oabsorb4,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( sup_sup_rat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_260_sup_Oabsorb4,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( sup_sup_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_261_sup_Oabsorb4,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( sup_sup_int @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_262_sup_Oabsorb3,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ B2 @ A2 )
     => ( ( sup_sup_set_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_263_sup_Oabsorb3,axiom,
    ! [B2: extended_enat,A2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B2 @ A2 )
     => ( ( sup_su3973961784419623482d_enat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_264_sup_Oabsorb3,axiom,
    ! [B2: assn,A2: assn] :
      ( ( ord_less_assn @ B2 @ A2 )
     => ( ( sup_sup_assn @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_265_sup_Oabsorb3,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( ( ord_less_set_complex @ B2 @ A2 )
     => ( ( sup_sup_set_complex @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_266_sup_Oabsorb3,axiom,
    ! [B2: set_int,A2: set_int] :
      ( ( ord_less_set_int @ B2 @ A2 )
     => ( ( sup_sup_set_int @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_267_sup_Oabsorb3,axiom,
    ! [B2: set_real,A2: set_real] :
      ( ( ord_less_set_real @ B2 @ A2 )
     => ( ( sup_sup_set_real @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_268_sup_Oabsorb3,axiom,
    ! [B2: real,A2: real] :
      ( ( ord_less_real @ B2 @ A2 )
     => ( ( sup_sup_real @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_269_sup_Oabsorb3,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ( ( sup_sup_rat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_270_sup_Oabsorb3,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( sup_sup_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_271_sup_Oabsorb3,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( sup_sup_int @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_272_less__supI2,axiom,
    ! [X: set_nat,B2: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ X @ B2 )
     => ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_273_less__supI2,axiom,
    ! [X: extended_enat,B2: extended_enat,A2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ B2 )
     => ( ord_le72135733267957522d_enat @ X @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_274_less__supI2,axiom,
    ! [X: assn,B2: assn,A2: assn] :
      ( ( ord_less_assn @ X @ B2 )
     => ( ord_less_assn @ X @ ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_275_less__supI2,axiom,
    ! [X: set_complex,B2: set_complex,A2: set_complex] :
      ( ( ord_less_set_complex @ X @ B2 )
     => ( ord_less_set_complex @ X @ ( sup_sup_set_complex @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_276_less__supI2,axiom,
    ! [X: set_int,B2: set_int,A2: set_int] :
      ( ( ord_less_set_int @ X @ B2 )
     => ( ord_less_set_int @ X @ ( sup_sup_set_int @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_277_less__supI2,axiom,
    ! [X: set_real,B2: set_real,A2: set_real] :
      ( ( ord_less_set_real @ X @ B2 )
     => ( ord_less_set_real @ X @ ( sup_sup_set_real @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_278_less__supI2,axiom,
    ! [X: real,B2: real,A2: real] :
      ( ( ord_less_real @ X @ B2 )
     => ( ord_less_real @ X @ ( sup_sup_real @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_279_less__supI2,axiom,
    ! [X: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ X @ B2 )
     => ( ord_less_rat @ X @ ( sup_sup_rat @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_280_less__supI2,axiom,
    ! [X: nat,B2: nat,A2: nat] :
      ( ( ord_less_nat @ X @ B2 )
     => ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_281_less__supI2,axiom,
    ! [X: int,B2: int,A2: int] :
      ( ( ord_less_int @ X @ B2 )
     => ( ord_less_int @ X @ ( sup_sup_int @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_282_less__supI1,axiom,
    ! [X: set_nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ X @ A2 )
     => ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_283_less__supI1,axiom,
    ! [X: extended_enat,A2: extended_enat,B2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ A2 )
     => ( ord_le72135733267957522d_enat @ X @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_284_less__supI1,axiom,
    ! [X: assn,A2: assn,B2: assn] :
      ( ( ord_less_assn @ X @ A2 )
     => ( ord_less_assn @ X @ ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_285_less__supI1,axiom,
    ! [X: set_complex,A2: set_complex,B2: set_complex] :
      ( ( ord_less_set_complex @ X @ A2 )
     => ( ord_less_set_complex @ X @ ( sup_sup_set_complex @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_286_less__supI1,axiom,
    ! [X: set_int,A2: set_int,B2: set_int] :
      ( ( ord_less_set_int @ X @ A2 )
     => ( ord_less_set_int @ X @ ( sup_sup_set_int @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_287_less__supI1,axiom,
    ! [X: set_real,A2: set_real,B2: set_real] :
      ( ( ord_less_set_real @ X @ A2 )
     => ( ord_less_set_real @ X @ ( sup_sup_set_real @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_288_less__supI1,axiom,
    ! [X: real,A2: real,B2: real] :
      ( ( ord_less_real @ X @ A2 )
     => ( ord_less_real @ X @ ( sup_sup_real @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_289_less__supI1,axiom,
    ! [X: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ X @ A2 )
     => ( ord_less_rat @ X @ ( sup_sup_rat @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_290_less__supI1,axiom,
    ! [X: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ X @ A2 )
     => ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_291_less__supI1,axiom,
    ! [X: int,A2: int,B2: int] :
      ( ( ord_less_int @ X @ A2 )
     => ( ord_less_int @ X @ ( sup_sup_int @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_292_enat__ord__number_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).

% enat_ord_number(2)
thf(fact_293_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y22: num] :
      ( ( ( bit0 @ X22 )
        = ( bit0 @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% verit_eq_simplify(8)
thf(fact_294_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] :
      ( one
     != ( bit0 @ X22 ) ) ).

% verit_eq_simplify(10)
thf(fact_295_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu7865238048354675525l_num1 @ ( numera7442385471795722001l_num1 @ K ) )
      = ( numera7442385471795722001l_num1 @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_296_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_297_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_298_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_299_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_300_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_301_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_302_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_303_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_304_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_305_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_306_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_307_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_308_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_309_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_310_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_311_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_312_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_313_sup__Un__eq,axiom,
    ! [R: set_nat,S: set_nat] :
      ( ( sup_sup_nat_o
        @ ^ [X2: nat] : ( member_nat @ X2 @ R )
        @ ^ [X2: nat] : ( member_nat @ X2 @ S ) )
      = ( ^ [X2: nat] : ( member_nat @ X2 @ ( sup_sup_set_nat @ R @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_314_sup__Un__eq,axiom,
    ! [R: set_complex,S: set_complex] :
      ( ( sup_sup_complex_o
        @ ^ [X2: complex] : ( member_complex @ X2 @ R )
        @ ^ [X2: complex] : ( member_complex @ X2 @ S ) )
      = ( ^ [X2: complex] : ( member_complex @ X2 @ ( sup_sup_set_complex @ R @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_315_sup__Un__eq,axiom,
    ! [R: set_int,S: set_int] :
      ( ( sup_sup_int_o
        @ ^ [X2: int] : ( member_int @ X2 @ R )
        @ ^ [X2: int] : ( member_int @ X2 @ S ) )
      = ( ^ [X2: int] : ( member_int @ X2 @ ( sup_sup_set_int @ R @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_316_sup__Un__eq,axiom,
    ! [R: set_real,S: set_real] :
      ( ( sup_sup_real_o
        @ ^ [X2: real] : ( member_real @ X2 @ R )
        @ ^ [X2: real] : ( member_real @ X2 @ S ) )
      = ( ^ [X2: real] : ( member_real @ X2 @ ( sup_sup_set_real @ R @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_317_vebt__heap__rules_I3_J,axiom,
    ! [X: nat,N: nat,S2: set_nat,Ti: vEBT_VEBTi] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( hoare_1429296392585015714_VEBTi @ ( vEBT_Intf_vebt_assn @ N @ S2 @ Ti ) @ ( vEBT_vebt_inserti @ Ti @ X ) @ ( vEBT_Intf_vebt_assn @ N @ ( sup_sup_set_nat @ S2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).

% vebt_heap_rules(3)
thf(fact_318_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_power_complex @ ( numera6690914467698888265omplex @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numera6690914467698888265omplex @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_319_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numera6620942414471956472nteger @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_320_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numera7442385471795722001l_num1 @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_321_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_power_real @ ( numeral_numeral_real @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numeral_numeral_real @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_322_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_power_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numeral_numeral_rat @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_323_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_power_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numeral_numeral_nat @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_324_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_power_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numeral_numeral_int @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_325_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_326_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_327_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_328_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_329_set__union,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( set_nat2 @ ( union_nat @ Xs @ Ys ) )
      = ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).

% set_union
thf(fact_330_set__union,axiom,
    ! [Xs: list_complex,Ys: list_complex] :
      ( ( set_complex2 @ ( union_complex @ Xs @ Ys ) )
      = ( sup_sup_set_complex @ ( set_complex2 @ Xs ) @ ( set_complex2 @ Ys ) ) ) ).

% set_union
thf(fact_331_set__union,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( set_int2 @ ( union_int @ Xs @ Ys ) )
      = ( sup_sup_set_int @ ( set_int2 @ Xs ) @ ( set_int2 @ Ys ) ) ) ).

% set_union
thf(fact_332_set__union,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( set_real2 @ ( union_real @ Xs @ Ys ) )
      = ( sup_sup_set_real @ ( set_real2 @ Xs ) @ ( set_real2 @ Ys ) ) ) ).

% set_union
thf(fact_333_power__one__right,axiom,
    ! [A2: nat] :
      ( ( power_power_nat @ A2 @ one_one_nat )
      = A2 ) ).

% power_one_right
thf(fact_334_power__one__right,axiom,
    ! [A2: real] :
      ( ( power_power_real @ A2 @ one_one_nat )
      = A2 ) ).

% power_one_right
thf(fact_335_power__one__right,axiom,
    ! [A2: int] :
      ( ( power_power_int @ A2 @ one_one_nat )
      = A2 ) ).

% power_one_right
thf(fact_336_power__one__right,axiom,
    ! [A2: complex] :
      ( ( power_power_complex @ A2 @ one_one_nat )
      = A2 ) ).

% power_one_right
thf(fact_337_power__one__right,axiom,
    ! [A2: code_integer] :
      ( ( power_8256067586552552935nteger @ A2 @ one_one_nat )
      = A2 ) ).

% power_one_right
thf(fact_338_empty__iff,axiom,
    ! [C: complex] :
      ~ ( member_complex @ C @ bot_bot_set_complex ) ).

% empty_iff
thf(fact_339_empty__iff,axiom,
    ! [C: real] :
      ~ ( member_real @ C @ bot_bot_set_real ) ).

% empty_iff
thf(fact_340_empty__iff,axiom,
    ! [C: $o] :
      ~ ( member_o @ C @ bot_bot_set_o ) ).

% empty_iff
thf(fact_341_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_342_empty__iff,axiom,
    ! [C: int] :
      ~ ( member_int @ C @ bot_bot_set_int ) ).

% empty_iff
thf(fact_343_all__not__in__conv,axiom,
    ! [A: set_complex] :
      ( ( ! [X2: complex] :
            ~ ( member_complex @ X2 @ A ) )
      = ( A = bot_bot_set_complex ) ) ).

% all_not_in_conv
thf(fact_344_all__not__in__conv,axiom,
    ! [A: set_real] :
      ( ( ! [X2: real] :
            ~ ( member_real @ X2 @ A ) )
      = ( A = bot_bot_set_real ) ) ).

% all_not_in_conv
thf(fact_345_all__not__in__conv,axiom,
    ! [A: set_o] :
      ( ( ! [X2: $o] :
            ~ ( member_o @ X2 @ A ) )
      = ( A = bot_bot_set_o ) ) ).

% all_not_in_conv
thf(fact_346_all__not__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ! [X2: nat] :
            ~ ( member_nat @ X2 @ A ) )
      = ( A = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_347_all__not__in__conv,axiom,
    ! [A: set_int] :
      ( ( ! [X2: int] :
            ~ ( member_int @ X2 @ A ) )
      = ( A = bot_bot_set_int ) ) ).

% all_not_in_conv
thf(fact_348_Collect__empty__eq,axiom,
    ! [P: complex > $o] :
      ( ( ( collect_complex @ P )
        = bot_bot_set_complex )
      = ( ! [X2: complex] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_349_Collect__empty__eq,axiom,
    ! [P: product_prod_int_int > $o] :
      ( ( ( collec213857154873943460nt_int @ P )
        = bot_bo1796632182523588997nt_int )
      = ( ! [X2: product_prod_int_int] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_350_Collect__empty__eq,axiom,
    ! [P: real > $o] :
      ( ( ( collect_real @ P )
        = bot_bot_set_real )
      = ( ! [X2: real] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_351_Collect__empty__eq,axiom,
    ! [P: $o > $o] :
      ( ( ( collect_o @ P )
        = bot_bot_set_o )
      = ( ! [X2: $o] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_352_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X2: nat] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_353_Collect__empty__eq,axiom,
    ! [P: int > $o] :
      ( ( ( collect_int @ P )
        = bot_bot_set_int )
      = ( ! [X2: int] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_354_empty__Collect__eq,axiom,
    ! [P: complex > $o] :
      ( ( bot_bot_set_complex
        = ( collect_complex @ P ) )
      = ( ! [X2: complex] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_355_empty__Collect__eq,axiom,
    ! [P: product_prod_int_int > $o] :
      ( ( bot_bo1796632182523588997nt_int
        = ( collec213857154873943460nt_int @ P ) )
      = ( ! [X2: product_prod_int_int] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_356_empty__Collect__eq,axiom,
    ! [P: real > $o] :
      ( ( bot_bot_set_real
        = ( collect_real @ P ) )
      = ( ! [X2: real] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_357_empty__Collect__eq,axiom,
    ! [P: $o > $o] :
      ( ( bot_bot_set_o
        = ( collect_o @ P ) )
      = ( ! [X2: $o] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_358_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X2: nat] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_359_empty__Collect__eq,axiom,
    ! [P: int > $o] :
      ( ( bot_bot_set_int
        = ( collect_int @ P ) )
      = ( ! [X2: int] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_360_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_361_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_362_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_363_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_364_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_365_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_366_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_367_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_368_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_369_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_370_insert__absorb2,axiom,
    ! [X: nat,A: set_nat] :
      ( ( insert_nat @ X @ ( insert_nat @ X @ A ) )
      = ( insert_nat @ X @ A ) ) ).

% insert_absorb2
thf(fact_371_insert__absorb2,axiom,
    ! [X: int,A: set_int] :
      ( ( insert_int @ X @ ( insert_int @ X @ A ) )
      = ( insert_int @ X @ A ) ) ).

% insert_absorb2
thf(fact_372_insert__absorb2,axiom,
    ! [X: real,A: set_real] :
      ( ( insert_real @ X @ ( insert_real @ X @ A ) )
      = ( insert_real @ X @ A ) ) ).

% insert_absorb2
thf(fact_373_insert__absorb2,axiom,
    ! [X: $o,A: set_o] :
      ( ( insert_o @ X @ ( insert_o @ X @ A ) )
      = ( insert_o @ X @ A ) ) ).

% insert_absorb2
thf(fact_374_i0__less,axiom,
    ! [N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
      = ( N != zero_z5237406670263579293d_enat ) ) ).

% i0_less
thf(fact_375_power__one,axiom,
    ! [N: nat] :
      ( ( power_2184487114949457152l_num1 @ one_on7727431528512463931l_num1 @ N )
      = one_on7727431528512463931l_num1 ) ).

% power_one
thf(fact_376_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ one_one_rat @ N )
      = one_one_rat ) ).

% power_one
thf(fact_377_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ one_one_nat @ N )
      = one_one_nat ) ).

% power_one
thf(fact_378_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_real @ one_one_real @ N )
      = one_one_real ) ).

% power_one
thf(fact_379_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_int @ one_one_int @ N )
      = one_one_int ) ).

% power_one
thf(fact_380_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ one_one_complex @ N )
      = one_one_complex ) ).

% power_one
thf(fact_381_power__one,axiom,
    ! [N: nat] :
      ( ( power_8256067586552552935nteger @ one_one_Code_integer @ N )
      = one_one_Code_integer ) ).

% power_one
thf(fact_382_singletonI,axiom,
    ! [A2: complex] : ( member_complex @ A2 @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ).

% singletonI
thf(fact_383_singletonI,axiom,
    ! [A2: real] : ( member_real @ A2 @ ( insert_real @ A2 @ bot_bot_set_real ) ) ).

% singletonI
thf(fact_384_singletonI,axiom,
    ! [A2: $o] : ( member_o @ A2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) ).

% singletonI
thf(fact_385_singletonI,axiom,
    ! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_386_singletonI,axiom,
    ! [A2: int] : ( member_int @ A2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) ).

% singletonI
thf(fact_387_sup__bot__left,axiom,
    ! [X: extended_enat] :
      ( ( sup_su3973961784419623482d_enat @ bot_bo4199563552545308370d_enat @ X )
      = X ) ).

% sup_bot_left
thf(fact_388_sup__bot__left,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ bot_bot_assn @ X )
      = X ) ).

% sup_bot_left
thf(fact_389_sup__bot__left,axiom,
    ! [X: set_complex] :
      ( ( sup_sup_set_complex @ bot_bot_set_complex @ X )
      = X ) ).

% sup_bot_left
thf(fact_390_sup__bot__left,axiom,
    ! [X: set_real] :
      ( ( sup_sup_set_real @ bot_bot_set_real @ X )
      = X ) ).

% sup_bot_left
thf(fact_391_sup__bot__left,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ X )
      = X ) ).

% sup_bot_left
thf(fact_392_sup__bot__left,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
      = X ) ).

% sup_bot_left
thf(fact_393_sup__bot__left,axiom,
    ! [X: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ X )
      = X ) ).

% sup_bot_left
thf(fact_394_sup__bot__right,axiom,
    ! [X: extended_enat] :
      ( ( sup_su3973961784419623482d_enat @ X @ bot_bo4199563552545308370d_enat )
      = X ) ).

% sup_bot_right
thf(fact_395_sup__bot__right,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ bot_bot_assn )
      = X ) ).

% sup_bot_right
thf(fact_396_sup__bot__right,axiom,
    ! [X: set_complex] :
      ( ( sup_sup_set_complex @ X @ bot_bot_set_complex )
      = X ) ).

% sup_bot_right
thf(fact_397_sup__bot__right,axiom,
    ! [X: set_real] :
      ( ( sup_sup_set_real @ X @ bot_bot_set_real )
      = X ) ).

% sup_bot_right
thf(fact_398_sup__bot__right,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ X @ bot_bot_set_o )
      = X ) ).

% sup_bot_right
thf(fact_399_sup__bot__right,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
      = X ) ).

% sup_bot_right
thf(fact_400_sup__bot__right,axiom,
    ! [X: set_int] :
      ( ( sup_sup_set_int @ X @ bot_bot_set_int )
      = X ) ).

% sup_bot_right
thf(fact_401_bot__eq__sup__iff,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( bot_bo4199563552545308370d_enat
        = ( sup_su3973961784419623482d_enat @ X @ Y ) )
      = ( ( X = bot_bo4199563552545308370d_enat )
        & ( Y = bot_bo4199563552545308370d_enat ) ) ) ).

% bot_eq_sup_iff
thf(fact_402_bot__eq__sup__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( bot_bot_assn
        = ( sup_sup_assn @ X @ Y ) )
      = ( ( X = bot_bot_assn )
        & ( Y = bot_bot_assn ) ) ) ).

% bot_eq_sup_iff
thf(fact_403_bot__eq__sup__iff,axiom,
    ! [X: set_complex,Y: set_complex] :
      ( ( bot_bot_set_complex
        = ( sup_sup_set_complex @ X @ Y ) )
      = ( ( X = bot_bot_set_complex )
        & ( Y = bot_bot_set_complex ) ) ) ).

% bot_eq_sup_iff
thf(fact_404_bot__eq__sup__iff,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( bot_bot_set_real
        = ( sup_sup_set_real @ X @ Y ) )
      = ( ( X = bot_bot_set_real )
        & ( Y = bot_bot_set_real ) ) ) ).

% bot_eq_sup_iff
thf(fact_405_bot__eq__sup__iff,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( bot_bot_set_o
        = ( sup_sup_set_o @ X @ Y ) )
      = ( ( X = bot_bot_set_o )
        & ( Y = bot_bot_set_o ) ) ) ).

% bot_eq_sup_iff
thf(fact_406_bot__eq__sup__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( bot_bot_set_nat
        = ( sup_sup_set_nat @ X @ Y ) )
      = ( ( X = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% bot_eq_sup_iff
thf(fact_407_bot__eq__sup__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( bot_bot_set_int
        = ( sup_sup_set_int @ X @ Y ) )
      = ( ( X = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% bot_eq_sup_iff
thf(fact_408_sup__eq__bot__iff,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ( sup_su3973961784419623482d_enat @ X @ Y )
        = bot_bo4199563552545308370d_enat )
      = ( ( X = bot_bo4199563552545308370d_enat )
        & ( Y = bot_bo4199563552545308370d_enat ) ) ) ).

% sup_eq_bot_iff
thf(fact_409_sup__eq__bot__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( sup_sup_assn @ X @ Y )
        = bot_bot_assn )
      = ( ( X = bot_bot_assn )
        & ( Y = bot_bot_assn ) ) ) ).

% sup_eq_bot_iff
thf(fact_410_sup__eq__bot__iff,axiom,
    ! [X: set_complex,Y: set_complex] :
      ( ( ( sup_sup_set_complex @ X @ Y )
        = bot_bot_set_complex )
      = ( ( X = bot_bot_set_complex )
        & ( Y = bot_bot_set_complex ) ) ) ).

% sup_eq_bot_iff
thf(fact_411_sup__eq__bot__iff,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( ( sup_sup_set_real @ X @ Y )
        = bot_bot_set_real )
      = ( ( X = bot_bot_set_real )
        & ( Y = bot_bot_set_real ) ) ) ).

% sup_eq_bot_iff
thf(fact_412_sup__eq__bot__iff,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( ( sup_sup_set_o @ X @ Y )
        = bot_bot_set_o )
      = ( ( X = bot_bot_set_o )
        & ( Y = bot_bot_set_o ) ) ) ).

% sup_eq_bot_iff
thf(fact_413_sup__eq__bot__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ( sup_sup_set_nat @ X @ Y )
        = bot_bot_set_nat )
      = ( ( X = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% sup_eq_bot_iff
thf(fact_414_sup__eq__bot__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ( sup_sup_set_int @ X @ Y )
        = bot_bot_set_int )
      = ( ( X = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% sup_eq_bot_iff
thf(fact_415_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( ( sup_su3973961784419623482d_enat @ A2 @ B2 )
        = bot_bo4199563552545308370d_enat )
      = ( ( A2 = bot_bo4199563552545308370d_enat )
        & ( B2 = bot_bo4199563552545308370d_enat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_416_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ( sup_sup_assn @ A2 @ B2 )
        = bot_bot_assn )
      = ( ( A2 = bot_bot_assn )
        & ( B2 = bot_bot_assn ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_417_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ( ( sup_sup_set_complex @ A2 @ B2 )
        = bot_bot_set_complex )
      = ( ( A2 = bot_bot_set_complex )
        & ( B2 = bot_bot_set_complex ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_418_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( ( sup_sup_set_real @ A2 @ B2 )
        = bot_bot_set_real )
      = ( ( A2 = bot_bot_set_real )
        & ( B2 = bot_bot_set_real ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_419_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( ( sup_sup_set_o @ A2 @ B2 )
        = bot_bot_set_o )
      = ( ( A2 = bot_bot_set_o )
        & ( B2 = bot_bot_set_o ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_420_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ( sup_sup_set_nat @ A2 @ B2 )
        = bot_bot_set_nat )
      = ( ( A2 = bot_bot_set_nat )
        & ( B2 = bot_bot_set_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_421_sup__bot_Oeq__neutr__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ( sup_sup_set_int @ A2 @ B2 )
        = bot_bot_set_int )
      = ( ( A2 = bot_bot_set_int )
        & ( B2 = bot_bot_set_int ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_422_sup__bot_Oleft__neutral,axiom,
    ! [A2: extended_enat] :
      ( ( sup_su3973961784419623482d_enat @ bot_bo4199563552545308370d_enat @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_423_sup__bot_Oleft__neutral,axiom,
    ! [A2: assn] :
      ( ( sup_sup_assn @ bot_bot_assn @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_424_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_complex] :
      ( ( sup_sup_set_complex @ bot_bot_set_complex @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_425_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_real] :
      ( ( sup_sup_set_real @ bot_bot_set_real @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_426_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_427_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_428_sup__bot_Oleft__neutral,axiom,
    ! [A2: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ A2 )
      = A2 ) ).

% sup_bot.left_neutral
thf(fact_429_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( bot_bo4199563552545308370d_enat
        = ( sup_su3973961784419623482d_enat @ A2 @ B2 ) )
      = ( ( A2 = bot_bo4199563552545308370d_enat )
        & ( B2 = bot_bo4199563552545308370d_enat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_430_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: assn,B2: assn] :
      ( ( bot_bot_assn
        = ( sup_sup_assn @ A2 @ B2 ) )
      = ( ( A2 = bot_bot_assn )
        & ( B2 = bot_bot_assn ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_431_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ( bot_bot_set_complex
        = ( sup_sup_set_complex @ A2 @ B2 ) )
      = ( ( A2 = bot_bot_set_complex )
        & ( B2 = bot_bot_set_complex ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_432_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( bot_bot_set_real
        = ( sup_sup_set_real @ A2 @ B2 ) )
      = ( ( A2 = bot_bot_set_real )
        & ( B2 = bot_bot_set_real ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_433_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( bot_bot_set_o
        = ( sup_sup_set_o @ A2 @ B2 ) )
      = ( ( A2 = bot_bot_set_o )
        & ( B2 = bot_bot_set_o ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_434_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( bot_bot_set_nat
        = ( sup_sup_set_nat @ A2 @ B2 ) )
      = ( ( A2 = bot_bot_set_nat )
        & ( B2 = bot_bot_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_435_sup__bot_Oneutr__eq__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( bot_bot_set_int
        = ( sup_sup_set_int @ A2 @ B2 ) )
      = ( ( A2 = bot_bot_set_int )
        & ( B2 = bot_bot_set_int ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_436_sup__bot_Oright__neutral,axiom,
    ! [A2: extended_enat] :
      ( ( sup_su3973961784419623482d_enat @ A2 @ bot_bo4199563552545308370d_enat )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_437_sup__bot_Oright__neutral,axiom,
    ! [A2: assn] :
      ( ( sup_sup_assn @ A2 @ bot_bot_assn )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_438_sup__bot_Oright__neutral,axiom,
    ! [A2: set_complex] :
      ( ( sup_sup_set_complex @ A2 @ bot_bot_set_complex )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_439_sup__bot_Oright__neutral,axiom,
    ! [A2: set_real] :
      ( ( sup_sup_set_real @ A2 @ bot_bot_set_real )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_440_sup__bot_Oright__neutral,axiom,
    ! [A2: set_o] :
      ( ( sup_sup_set_o @ A2 @ bot_bot_set_o )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_441_sup__bot_Oright__neutral,axiom,
    ! [A2: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_442_sup__bot_Oright__neutral,axiom,
    ! [A2: set_int] :
      ( ( sup_sup_set_int @ A2 @ bot_bot_set_int )
      = A2 ) ).

% sup_bot.right_neutral
thf(fact_443_nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
thf(fact_444_Un__empty,axiom,
    ! [A: set_complex,B: set_complex] :
      ( ( ( sup_sup_set_complex @ A @ B )
        = bot_bot_set_complex )
      = ( ( A = bot_bot_set_complex )
        & ( B = bot_bot_set_complex ) ) ) ).

% Un_empty
thf(fact_445_Un__empty,axiom,
    ! [A: set_real,B: set_real] :
      ( ( ( sup_sup_set_real @ A @ B )
        = bot_bot_set_real )
      = ( ( A = bot_bot_set_real )
        & ( B = bot_bot_set_real ) ) ) ).

% Un_empty
thf(fact_446_Un__empty,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ( sup_sup_set_o @ A @ B )
        = bot_bot_set_o )
      = ( ( A = bot_bot_set_o )
        & ( B = bot_bot_set_o ) ) ) ).

% Un_empty
thf(fact_447_Un__empty,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( sup_sup_set_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% Un_empty
thf(fact_448_Un__empty,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( sup_sup_set_int @ A @ B )
        = bot_bot_set_int )
      = ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% Un_empty
thf(fact_449_Un__insert__left,axiom,
    ! [A2: $o,B: set_o,C2: set_o] :
      ( ( sup_sup_set_o @ ( insert_o @ A2 @ B ) @ C2 )
      = ( insert_o @ A2 @ ( sup_sup_set_o @ B @ C2 ) ) ) ).

% Un_insert_left
thf(fact_450_Un__insert__left,axiom,
    ! [A2: nat,B: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
      = ( insert_nat @ A2 @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).

% Un_insert_left
thf(fact_451_Un__insert__left,axiom,
    ! [A2: complex,B: set_complex,C2: set_complex] :
      ( ( sup_sup_set_complex @ ( insert_complex @ A2 @ B ) @ C2 )
      = ( insert_complex @ A2 @ ( sup_sup_set_complex @ B @ C2 ) ) ) ).

% Un_insert_left
thf(fact_452_Un__insert__left,axiom,
    ! [A2: int,B: set_int,C2: set_int] :
      ( ( sup_sup_set_int @ ( insert_int @ A2 @ B ) @ C2 )
      = ( insert_int @ A2 @ ( sup_sup_set_int @ B @ C2 ) ) ) ).

% Un_insert_left
thf(fact_453_Un__insert__left,axiom,
    ! [A2: real,B: set_real,C2: set_real] :
      ( ( sup_sup_set_real @ ( insert_real @ A2 @ B ) @ C2 )
      = ( insert_real @ A2 @ ( sup_sup_set_real @ B @ C2 ) ) ) ).

% Un_insert_left
thf(fact_454_Un__insert__right,axiom,
    ! [A: set_o,A2: $o,B: set_o] :
      ( ( sup_sup_set_o @ A @ ( insert_o @ A2 @ B ) )
      = ( insert_o @ A2 @ ( sup_sup_set_o @ A @ B ) ) ) ).

% Un_insert_right
thf(fact_455_Un__insert__right,axiom,
    ! [A: set_nat,A2: nat,B: set_nat] :
      ( ( sup_sup_set_nat @ A @ ( insert_nat @ A2 @ B ) )
      = ( insert_nat @ A2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% Un_insert_right
thf(fact_456_Un__insert__right,axiom,
    ! [A: set_complex,A2: complex,B: set_complex] :
      ( ( sup_sup_set_complex @ A @ ( insert_complex @ A2 @ B ) )
      = ( insert_complex @ A2 @ ( sup_sup_set_complex @ A @ B ) ) ) ).

% Un_insert_right
thf(fact_457_Un__insert__right,axiom,
    ! [A: set_int,A2: int,B: set_int] :
      ( ( sup_sup_set_int @ A @ ( insert_int @ A2 @ B ) )
      = ( insert_int @ A2 @ ( sup_sup_set_int @ A @ B ) ) ) ).

% Un_insert_right
thf(fact_458_Un__insert__right,axiom,
    ! [A: set_real,A2: real,B: set_real] :
      ( ( sup_sup_set_real @ A @ ( insert_real @ A2 @ B ) )
      = ( insert_real @ A2 @ ( sup_sup_set_real @ A @ B ) ) ) ).

% Un_insert_right
thf(fact_459_singleton__conv,axiom,
    ! [A2: complex] :
      ( ( collect_complex
        @ ^ [X2: complex] : X2 = A2 )
      = ( insert_complex @ A2 @ bot_bot_set_complex ) ) ).

% singleton_conv
thf(fact_460_singleton__conv,axiom,
    ! [A2: product_prod_int_int] :
      ( ( collec213857154873943460nt_int
        @ ^ [X2: product_prod_int_int] : X2 = A2 )
      = ( insert5033312907999012233nt_int @ A2 @ bot_bo1796632182523588997nt_int ) ) ).

% singleton_conv
thf(fact_461_singleton__conv,axiom,
    ! [A2: real] :
      ( ( collect_real
        @ ^ [X2: real] : X2 = A2 )
      = ( insert_real @ A2 @ bot_bot_set_real ) ) ).

% singleton_conv
thf(fact_462_singleton__conv,axiom,
    ! [A2: $o] :
      ( ( collect_o
        @ ^ [X2: $o] : X2 = A2 )
      = ( insert_o @ A2 @ bot_bot_set_o ) ) ).

% singleton_conv
thf(fact_463_singleton__conv,axiom,
    ! [A2: nat] :
      ( ( collect_nat
        @ ^ [X2: nat] : X2 = A2 )
      = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_464_singleton__conv,axiom,
    ! [A2: int] :
      ( ( collect_int
        @ ^ [X2: int] : X2 = A2 )
      = ( insert_int @ A2 @ bot_bot_set_int ) ) ).

% singleton_conv
thf(fact_465_singleton__conv2,axiom,
    ! [A2: complex] :
      ( ( collect_complex
        @ ( ^ [Y3: complex,Z2: complex] : Y3 = Z2
          @ A2 ) )
      = ( insert_complex @ A2 @ bot_bot_set_complex ) ) ).

% singleton_conv2
thf(fact_466_singleton__conv2,axiom,
    ! [A2: product_prod_int_int] :
      ( ( collec213857154873943460nt_int
        @ ( ^ [Y3: product_prod_int_int,Z2: product_prod_int_int] : Y3 = Z2
          @ A2 ) )
      = ( insert5033312907999012233nt_int @ A2 @ bot_bo1796632182523588997nt_int ) ) ).

% singleton_conv2
thf(fact_467_singleton__conv2,axiom,
    ! [A2: real] :
      ( ( collect_real
        @ ( ^ [Y3: real,Z2: real] : Y3 = Z2
          @ A2 ) )
      = ( insert_real @ A2 @ bot_bot_set_real ) ) ).

% singleton_conv2
thf(fact_468_singleton__conv2,axiom,
    ! [A2: $o] :
      ( ( collect_o
        @ ( ^ [Y3: $o,Z2: $o] : Y3 = Z2
          @ A2 ) )
      = ( insert_o @ A2 @ bot_bot_set_o ) ) ).

% singleton_conv2
thf(fact_469_singleton__conv2,axiom,
    ! [A2: nat] :
      ( ( collect_nat
        @ ( ^ [Y3: nat,Z2: nat] : Y3 = Z2
          @ A2 ) )
      = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_470_singleton__conv2,axiom,
    ! [A2: int] :
      ( ( collect_int
        @ ( ^ [Y3: int,Z2: int] : Y3 = Z2
          @ A2 ) )
      = ( insert_int @ A2 @ bot_bot_set_int ) ) ).

% singleton_conv2
thf(fact_471_dbl__simps_I2_J,axiom,
    ( ( neg_nu7865238048354675525l_num1 @ zero_z3563351764282998399l_num1 )
    = zero_z3563351764282998399l_num1 ) ).

% dbl_simps(2)
thf(fact_472_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_real @ zero_zero_real )
    = zero_zero_real ) ).

% dbl_simps(2)
thf(fact_473_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% dbl_simps(2)
thf(fact_474_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_int @ zero_zero_int )
    = zero_zero_int ) ).

% dbl_simps(2)
thf(fact_475_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_real @ N )
        = one_one_real )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_476_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_rat @ N )
        = one_one_rat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_477_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_nat @ N )
        = one_one_nat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_478_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_int @ N )
        = one_one_int )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_479_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_real
        = ( numeral_numeral_real @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_480_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_rat
        = ( numeral_numeral_rat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_481_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_nat
        = ( numeral_numeral_nat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_482_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_int
        = ( numeral_numeral_int @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_483_power__inject__exp,axiom,
    ! [A2: code_integer,M: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
     => ( ( ( power_8256067586552552935nteger @ A2 @ M )
          = ( power_8256067586552552935nteger @ A2 @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_484_power__inject__exp,axiom,
    ! [A2: real,M: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ( power_power_real @ A2 @ M )
          = ( power_power_real @ A2 @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_485_power__inject__exp,axiom,
    ! [A2: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ( ( power_power_rat @ A2 @ M )
          = ( power_power_rat @ A2 @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_486_power__inject__exp,axiom,
    ! [A2: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ( ( power_power_nat @ A2 @ M )
          = ( power_power_nat @ A2 @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_487_power__inject__exp,axiom,
    ! [A2: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ( ( power_power_int @ A2 @ M )
          = ( power_power_int @ A2 @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_488_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_2184487114949457152l_num1 @ zero_z3563351764282998399l_num1 @ ( numeral_numeral_nat @ K ) )
      = zero_z3563351764282998399l_num1 ) ).

% power_zero_numeral
thf(fact_489_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_rat ) ).

% power_zero_numeral
thf(fact_490_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_nat ) ).

% power_zero_numeral
thf(fact_491_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
      = zero_zero_real ) ).

% power_zero_numeral
thf(fact_492_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
      = zero_zero_int ) ).

% power_zero_numeral
thf(fact_493_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
      = zero_zero_complex ) ).

% power_zero_numeral
thf(fact_494_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ ( numeral_numeral_nat @ K ) )
      = zero_z3403309356797280102nteger ) ).

% power_zero_numeral
thf(fact_495_power__eq__0__iff,axiom,
    ! [A2: rat,N: nat] :
      ( ( ( power_power_rat @ A2 @ N )
        = zero_zero_rat )
      = ( ( A2 = zero_zero_rat )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_496_power__eq__0__iff,axiom,
    ! [A2: nat,N: nat] :
      ( ( ( power_power_nat @ A2 @ N )
        = zero_zero_nat )
      = ( ( A2 = zero_zero_nat )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_497_power__eq__0__iff,axiom,
    ! [A2: real,N: nat] :
      ( ( ( power_power_real @ A2 @ N )
        = zero_zero_real )
      = ( ( A2 = zero_zero_real )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_498_power__eq__0__iff,axiom,
    ! [A2: int,N: nat] :
      ( ( ( power_power_int @ A2 @ N )
        = zero_zero_int )
      = ( ( A2 = zero_zero_int )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_499_power__eq__0__iff,axiom,
    ! [A2: complex,N: nat] :
      ( ( ( power_power_complex @ A2 @ N )
        = zero_zero_complex )
      = ( ( A2 = zero_zero_complex )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_500_power__eq__0__iff,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ( power_8256067586552552935nteger @ A2 @ N )
        = zero_z3403309356797280102nteger )
      = ( ( A2 = zero_z3403309356797280102nteger )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_501_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri8819519690708144855l_num1 @ ( numeral_numeral_nat @ N ) )
      = ( numera7442385471795722001l_num1 @ N ) ) ).

% of_nat_numeral
thf(fact_502_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_rat @ N ) ) ).

% of_nat_numeral
thf(fact_503_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% of_nat_numeral
thf(fact_504_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_real @ N ) ) ).

% of_nat_numeral
thf(fact_505_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ N ) ) ).

% of_nat_numeral
thf(fact_506_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri4939895301339042750nteger @ ( numeral_numeral_nat @ N ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% of_nat_numeral
thf(fact_507_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
      = ( numera6690914467698888265omplex @ N ) ) ).

% of_nat_numeral
thf(fact_508_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
      = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).

% semiring_1_class.of_nat_power
thf(fact_509_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
      = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).

% semiring_1_class.of_nat_power
thf(fact_510_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
      = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).

% semiring_1_class.of_nat_power
thf(fact_511_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4939895301339042750nteger @ ( power_power_nat @ M @ N ) )
      = ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ M ) @ N ) ) ).

% semiring_1_class.of_nat_power
thf(fact_512_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N ) )
      = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N ) ) ).

% semiring_1_class.of_nat_power
thf(fact_513_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_514_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_515_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_516_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_517_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_518_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_519_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_520_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_521_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_522_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_523_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_524_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_525_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_526_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_527_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_528_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_529_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_530_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_531_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_532_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_533_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_534_power__strict__decreasing__iff,axiom,
    ! [B2: code_integer,M: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
     => ( ( ord_le6747313008572928689nteger @ B2 @ one_one_Code_integer )
       => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B2 @ M ) @ ( power_8256067586552552935nteger @ B2 @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_535_power__strict__decreasing__iff,axiom,
    ! [B2: real,M: nat,N: 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 @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_536_power__strict__decreasing__iff,axiom,
    ! [B2: rat,M: nat,N: 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 @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_537_power__strict__decreasing__iff,axiom,
    ! [B2: nat,M: nat,N: 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 @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_538_power__strict__decreasing__iff,axiom,
    ! [B2: int,M: nat,N: 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 @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_539_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_540_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_541_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_542_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_543_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_544_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
        = ( semiri681578069525770553at_rat @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_545_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = ( semiri1314217659103216013at_int @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_546_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
        = ( semiri5074537144036343181t_real @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_547_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = ( semiri1316708129612266289at_nat @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_548_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N )
        = ( semiri4939895301339042750nteger @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_549_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
        = ( semiri8010041392384452111omplex @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_550_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( ( semiri681578069525770553at_rat @ Y )
        = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_551_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ Y )
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_552_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ Y )
        = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_553_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ Y )
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_554_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( ( semiri4939895301339042750nteger @ Y )
        = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_555_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( ( semiri8010041392384452111omplex @ Y )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_556_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_557_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_558_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_559_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_560_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_561_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_562_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_563_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_564_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_565_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_566_dbl__simps_I3_J,axiom,
    ( ( neg_nu7865238048354675525l_num1 @ one_on7727431528512463931l_num1 )
    = ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_567_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_568_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_569_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_570_emptyE,axiom,
    ! [A2: complex] :
      ~ ( member_complex @ A2 @ bot_bot_set_complex ) ).

% emptyE
thf(fact_571_emptyE,axiom,
    ! [A2: real] :
      ~ ( member_real @ A2 @ bot_bot_set_real ) ).

% emptyE
thf(fact_572_emptyE,axiom,
    ! [A2: $o] :
      ~ ( member_o @ A2 @ bot_bot_set_o ) ).

% emptyE
thf(fact_573_emptyE,axiom,
    ! [A2: nat] :
      ~ ( member_nat @ A2 @ bot_bot_set_nat ) ).

% emptyE
thf(fact_574_emptyE,axiom,
    ! [A2: int] :
      ~ ( member_int @ A2 @ bot_bot_set_int ) ).

% emptyE
thf(fact_575_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_576_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_577_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_578_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_579_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_580_equals0D,axiom,
    ! [A: set_complex,A2: complex] :
      ( ( A = bot_bot_set_complex )
     => ~ ( member_complex @ A2 @ A ) ) ).

% equals0D
thf(fact_581_equals0D,axiom,
    ! [A: set_real,A2: real] :
      ( ( A = bot_bot_set_real )
     => ~ ( member_real @ A2 @ A ) ) ).

% equals0D
thf(fact_582_equals0D,axiom,
    ! [A: set_o,A2: $o] :
      ( ( A = bot_bot_set_o )
     => ~ ( member_o @ A2 @ A ) ) ).

% equals0D
thf(fact_583_equals0D,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( A = bot_bot_set_nat )
     => ~ ( member_nat @ A2 @ A ) ) ).

% equals0D
thf(fact_584_equals0D,axiom,
    ! [A: set_int,A2: int] :
      ( ( A = bot_bot_set_int )
     => ~ ( member_int @ A2 @ A ) ) ).

% equals0D
thf(fact_585_equals0I,axiom,
    ! [A: set_complex] :
      ( ! [Y4: complex] :
          ~ ( member_complex @ Y4 @ A )
     => ( A = bot_bot_set_complex ) ) ).

% equals0I
thf(fact_586_equals0I,axiom,
    ! [A: set_real] :
      ( ! [Y4: real] :
          ~ ( member_real @ Y4 @ A )
     => ( A = bot_bot_set_real ) ) ).

% equals0I
thf(fact_587_equals0I,axiom,
    ! [A: set_o] :
      ( ! [Y4: $o] :
          ~ ( member_o @ Y4 @ A )
     => ( A = bot_bot_set_o ) ) ).

% equals0I
thf(fact_588_equals0I,axiom,
    ! [A: set_nat] :
      ( ! [Y4: nat] :
          ~ ( member_nat @ Y4 @ A )
     => ( A = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_589_equals0I,axiom,
    ! [A: set_int] :
      ( ! [Y4: int] :
          ~ ( member_int @ Y4 @ A )
     => ( A = bot_bot_set_int ) ) ).

% equals0I
thf(fact_590_insertI1,axiom,
    ! [A2: $o,B: set_o] : ( member_o @ A2 @ ( insert_o @ A2 @ B ) ) ).

% insertI1
thf(fact_591_insertI1,axiom,
    ! [A2: real,B: set_real] : ( member_real @ A2 @ ( insert_real @ A2 @ B ) ) ).

% insertI1
thf(fact_592_insertI1,axiom,
    ! [A2: nat,B: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B ) ) ).

% insertI1
thf(fact_593_insertI1,axiom,
    ! [A2: int,B: set_int] : ( member_int @ A2 @ ( insert_int @ A2 @ B ) ) ).

% insertI1
thf(fact_594_insertI1,axiom,
    ! [A2: complex,B: set_complex] : ( member_complex @ A2 @ ( insert_complex @ A2 @ B ) ) ).

% insertI1
thf(fact_595_insertI2,axiom,
    ! [A2: $o,B: set_o,B2: $o] :
      ( ( member_o @ A2 @ B )
     => ( member_o @ A2 @ ( insert_o @ B2 @ B ) ) ) ).

% insertI2
thf(fact_596_insertI2,axiom,
    ! [A2: real,B: set_real,B2: real] :
      ( ( member_real @ A2 @ B )
     => ( member_real @ A2 @ ( insert_real @ B2 @ B ) ) ) ).

% insertI2
thf(fact_597_insertI2,axiom,
    ! [A2: nat,B: set_nat,B2: nat] :
      ( ( member_nat @ A2 @ B )
     => ( member_nat @ A2 @ ( insert_nat @ B2 @ B ) ) ) ).

% insertI2
thf(fact_598_insertI2,axiom,
    ! [A2: int,B: set_int,B2: int] :
      ( ( member_int @ A2 @ B )
     => ( member_int @ A2 @ ( insert_int @ B2 @ B ) ) ) ).

% insertI2
thf(fact_599_insertI2,axiom,
    ! [A2: complex,B: set_complex,B2: complex] :
      ( ( member_complex @ A2 @ B )
     => ( member_complex @ A2 @ ( insert_complex @ B2 @ B ) ) ) ).

% insertI2
thf(fact_600_ex__in__conv,axiom,
    ! [A: set_complex] :
      ( ( ? [X2: complex] : ( member_complex @ X2 @ A ) )
      = ( A != bot_bot_set_complex ) ) ).

% ex_in_conv
thf(fact_601_ex__in__conv,axiom,
    ! [A: set_real] :
      ( ( ? [X2: real] : ( member_real @ X2 @ A ) )
      = ( A != bot_bot_set_real ) ) ).

% ex_in_conv
thf(fact_602_ex__in__conv,axiom,
    ! [A: set_o] :
      ( ( ? [X2: $o] : ( member_o @ X2 @ A ) )
      = ( A != bot_bot_set_o ) ) ).

% ex_in_conv
thf(fact_603_ex__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ? [X2: nat] : ( member_nat @ X2 @ A ) )
      = ( A != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_604_ex__in__conv,axiom,
    ! [A: set_int] :
      ( ( ? [X2: int] : ( member_int @ X2 @ A ) )
      = ( A != bot_bot_set_int ) ) ).

% ex_in_conv
thf(fact_605_Set_Oset__insert,axiom,
    ! [X: $o,A: set_o] :
      ( ( member_o @ X @ A )
     => ~ ! [B5: set_o] :
            ( ( A
              = ( insert_o @ X @ B5 ) )
           => ( member_o @ X @ B5 ) ) ) ).

% Set.set_insert
thf(fact_606_Set_Oset__insert,axiom,
    ! [X: real,A: set_real] :
      ( ( member_real @ X @ A )
     => ~ ! [B5: set_real] :
            ( ( A
              = ( insert_real @ X @ B5 ) )
           => ( member_real @ X @ B5 ) ) ) ).

% Set.set_insert
thf(fact_607_Set_Oset__insert,axiom,
    ! [X: nat,A: set_nat] :
      ( ( member_nat @ X @ A )
     => ~ ! [B5: set_nat] :
            ( ( A
              = ( insert_nat @ X @ B5 ) )
           => ( member_nat @ X @ B5 ) ) ) ).

% Set.set_insert
thf(fact_608_Set_Oset__insert,axiom,
    ! [X: int,A: set_int] :
      ( ( member_int @ X @ A )
     => ~ ! [B5: set_int] :
            ( ( A
              = ( insert_int @ X @ B5 ) )
           => ( member_int @ X @ B5 ) ) ) ).

% Set.set_insert
thf(fact_609_Set_Oset__insert,axiom,
    ! [X: complex,A: set_complex] :
      ( ( member_complex @ X @ A )
     => ~ ! [B5: set_complex] :
            ( ( A
              = ( insert_complex @ X @ B5 ) )
           => ( member_complex @ X @ B5 ) ) ) ).

% Set.set_insert
thf(fact_610_singletonD,axiom,
    ! [B2: complex,A2: complex] :
      ( ( member_complex @ B2 @ ( insert_complex @ A2 @ bot_bot_set_complex ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_611_singletonD,axiom,
    ! [B2: real,A2: real] :
      ( ( member_real @ B2 @ ( insert_real @ A2 @ bot_bot_set_real ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_612_singletonD,axiom,
    ! [B2: $o,A2: $o] :
      ( ( member_o @ B2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_613_singletonD,axiom,
    ! [B2: nat,A2: nat] :
      ( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_614_singletonD,axiom,
    ! [B2: int,A2: int] :
      ( ( member_int @ B2 @ ( insert_int @ A2 @ bot_bot_set_int ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_615_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_616_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_617_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_618_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_619_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_620_insert__absorb,axiom,
    ! [A2: $o,A: set_o] :
      ( ( member_o @ A2 @ A )
     => ( ( insert_o @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_621_insert__absorb,axiom,
    ! [A2: real,A: set_real] :
      ( ( member_real @ A2 @ A )
     => ( ( insert_real @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_622_insert__absorb,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( member_nat @ A2 @ A )
     => ( ( insert_nat @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_623_insert__absorb,axiom,
    ! [A2: int,A: set_int] :
      ( ( member_int @ A2 @ A )
     => ( ( insert_int @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_624_insert__absorb,axiom,
    ! [A2: complex,A: set_complex] :
      ( ( member_complex @ A2 @ A )
     => ( ( insert_complex @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_625_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_626_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_627_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_628_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_629_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_630_singleton__iff,axiom,
    ! [B2: complex,A2: complex] :
      ( ( member_complex @ B2 @ ( insert_complex @ A2 @ bot_bot_set_complex ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_631_singleton__iff,axiom,
    ! [B2: real,A2: real] :
      ( ( member_real @ B2 @ ( insert_real @ A2 @ bot_bot_set_real ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_632_singleton__iff,axiom,
    ! [B2: $o,A2: $o] :
      ( ( member_o @ B2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_633_singleton__iff,axiom,
    ! [B2: nat,A2: nat] :
      ( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_634_singleton__iff,axiom,
    ! [B2: int,A2: int] :
      ( ( member_int @ B2 @ ( insert_int @ A2 @ bot_bot_set_int ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_635_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_636_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_637_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_638_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_639_Collect__conv__if,axiom,
    ! [P: complex > $o,A2: complex] :
      ( ( ( P @ A2 )
       => ( ( collect_complex
            @ ^ [X2: complex] :
                ( ( X2 = A2 )
                & ( P @ X2 ) ) )
          = ( insert_complex @ A2 @ bot_bot_set_complex ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_complex
            @ ^ [X2: complex] :
                ( ( X2 = A2 )
                & ( P @ X2 ) ) )
          = bot_bot_set_complex ) ) ) ).

% Collect_conv_if
thf(fact_640_Collect__conv__if,axiom,
    ! [P: product_prod_int_int > $o,A2: product_prod_int_int] :
      ( ( ( P @ A2 )
       => ( ( collec213857154873943460nt_int
            @ ^ [X2: product_prod_int_int] :
                ( ( X2 = A2 )
                & ( P @ X2 ) ) )
          = ( insert5033312907999012233nt_int @ A2 @ bot_bo1796632182523588997nt_int ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collec213857154873943460nt_int
            @ ^ [X2: product_prod_int_int] :
                ( ( X2 = A2 )
                & ( P @ X2 ) ) )
          = bot_bo1796632182523588997nt_int ) ) ) ).

% Collect_conv_if
thf(fact_641_Collect__conv__if,axiom,
    ! [P: real > $o,A2: real] :
      ( ( ( P @ A2 )
       => ( ( collect_real
            @ ^ [X2: real] :
                ( ( X2 = A2 )
                & ( P @ X2 ) ) )
          = ( insert_real @ A2 @ bot_bot_set_real ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_real
            @ ^ [X2: real] :
                ( ( X2 = A2 )
                & ( P @ X2 ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if
thf(fact_642_Collect__conv__if,axiom,
    ! [P: $o > $o,A2: $o] :
      ( ( ( P @ A2 )
       => ( ( collect_o
            @ ^ [X2: $o] :
                ( ( X2 = A2 )
                & ( P @ X2 ) ) )
          = ( insert_o @ A2 @ bot_bot_set_o ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_o
            @ ^ [X2: $o] :
                ( ( X2 = A2 )
                & ( P @ X2 ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if
thf(fact_643_Collect__conv__if,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ( ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X2: nat] :
                ( ( X2 = A2 )
                & ( P @ X2 ) ) )
          = ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X2: nat] :
                ( ( X2 = A2 )
                & ( P @ X2 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_644_Collect__conv__if,axiom,
    ! [P: int > $o,A2: int] :
      ( ( ( P @ A2 )
       => ( ( collect_int
            @ ^ [X2: int] :
                ( ( X2 = A2 )
                & ( P @ X2 ) ) )
          = ( insert_int @ A2 @ bot_bot_set_int ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_int
            @ ^ [X2: int] :
                ( ( X2 = A2 )
                & ( P @ X2 ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if
thf(fact_645_Collect__conv__if2,axiom,
    ! [P: complex > $o,A2: complex] :
      ( ( ( P @ A2 )
       => ( ( collect_complex
            @ ^ [X2: complex] :
                ( ( A2 = X2 )
                & ( P @ X2 ) ) )
          = ( insert_complex @ A2 @ bot_bot_set_complex ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_complex
            @ ^ [X2: complex] :
                ( ( A2 = X2 )
                & ( P @ X2 ) ) )
          = bot_bot_set_complex ) ) ) ).

% Collect_conv_if2
thf(fact_646_Collect__conv__if2,axiom,
    ! [P: product_prod_int_int > $o,A2: product_prod_int_int] :
      ( ( ( P @ A2 )
       => ( ( collec213857154873943460nt_int
            @ ^ [X2: product_prod_int_int] :
                ( ( A2 = X2 )
                & ( P @ X2 ) ) )
          = ( insert5033312907999012233nt_int @ A2 @ bot_bo1796632182523588997nt_int ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collec213857154873943460nt_int
            @ ^ [X2: product_prod_int_int] :
                ( ( A2 = X2 )
                & ( P @ X2 ) ) )
          = bot_bo1796632182523588997nt_int ) ) ) ).

% Collect_conv_if2
thf(fact_647_Collect__conv__if2,axiom,
    ! [P: real > $o,A2: real] :
      ( ( ( P @ A2 )
       => ( ( collect_real
            @ ^ [X2: real] :
                ( ( A2 = X2 )
                & ( P @ X2 ) ) )
          = ( insert_real @ A2 @ bot_bot_set_real ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_real
            @ ^ [X2: real] :
                ( ( A2 = X2 )
                & ( P @ X2 ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if2
thf(fact_648_Collect__conv__if2,axiom,
    ! [P: $o > $o,A2: $o] :
      ( ( ( P @ A2 )
       => ( ( collect_o
            @ ^ [X2: $o] :
                ( ( A2 = X2 )
                & ( P @ X2 ) ) )
          = ( insert_o @ A2 @ bot_bot_set_o ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_o
            @ ^ [X2: $o] :
                ( ( A2 = X2 )
                & ( P @ X2 ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if2
thf(fact_649_Collect__conv__if2,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ( ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X2: nat] :
                ( ( A2 = X2 )
                & ( P @ X2 ) ) )
          = ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X2: nat] :
                ( ( A2 = X2 )
                & ( P @ X2 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_650_Collect__conv__if2,axiom,
    ! [P: int > $o,A2: int] :
      ( ( ( P @ A2 )
       => ( ( collect_int
            @ ^ [X2: int] :
                ( ( A2 = X2 )
                & ( P @ X2 ) ) )
          = ( insert_int @ A2 @ bot_bot_set_int ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_int
            @ ^ [X2: int] :
                ( ( A2 = X2 )
                & ( P @ X2 ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if2
thf(fact_651_doubleton__eq__iff,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( ( ( insert_real @ A2 @ ( insert_real @ B2 @ bot_bot_set_real ) )
        = ( insert_real @ C @ ( insert_real @ D @ bot_bot_set_real ) ) )
      = ( ( ( A2 = C )
          & ( B2 = D ) )
        | ( ( A2 = D )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_652_doubleton__eq__iff,axiom,
    ! [A2: $o,B2: $o,C: $o,D: $o] :
      ( ( ( insert_o @ A2 @ ( insert_o @ B2 @ bot_bot_set_o ) )
        = ( insert_o @ C @ ( insert_o @ D @ bot_bot_set_o ) ) )
      = ( ( ( A2 = C )
          & ( B2 = D ) )
        | ( ( A2 = D )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_653_doubleton__eq__iff,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( ( ( insert_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) )
        = ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
      = ( ( ( A2 = C )
          & ( B2 = D ) )
        | ( ( A2 = D )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_654_doubleton__eq__iff,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( ( ( insert_int @ A2 @ ( insert_int @ B2 @ bot_bot_set_int ) )
        = ( insert_int @ C @ ( insert_int @ D @ bot_bot_set_int ) ) )
      = ( ( ( A2 = C )
          & ( B2 = D ) )
        | ( ( A2 = D )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_655_insert__not__empty,axiom,
    ! [A2: real,A: set_real] :
      ( ( insert_real @ A2 @ A )
     != bot_bot_set_real ) ).

% insert_not_empty
thf(fact_656_insert__not__empty,axiom,
    ! [A2: $o,A: set_o] :
      ( ( insert_o @ A2 @ A )
     != bot_bot_set_o ) ).

% insert_not_empty
thf(fact_657_insert__not__empty,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( insert_nat @ A2 @ A )
     != bot_bot_set_nat ) ).

% insert_not_empty
thf(fact_658_insert__not__empty,axiom,
    ! [A2: int,A: set_int] :
      ( ( insert_int @ A2 @ A )
     != bot_bot_set_int ) ).

% insert_not_empty
thf(fact_659_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_660_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_661_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_662_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_663_not__psubset__empty,axiom,
    ! [A: set_real] :
      ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).

% not_psubset_empty
thf(fact_664_not__psubset__empty,axiom,
    ! [A: set_o] :
      ~ ( ord_less_set_o @ A @ bot_bot_set_o ) ).

% not_psubset_empty
thf(fact_665_not__psubset__empty,axiom,
    ! [A: set_nat] :
      ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).

% not_psubset_empty
thf(fact_666_not__psubset__empty,axiom,
    ! [A: set_int] :
      ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).

% not_psubset_empty
thf(fact_667_mk__disjoint__insert,axiom,
    ! [A2: $o,A: set_o] :
      ( ( member_o @ A2 @ A )
     => ? [B5: set_o] :
          ( ( A
            = ( insert_o @ A2 @ B5 ) )
          & ~ ( member_o @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_668_mk__disjoint__insert,axiom,
    ! [A2: real,A: set_real] :
      ( ( member_real @ A2 @ A )
     => ? [B5: set_real] :
          ( ( A
            = ( insert_real @ A2 @ B5 ) )
          & ~ ( member_real @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_669_mk__disjoint__insert,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( member_nat @ A2 @ A )
     => ? [B5: set_nat] :
          ( ( A
            = ( insert_nat @ A2 @ B5 ) )
          & ~ ( member_nat @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_670_mk__disjoint__insert,axiom,
    ! [A2: int,A: set_int] :
      ( ( member_int @ A2 @ A )
     => ? [B5: set_int] :
          ( ( A
            = ( insert_int @ A2 @ B5 ) )
          & ~ ( member_int @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_671_mk__disjoint__insert,axiom,
    ! [A2: complex,A: set_complex] :
      ( ( member_complex @ A2 @ A )
     => ? [B5: set_complex] :
          ( ( A
            = ( insert_complex @ A2 @ B5 ) )
          & ~ ( member_complex @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_672_not__iless0,axiom,
    ! [N: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).

% not_iless0
thf(fact_673_enat__less__induct,axiom,
    ! [P: extended_enat > $o,N: extended_enat] :
      ( ! [N2: extended_enat] :
          ( ! [M2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ M2 @ N2 )
             => ( P @ M2 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% enat_less_induct
thf(fact_674_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_2184487114949457152l_num1 @ zero_z3563351764282998399l_num1 @ N )
          = one_on7727431528512463931l_num1 ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_2184487114949457152l_num1 @ zero_z3563351764282998399l_num1 @ N )
          = zero_z3563351764282998399l_num1 ) ) ) ).

% power_0_left
thf(fact_675_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N )
          = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N )
          = zero_zero_rat ) ) ) ).

% power_0_left
thf(fact_676_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N )
          = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N )
          = zero_zero_nat ) ) ) ).

% power_0_left
thf(fact_677_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N )
          = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N )
          = zero_zero_real ) ) ) ).

% power_0_left
thf(fact_678_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N )
          = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N )
          = zero_zero_int ) ) ) ).

% power_0_left
thf(fact_679_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N )
          = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N )
          = zero_zero_complex ) ) ) ).

% power_0_left
thf(fact_680_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N )
          = one_one_Code_integer ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N )
          = zero_z3403309356797280102nteger ) ) ) ).

% power_0_left
thf(fact_681_power__0,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( power_2184487114949457152l_num1 @ A2 @ zero_zero_nat )
      = one_on7727431528512463931l_num1 ) ).

% power_0
thf(fact_682_power__0,axiom,
    ! [A2: rat] :
      ( ( power_power_rat @ A2 @ zero_zero_nat )
      = one_one_rat ) ).

% power_0
thf(fact_683_power__0,axiom,
    ! [A2: nat] :
      ( ( power_power_nat @ A2 @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_684_power__0,axiom,
    ! [A2: real] :
      ( ( power_power_real @ A2 @ zero_zero_nat )
      = one_one_real ) ).

% power_0
thf(fact_685_power__0,axiom,
    ! [A2: int] :
      ( ( power_power_int @ A2 @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_686_power__0,axiom,
    ! [A2: complex] :
      ( ( power_power_complex @ A2 @ zero_zero_nat )
      = one_one_complex ) ).

% power_0
thf(fact_687_power__0,axiom,
    ! [A2: code_integer] :
      ( ( power_8256067586552552935nteger @ A2 @ zero_zero_nat )
      = one_one_Code_integer ) ).

% power_0
thf(fact_688_less__numeral__extra_I1_J,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% less_numeral_extra(1)
thf(fact_689_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_690_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_691_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_692_singleton__Un__iff,axiom,
    ! [X: complex,A: set_complex,B: set_complex] :
      ( ( ( insert_complex @ X @ bot_bot_set_complex )
        = ( sup_sup_set_complex @ A @ B ) )
      = ( ( ( A = bot_bot_set_complex )
          & ( B
            = ( insert_complex @ X @ bot_bot_set_complex ) ) )
        | ( ( A
            = ( insert_complex @ X @ bot_bot_set_complex ) )
          & ( B = bot_bot_set_complex ) )
        | ( ( A
            = ( insert_complex @ X @ bot_bot_set_complex ) )
          & ( B
            = ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_693_singleton__Un__iff,axiom,
    ! [X: real,A: set_real,B: set_real] :
      ( ( ( insert_real @ X @ bot_bot_set_real )
        = ( sup_sup_set_real @ A @ B ) )
      = ( ( ( A = bot_bot_set_real )
          & ( B
            = ( insert_real @ X @ bot_bot_set_real ) ) )
        | ( ( A
            = ( insert_real @ X @ bot_bot_set_real ) )
          & ( B = bot_bot_set_real ) )
        | ( ( A
            = ( insert_real @ X @ bot_bot_set_real ) )
          & ( B
            = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_694_singleton__Un__iff,axiom,
    ! [X: $o,A: set_o,B: set_o] :
      ( ( ( insert_o @ X @ bot_bot_set_o )
        = ( sup_sup_set_o @ A @ B ) )
      = ( ( ( A = bot_bot_set_o )
          & ( B
            = ( insert_o @ X @ bot_bot_set_o ) ) )
        | ( ( A
            = ( insert_o @ X @ bot_bot_set_o ) )
          & ( B = bot_bot_set_o ) )
        | ( ( A
            = ( insert_o @ X @ bot_bot_set_o ) )
          & ( B
            = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_695_singleton__Un__iff,axiom,
    ! [X: nat,A: set_nat,B: set_nat] :
      ( ( ( insert_nat @ X @ bot_bot_set_nat )
        = ( sup_sup_set_nat @ A @ B ) )
      = ( ( ( A = bot_bot_set_nat )
          & ( B
            = ( insert_nat @ X @ bot_bot_set_nat ) ) )
        | ( ( A
            = ( insert_nat @ X @ bot_bot_set_nat ) )
          & ( B = bot_bot_set_nat ) )
        | ( ( A
            = ( insert_nat @ X @ bot_bot_set_nat ) )
          & ( B
            = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_696_singleton__Un__iff,axiom,
    ! [X: int,A: set_int,B: set_int] :
      ( ( ( insert_int @ X @ bot_bot_set_int )
        = ( sup_sup_set_int @ A @ B ) )
      = ( ( ( A = bot_bot_set_int )
          & ( B
            = ( insert_int @ X @ bot_bot_set_int ) ) )
        | ( ( A
            = ( insert_int @ X @ bot_bot_set_int ) )
          & ( B = bot_bot_set_int ) )
        | ( ( A
            = ( insert_int @ X @ bot_bot_set_int ) )
          & ( B
            = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_697_Un__singleton__iff,axiom,
    ! [A: set_complex,B: set_complex,X: complex] :
      ( ( ( sup_sup_set_complex @ A @ B )
        = ( insert_complex @ X @ bot_bot_set_complex ) )
      = ( ( ( A = bot_bot_set_complex )
          & ( B
            = ( insert_complex @ X @ bot_bot_set_complex ) ) )
        | ( ( A
            = ( insert_complex @ X @ bot_bot_set_complex ) )
          & ( B = bot_bot_set_complex ) )
        | ( ( A
            = ( insert_complex @ X @ bot_bot_set_complex ) )
          & ( B
            = ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_698_Un__singleton__iff,axiom,
    ! [A: set_real,B: set_real,X: real] :
      ( ( ( sup_sup_set_real @ A @ B )
        = ( insert_real @ X @ bot_bot_set_real ) )
      = ( ( ( A = bot_bot_set_real )
          & ( B
            = ( insert_real @ X @ bot_bot_set_real ) ) )
        | ( ( A
            = ( insert_real @ X @ bot_bot_set_real ) )
          & ( B = bot_bot_set_real ) )
        | ( ( A
            = ( insert_real @ X @ bot_bot_set_real ) )
          & ( B
            = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_699_Un__singleton__iff,axiom,
    ! [A: set_o,B: set_o,X: $o] :
      ( ( ( sup_sup_set_o @ A @ B )
        = ( insert_o @ X @ bot_bot_set_o ) )
      = ( ( ( A = bot_bot_set_o )
          & ( B
            = ( insert_o @ X @ bot_bot_set_o ) ) )
        | ( ( A
            = ( insert_o @ X @ bot_bot_set_o ) )
          & ( B = bot_bot_set_o ) )
        | ( ( A
            = ( insert_o @ X @ bot_bot_set_o ) )
          & ( B
            = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_700_Un__singleton__iff,axiom,
    ! [A: set_nat,B: set_nat,X: nat] :
      ( ( ( sup_sup_set_nat @ A @ B )
        = ( insert_nat @ X @ bot_bot_set_nat ) )
      = ( ( ( A = bot_bot_set_nat )
          & ( B
            = ( insert_nat @ X @ bot_bot_set_nat ) ) )
        | ( ( A
            = ( insert_nat @ X @ bot_bot_set_nat ) )
          & ( B = bot_bot_set_nat ) )
        | ( ( A
            = ( insert_nat @ X @ bot_bot_set_nat ) )
          & ( B
            = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_701_Un__singleton__iff,axiom,
    ! [A: set_int,B: set_int,X: int] :
      ( ( ( sup_sup_set_int @ A @ B )
        = ( insert_int @ X @ bot_bot_set_int ) )
      = ( ( ( A = bot_bot_set_int )
          & ( B
            = ( insert_int @ X @ bot_bot_set_int ) ) )
        | ( ( A
            = ( insert_int @ X @ bot_bot_set_int ) )
          & ( B = bot_bot_set_int ) )
        | ( ( A
            = ( insert_int @ X @ bot_bot_set_int ) )
          & ( B
            = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_702_insert__is__Un,axiom,
    ( insert_complex
    = ( ^ [A4: complex] : ( sup_sup_set_complex @ ( insert_complex @ A4 @ bot_bot_set_complex ) ) ) ) ).

% insert_is_Un
thf(fact_703_insert__is__Un,axiom,
    ( insert_real
    = ( ^ [A4: real] : ( sup_sup_set_real @ ( insert_real @ A4 @ bot_bot_set_real ) ) ) ) ).

% insert_is_Un
thf(fact_704_insert__is__Un,axiom,
    ( insert_o
    = ( ^ [A4: $o] : ( sup_sup_set_o @ ( insert_o @ A4 @ bot_bot_set_o ) ) ) ) ).

% insert_is_Un
thf(fact_705_insert__is__Un,axiom,
    ( insert_nat
    = ( ^ [A4: nat] : ( sup_sup_set_nat @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ).

% insert_is_Un
thf(fact_706_insert__is__Un,axiom,
    ( insert_int
    = ( ^ [A4: int] : ( sup_sup_set_int @ ( insert_int @ A4 @ bot_bot_set_int ) ) ) ) ).

% insert_is_Un
thf(fact_707_int__ops_I3_J,axiom,
    ! [N: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% int_ops(3)
thf(fact_708_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_709_insert__Collect,axiom,
    ! [A2: real,P: real > $o] :
      ( ( insert_real @ A2 @ ( collect_real @ P ) )
      = ( collect_real
        @ ^ [U: real] :
            ( ( U != A2 )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_710_insert__Collect,axiom,
    ! [A2: $o,P: $o > $o] :
      ( ( insert_o @ A2 @ ( collect_o @ P ) )
      = ( collect_o
        @ ^ [U: $o] :
            ( ( U != A2 )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_711_insert__Collect,axiom,
    ! [A2: nat,P: nat > $o] :
      ( ( insert_nat @ A2 @ ( collect_nat @ P ) )
      = ( collect_nat
        @ ^ [U: nat] :
            ( ( U != A2 )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_712_insert__Collect,axiom,
    ! [A2: complex,P: complex > $o] :
      ( ( insert_complex @ A2 @ ( collect_complex @ P ) )
      = ( collect_complex
        @ ^ [U: complex] :
            ( ( U != A2 )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_713_insert__Collect,axiom,
    ! [A2: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( insert5033312907999012233nt_int @ A2 @ ( collec213857154873943460nt_int @ P ) )
      = ( collec213857154873943460nt_int
        @ ^ [U: product_prod_int_int] :
            ( ( U != A2 )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_714_insert__Collect,axiom,
    ! [A2: int,P: int > $o] :
      ( ( insert_int @ A2 @ ( collect_int @ P ) )
      = ( collect_int
        @ ^ [U: int] :
            ( ( U != A2 )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_715_insert__compr,axiom,
    ( insert_o
    = ( ^ [A4: $o,B3: set_o] :
          ( collect_o
          @ ^ [X2: $o] :
              ( ( X2 = A4 )
              | ( member_o @ X2 @ B3 ) ) ) ) ) ).

% insert_compr
thf(fact_716_insert__compr,axiom,
    ( insert_real
    = ( ^ [A4: real,B3: set_real] :
          ( collect_real
          @ ^ [X2: real] :
              ( ( X2 = A4 )
              | ( member_real @ X2 @ B3 ) ) ) ) ) ).

% insert_compr
thf(fact_717_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A4: nat,B3: set_nat] :
          ( collect_nat
          @ ^ [X2: nat] :
              ( ( X2 = A4 )
              | ( member_nat @ X2 @ B3 ) ) ) ) ) ).

% insert_compr
thf(fact_718_insert__compr,axiom,
    ( insert_complex
    = ( ^ [A4: complex,B3: set_complex] :
          ( collect_complex
          @ ^ [X2: complex] :
              ( ( X2 = A4 )
              | ( member_complex @ X2 @ B3 ) ) ) ) ) ).

% insert_compr
thf(fact_719_insert__compr,axiom,
    ( insert5033312907999012233nt_int
    = ( ^ [A4: product_prod_int_int,B3: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ^ [X2: product_prod_int_int] :
              ( ( X2 = A4 )
              | ( member5262025264175285858nt_int @ X2 @ B3 ) ) ) ) ) ).

% insert_compr
thf(fact_720_insert__compr,axiom,
    ( insert_int
    = ( ^ [A4: int,B3: set_int] :
          ( collect_int
          @ ^ [X2: int] :
              ( ( X2 = A4 )
              | ( member_int @ X2 @ B3 ) ) ) ) ) ).

% insert_compr
thf(fact_721_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: rat,N: nat] :
      ( ( A2 != zero_zero_rat )
     => ( ( power_power_rat @ A2 @ N )
       != zero_zero_rat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_722_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: nat,N: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( power_power_nat @ A2 @ N )
       != zero_zero_nat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_723_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: real,N: nat] :
      ( ( A2 != zero_zero_real )
     => ( ( power_power_real @ A2 @ N )
       != zero_zero_real ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_724_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: int,N: nat] :
      ( ( A2 != zero_zero_int )
     => ( ( power_power_int @ A2 @ N )
       != zero_zero_int ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_725_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: complex,N: nat] :
      ( ( A2 != zero_zero_complex )
     => ( ( power_power_complex @ A2 @ N )
       != zero_zero_complex ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_726_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ A2 @ N )
       != zero_z3403309356797280102nteger ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_727_Set_Oempty__def,axiom,
    ( bot_bot_set_complex
    = ( collect_complex
      @ ^ [X2: complex] : $false ) ) ).

% Set.empty_def
thf(fact_728_Set_Oempty__def,axiom,
    ( bot_bo1796632182523588997nt_int
    = ( collec213857154873943460nt_int
      @ ^ [X2: product_prod_int_int] : $false ) ) ).

% Set.empty_def
thf(fact_729_Set_Oempty__def,axiom,
    ( bot_bot_set_real
    = ( collect_real
      @ ^ [X2: real] : $false ) ) ).

% Set.empty_def
thf(fact_730_Set_Oempty__def,axiom,
    ( bot_bot_set_o
    = ( collect_o
      @ ^ [X2: $o] : $false ) ) ).

% Set.empty_def
thf(fact_731_Set_Oempty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X2: nat] : $false ) ) ).

% Set.empty_def
thf(fact_732_Set_Oempty__def,axiom,
    ( bot_bot_set_int
    = ( collect_int
      @ ^ [X2: int] : $false ) ) ).

% Set.empty_def
thf(fact_733_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_2184487114949457152l_num1 @ zero_z3563351764282998399l_num1 @ N )
        = zero_z3563351764282998399l_num1 ) ) ).

% zero_power
thf(fact_734_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_rat @ zero_zero_rat @ N )
        = zero_zero_rat ) ) ).

% zero_power
thf(fact_735_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_nat @ zero_zero_nat @ N )
        = zero_zero_nat ) ) ).

% zero_power
thf(fact_736_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_real @ zero_zero_real @ N )
        = zero_zero_real ) ) ).

% zero_power
thf(fact_737_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_int @ zero_zero_int @ N )
        = zero_zero_int ) ) ).

% zero_power
thf(fact_738_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_complex @ zero_zero_complex @ N )
        = zero_zero_complex ) ) ).

% zero_power
thf(fact_739_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N )
        = zero_z3403309356797280102nteger ) ) ).

% zero_power
thf(fact_740_nat__less__as__int,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_less_as_int
thf(fact_741_power__strict__decreasing,axiom,
    ! [N: nat,N3: nat,A2: code_integer] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
       => ( ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer )
         => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_742_power__strict__decreasing,axiom,
    ! [N: nat,N3: nat,A2: real] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ A2 )
       => ( ( ord_less_real @ A2 @ one_one_real )
         => ( ord_less_real @ ( power_power_real @ A2 @ N3 ) @ ( power_power_real @ A2 @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_743_power__strict__decreasing,axiom,
    ! [N: nat,N3: nat,A2: rat] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_less_rat @ zero_zero_rat @ A2 )
       => ( ( ord_less_rat @ A2 @ one_one_rat )
         => ( ord_less_rat @ ( power_power_rat @ A2 @ N3 ) @ ( power_power_rat @ A2 @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_744_power__strict__decreasing,axiom,
    ! [N: nat,N3: nat,A2: nat] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_less_nat @ zero_zero_nat @ A2 )
       => ( ( ord_less_nat @ A2 @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A2 @ N3 ) @ ( power_power_nat @ A2 @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_745_power__strict__decreasing,axiom,
    ! [N: nat,N3: nat,A2: int] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_less_int @ zero_zero_int @ A2 )
       => ( ( ord_less_int @ A2 @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A2 @ N3 ) @ ( power_power_int @ A2 @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_746_one__less__power,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ) ).

% one_less_power
thf(fact_747_one__less__power,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_real @ one_one_real @ ( power_power_real @ A2 @ N ) ) ) ) ).

% one_less_power
thf(fact_748_one__less__power,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A2 @ N ) ) ) ) ).

% one_less_power
thf(fact_749_one__less__power,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A2 @ N ) ) ) ) ).

% one_less_power
thf(fact_750_one__less__power,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_int @ one_one_int @ ( power_power_int @ A2 @ N ) ) ) ) ).

% one_less_power
thf(fact_751_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ one_one_real ) ).

% less_numeral_extra(4)
thf(fact_752_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_753_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_754_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_755_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).

% less_numeral_extra(3)
thf(fact_756_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).

% less_numeral_extra(3)
thf(fact_757_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_758_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_759_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_real
     != ( numeral_numeral_real @ N ) ) ).

% zero_neq_numeral
thf(fact_760_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_rat
     != ( numeral_numeral_rat @ N ) ) ).

% zero_neq_numeral
thf(fact_761_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_nat
     != ( numeral_numeral_nat @ N ) ) ).

% zero_neq_numeral
thf(fact_762_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_int
     != ( numeral_numeral_int @ N ) ) ).

% zero_neq_numeral
thf(fact_763_Un__empty__left,axiom,
    ! [B: set_complex] :
      ( ( sup_sup_set_complex @ bot_bot_set_complex @ B )
      = B ) ).

% Un_empty_left
thf(fact_764_Un__empty__left,axiom,
    ! [B: set_real] :
      ( ( sup_sup_set_real @ bot_bot_set_real @ B )
      = B ) ).

% Un_empty_left
thf(fact_765_Un__empty__left,axiom,
    ! [B: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ B )
      = B ) ).

% Un_empty_left
thf(fact_766_Un__empty__left,axiom,
    ! [B: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
      = B ) ).

% Un_empty_left
thf(fact_767_Un__empty__left,axiom,
    ! [B: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ B )
      = B ) ).

% Un_empty_left
thf(fact_768_Un__empty__right,axiom,
    ! [A: set_complex] :
      ( ( sup_sup_set_complex @ A @ bot_bot_set_complex )
      = A ) ).

% Un_empty_right
thf(fact_769_Un__empty__right,axiom,
    ! [A: set_real] :
      ( ( sup_sup_set_real @ A @ bot_bot_set_real )
      = A ) ).

% Un_empty_right
thf(fact_770_Un__empty__right,axiom,
    ! [A: set_o] :
      ( ( sup_sup_set_o @ A @ bot_bot_set_o )
      = A ) ).

% Un_empty_right
thf(fact_771_Un__empty__right,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
      = A ) ).

% Un_empty_right
thf(fact_772_Un__empty__right,axiom,
    ! [A: set_int] :
      ( ( sup_sup_set_int @ A @ bot_bot_set_int )
      = A ) ).

% Un_empty_right
thf(fact_773_zero__less__power,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ).

% zero_less_power
thf(fact_774_zero__less__power,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A2 @ N ) ) ) ).

% zero_less_power
thf(fact_775_zero__less__power,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ N ) ) ) ).

% zero_less_power
thf(fact_776_zero__less__power,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A2 @ N ) ) ) ).

% zero_less_power
thf(fact_777_zero__less__power,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A2 @ N ) ) ) ).

% zero_less_power
thf(fact_778_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% nat_power_less_imp_less
thf(fact_779_insert__def,axiom,
    ( insert_o
    = ( ^ [A4: $o] :
          ( sup_sup_set_o
          @ ( collect_o
            @ ^ [X2: $o] : X2 = A4 ) ) ) ) ).

% insert_def
thf(fact_780_insert__def,axiom,
    ( insert5033312907999012233nt_int
    = ( ^ [A4: product_prod_int_int] :
          ( sup_su6024340866399070445nt_int
          @ ( collec213857154873943460nt_int
            @ ^ [X2: product_prod_int_int] : X2 = A4 ) ) ) ) ).

% insert_def
thf(fact_781_insert__def,axiom,
    ( insert_nat
    = ( ^ [A4: nat] :
          ( sup_sup_set_nat
          @ ( collect_nat
            @ ^ [X2: nat] : X2 = A4 ) ) ) ) ).

% insert_def
thf(fact_782_insert__def,axiom,
    ( insert_complex
    = ( ^ [A4: complex] :
          ( sup_sup_set_complex
          @ ( collect_complex
            @ ^ [X2: complex] : X2 = A4 ) ) ) ) ).

% insert_def
thf(fact_783_insert__def,axiom,
    ( insert_int
    = ( ^ [A4: int] :
          ( sup_sup_set_int
          @ ( collect_int
            @ ^ [X2: int] : X2 = A4 ) ) ) ) ).

% insert_def
thf(fact_784_insert__def,axiom,
    ( insert_real
    = ( ^ [A4: real] :
          ( sup_sup_set_real
          @ ( collect_real
            @ ^ [X2: real] : X2 = A4 ) ) ) ) ).

% insert_def
thf(fact_785_power__less__imp__less__exp,axiom,
    ! [A2: code_integer,M: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
     => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ M ) @ ( power_8256067586552552935nteger @ A2 @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_786_power__less__imp__less__exp,axiom,
    ! [A2: real,M: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_real @ ( power_power_real @ A2 @ M ) @ ( power_power_real @ A2 @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_787_power__less__imp__less__exp,axiom,
    ! [A2: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ( ord_less_rat @ ( power_power_rat @ A2 @ M ) @ ( power_power_rat @ A2 @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_788_power__less__imp__less__exp,axiom,
    ! [A2: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ( ord_less_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_789_power__less__imp__less__exp,axiom,
    ! [A2: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ( ord_less_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_790_power__strict__increasing,axiom,
    ! [N: nat,N3: nat,A2: code_integer] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
       => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_791_power__strict__increasing,axiom,
    ! [N: nat,N3: nat,A2: real] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_less_real @ one_one_real @ A2 )
       => ( ord_less_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ A2 @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_792_power__strict__increasing,axiom,
    ! [N: nat,N3: nat,A2: rat] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_less_rat @ one_one_rat @ A2 )
       => ( ord_less_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ A2 @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_793_power__strict__increasing,axiom,
    ! [N: nat,N3: nat,A2: nat] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_less_nat @ one_one_nat @ A2 )
       => ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ A2 @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_794_power__strict__increasing,axiom,
    ! [N: nat,N3: nat,A2: int] :
      ( ( ord_less_nat @ N @ N3 )
     => ( ( ord_less_int @ one_one_int @ A2 )
       => ( ord_less_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ A2 @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_795_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).

% not_numeral_less_one
thf(fact_796_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).

% not_numeral_less_one
thf(fact_797_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_798_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_799_numeral__One,axiom,
    ( ( numera7442385471795722001l_num1 @ one )
    = one_on7727431528512463931l_num1 ) ).

% numeral_One
thf(fact_800_numeral__One,axiom,
    ( ( numeral_numeral_real @ one )
    = one_one_real ) ).

% numeral_One
thf(fact_801_numeral__One,axiom,
    ( ( numeral_numeral_rat @ one )
    = one_one_rat ) ).

% numeral_One
thf(fact_802_numeral__One,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numeral_One
thf(fact_803_numeral__One,axiom,
    ( ( numeral_numeral_int @ one )
    = one_one_int ) ).

% numeral_One
thf(fact_804_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).

% zero_less_numeral
thf(fact_805_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).

% zero_less_numeral
thf(fact_806_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).

% zero_less_numeral
thf(fact_807_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).

% zero_less_numeral
thf(fact_808_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).

% not_numeral_less_zero
thf(fact_809_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).

% not_numeral_less_zero
thf(fact_810_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_811_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).

% not_numeral_less_zero
thf(fact_812_numerals_I1_J,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numerals(1)
thf(fact_813_pow_Osimps_I1_J,axiom,
    ! [X: num] :
      ( ( pow @ X @ one )
      = X ) ).

% pow.simps(1)
thf(fact_814_one__power2,axiom,
    ( ( power_2184487114949457152l_num1 @ one_on7727431528512463931l_num1 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_on7727431528512463931l_num1 ) ).

% one_power2
thf(fact_815_one__power2,axiom,
    ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_rat ) ).

% one_power2
thf(fact_816_one__power2,axiom,
    ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_power2
thf(fact_817_one__power2,axiom,
    ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_real ) ).

% one_power2
thf(fact_818_one__power2,axiom,
    ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_power2
thf(fact_819_one__power2,axiom,
    ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_complex ) ).

% one_power2
thf(fact_820_one__power2,axiom,
    ( ( power_8256067586552552935nteger @ one_one_Code_integer @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% one_power2
thf(fact_821_zero__power2,axiom,
    ( ( power_2184487114949457152l_num1 @ zero_z3563351764282998399l_num1 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_z3563351764282998399l_num1 ) ).

% zero_power2
thf(fact_822_zero__power2,axiom,
    ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_rat ) ).

% zero_power2
thf(fact_823_zero__power2,axiom,
    ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% zero_power2
thf(fact_824_zero__power2,axiom,
    ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_real ) ).

% zero_power2
thf(fact_825_zero__power2,axiom,
    ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% zero_power2
thf(fact_826_zero__power2,axiom,
    ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_complex ) ).

% zero_power2
thf(fact_827_zero__power2,axiom,
    ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_z3403309356797280102nteger ) ).

% zero_power2
thf(fact_828_verit__comp__simplify1_I1_J,axiom,
    ! [A2: real] :
      ~ ( ord_less_real @ A2 @ A2 ) ).

% verit_comp_simplify1(1)
thf(fact_829_verit__comp__simplify1_I1_J,axiom,
    ! [A2: rat] :
      ~ ( ord_less_rat @ A2 @ A2 ) ).

% verit_comp_simplify1(1)
thf(fact_830_verit__comp__simplify1_I1_J,axiom,
    ! [A2: num] :
      ~ ( ord_less_num @ A2 @ A2 ) ).

% verit_comp_simplify1(1)
thf(fact_831_verit__comp__simplify1_I1_J,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ A2 ) ).

% verit_comp_simplify1(1)
thf(fact_832_verit__comp__simplify1_I1_J,axiom,
    ! [A2: int] :
      ~ ( ord_less_int @ A2 @ A2 ) ).

% verit_comp_simplify1(1)
thf(fact_833_power2__less__0,axiom,
    ! [A2: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_z3403309356797280102nteger ) ).

% power2_less_0
thf(fact_834_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_835_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_836_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_837_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_838_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_839_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_840_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_841_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( semiri4939895301339042750nteger @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_842_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_843_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_844_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_845_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_846_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_847_of__nat__less__two__power,axiom,
    ! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).

% of_nat_less_two_power
thf(fact_848_of__nat__less__two__power,axiom,
    ! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% of_nat_less_two_power
thf(fact_849_of__nat__less__two__power,axiom,
    ! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).

% of_nat_less_two_power
thf(fact_850_of__nat__less__two__power,axiom,
    ! [N: nat] : ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ N ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ).

% of_nat_less_two_power
thf(fact_851_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri681578069525770553at_rat @ N )
        = one_one_rat )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_852_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri1314217659103216013at_int @ N )
        = one_one_int )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_853_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri5074537144036343181t_real @ N )
        = one_one_real )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_854_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ N )
        = one_one_nat )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_855_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri4939895301339042750nteger @ N )
        = one_one_Code_integer )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_856_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri8010041392384452111omplex @ N )
        = one_one_complex )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_857_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_rat
        = ( semiri681578069525770553at_rat @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_858_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_int
        = ( semiri1314217659103216013at_int @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_859_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_real
        = ( semiri5074537144036343181t_real @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_860_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_nat
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_861_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_Code_integer
        = ( semiri4939895301339042750nteger @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_862_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_complex
        = ( semiri8010041392384452111omplex @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_863_of__nat__1,axiom,
    ( ( semiri8819519690708144855l_num1 @ one_one_nat )
    = one_on7727431528512463931l_num1 ) ).

% of_nat_1
thf(fact_864_of__nat__1,axiom,
    ( ( semiri681578069525770553at_rat @ one_one_nat )
    = one_one_rat ) ).

% of_nat_1
thf(fact_865_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_866_of__nat__1,axiom,
    ( ( semiri5074537144036343181t_real @ one_one_nat )
    = one_one_real ) ).

% of_nat_1
thf(fact_867_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_868_of__nat__1,axiom,
    ( ( semiri4939895301339042750nteger @ one_one_nat )
    = one_one_Code_integer ) ).

% of_nat_1
thf(fact_869_of__nat__1,axiom,
    ( ( semiri8010041392384452111omplex @ one_one_nat )
    = one_one_complex ) ).

% of_nat_1
thf(fact_870_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,N: nat] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N ) ) ).

% numeral_less_real_of_nat_iff
thf(fact_871_real__of__nat__less__numeral__iff,axiom,
    ! [N: nat,W: num] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W ) )
      = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W ) ) ) ).

% real_of_nat_less_numeral_iff
thf(fact_872_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_873_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_874_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_875_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_876_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri4939895301339042750nteger @ M )
        = zero_z3403309356797280102nteger )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_877_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri8010041392384452111omplex @ M )
        = zero_zero_complex )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_878_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_rat
        = ( semiri681578069525770553at_rat @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_879_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_int
        = ( semiri1314217659103216013at_int @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_880_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_real
        = ( semiri5074537144036343181t_real @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_881_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_nat
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_882_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_z3403309356797280102nteger
        = ( semiri4939895301339042750nteger @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_883_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_complex
        = ( semiri8010041392384452111omplex @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_884_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri8819519690708144855l_num1 @ zero_zero_nat )
    = zero_z3563351764282998399l_num1 ) ).

% semiring_1_class.of_nat_0
thf(fact_885_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri681578069525770553at_rat @ zero_zero_nat )
    = zero_zero_rat ) ).

% semiring_1_class.of_nat_0
thf(fact_886_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% semiring_1_class.of_nat_0
thf(fact_887_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri5074537144036343181t_real @ zero_zero_nat )
    = zero_zero_real ) ).

% semiring_1_class.of_nat_0
thf(fact_888_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% semiring_1_class.of_nat_0
thf(fact_889_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri4939895301339042750nteger @ zero_zero_nat )
    = zero_z3403309356797280102nteger ) ).

% semiring_1_class.of_nat_0
thf(fact_890_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri8010041392384452111omplex @ zero_zero_nat )
    = zero_zero_complex ) ).

% semiring_1_class.of_nat_0
thf(fact_891_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_892_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_893_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( semiri5074537144036343181t_real @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_894_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_895_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri4939895301339042750nteger @ M )
        = ( semiri4939895301339042750nteger @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_896_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri8010041392384452111omplex @ M )
        = ( semiri8010041392384452111omplex @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_897_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_898_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_899_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_900_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_901_psubsetD,axiom,
    ! [A: set_real,B: set_real,C: real] :
      ( ( ord_less_set_real @ A @ B )
     => ( ( member_real @ C @ A )
       => ( member_real @ C @ B ) ) ) ).

% psubsetD
thf(fact_902_psubsetD,axiom,
    ! [A: set_nat,B: set_nat,C: nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ( member_nat @ C @ A )
       => ( member_nat @ C @ B ) ) ) ).

% psubsetD
thf(fact_903_psubsetD,axiom,
    ! [A: set_int,B: set_int,C: int] :
      ( ( ord_less_set_int @ A @ B )
     => ( ( member_int @ C @ A )
       => ( member_int @ C @ B ) ) ) ).

% psubsetD
thf(fact_904_psubsetD,axiom,
    ! [A: set_complex,B: set_complex,C: complex] :
      ( ( ord_less_set_complex @ A @ B )
     => ( ( member_complex @ C @ A )
       => ( member_complex @ C @ B ) ) ) ).

% psubsetD
thf(fact_905_less__set__def,axiom,
    ( ord_less_set_real
    = ( ^ [A3: set_real,B3: set_real] :
          ( ord_less_real_o
          @ ^ [X2: real] : ( member_real @ X2 @ A3 )
          @ ^ [X2: real] : ( member_real @ X2 @ B3 ) ) ) ) ).

% less_set_def
thf(fact_906_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ord_less_nat_o
          @ ^ [X2: nat] : ( member_nat @ X2 @ A3 )
          @ ^ [X2: nat] : ( member_nat @ X2 @ B3 ) ) ) ) ).

% less_set_def
thf(fact_907_less__set__def,axiom,
    ( ord_less_set_int
    = ( ^ [A3: set_int,B3: set_int] :
          ( ord_less_int_o
          @ ^ [X2: int] : ( member_int @ X2 @ A3 )
          @ ^ [X2: int] : ( member_int @ X2 @ B3 ) ) ) ) ).

% less_set_def
thf(fact_908_less__set__def,axiom,
    ( ord_less_set_complex
    = ( ^ [A3: set_complex,B3: set_complex] :
          ( ord_less_complex_o
          @ ^ [X2: complex] : ( member_complex @ X2 @ A3 )
          @ ^ [X2: complex] : ( member_complex @ X2 @ B3 ) ) ) ) ).

% less_set_def
thf(fact_909_real__arch__pow,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ? [N2: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N2 ) ) ) ).

% real_arch_pow
thf(fact_910_real__arch__pow__inv,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ X @ one_one_real )
       => ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X @ N2 ) @ Y ) ) ) ).

% real_arch_pow_inv
thf(fact_911_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
thf(fact_912_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_913_int__if,axiom,
    ! [P: $o,A2: nat,B2: nat] :
      ( ( P
       => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B2 ) )
          = ( semiri1314217659103216013at_int @ A2 ) ) )
      & ( ~ P
       => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B2 ) )
          = ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% int_if
thf(fact_914_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
thf(fact_915_nat__int__comparison_I1_J,axiom,
    ( ( ^ [Y3: nat,Z2: nat] : Y3 = Z2 )
    = ( ^ [A4: nat,B4: nat] :
          ( ( semiri1314217659103216013at_int @ A4 )
          = ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_int_comparison(1)
thf(fact_916_int__ops_I1_J,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% int_ops(1)
thf(fact_917_int__ops_I2_J,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% int_ops(2)
thf(fact_918_bot__empty__eq,axiom,
    ( bot_bot_complex_o
    = ( ^ [X2: complex] : ( member_complex @ X2 @ bot_bot_set_complex ) ) ) ).

% bot_empty_eq
thf(fact_919_bot__empty__eq,axiom,
    ( bot_bot_real_o
    = ( ^ [X2: real] : ( member_real @ X2 @ bot_bot_set_real ) ) ) ).

% bot_empty_eq
thf(fact_920_bot__empty__eq,axiom,
    ( bot_bot_o_o
    = ( ^ [X2: $o] : ( member_o @ X2 @ bot_bot_set_o ) ) ) ).

% bot_empty_eq
thf(fact_921_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X2: nat] : ( member_nat @ X2 @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_922_bot__empty__eq,axiom,
    ( bot_bot_int_o
    = ( ^ [X2: int] : ( member_int @ X2 @ bot_bot_set_int ) ) ) ).

% bot_empty_eq
thf(fact_923_bot__set__def,axiom,
    ( bot_bot_set_complex
    = ( collect_complex @ bot_bot_complex_o ) ) ).

% bot_set_def
thf(fact_924_bot__set__def,axiom,
    ( bot_bo1796632182523588997nt_int
    = ( collec213857154873943460nt_int @ bot_bo8147686125503663512_int_o ) ) ).

% bot_set_def
thf(fact_925_bot__set__def,axiom,
    ( bot_bot_set_real
    = ( collect_real @ bot_bot_real_o ) ) ).

% bot_set_def
thf(fact_926_bot__set__def,axiom,
    ( bot_bot_set_o
    = ( collect_o @ bot_bot_o_o ) ) ).

% bot_set_def
thf(fact_927_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_928_bot__set__def,axiom,
    ( bot_bot_set_int
    = ( collect_int @ bot_bot_int_o ) ) ).

% bot_set_def
thf(fact_929_zero__reorient,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( zero_z3563351764282998399l_num1 = X )
      = ( X = zero_z3563351764282998399l_num1 ) ) ).

% zero_reorient
thf(fact_930_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
      = ( X = zero_zero_real ) ) ).

% zero_reorient
thf(fact_931_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
      = ( X = zero_zero_rat ) ) ).

% zero_reorient
thf(fact_932_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_933_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
      = ( X = zero_zero_int ) ) ).

% zero_reorient
thf(fact_934_one__reorient,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( one_on7727431528512463931l_num1 = X )
      = ( X = one_on7727431528512463931l_num1 ) ) ).

% one_reorient
thf(fact_935_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
      = ( X = one_one_real ) ) ).

% one_reorient
thf(fact_936_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
      = ( X = one_one_rat ) ) ).

% one_reorient
thf(fact_937_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_938_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
      = ( X = one_one_int ) ) ).

% one_reorient
thf(fact_939_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_940_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ~ ( P @ N2 )
         => ? [M2: nat] :
              ( ( ord_less_nat @ M2 @ N2 )
              & ~ ( P @ M2 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_941_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N2 )
             => ( P @ M2 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_942_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_943_less__not__refl3,axiom,
    ! [S2: nat,T: nat] :
      ( ( ord_less_nat @ S2 @ T )
     => ( S2 != T ) ) ).

% less_not_refl3
thf(fact_944_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( M != N ) ) ).

% less_not_refl2
thf(fact_945_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_946_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
      = ( ( ord_less_nat @ M @ N )
        | ( ord_less_nat @ N @ M ) ) ) ).

% nat_neq_iff
thf(fact_947_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ~ ( P @ N2 )
             => ? [M2: nat] :
                  ( ( ord_less_nat @ M2 @ N2 )
                  & ~ ( P @ M2 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_948_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_949_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_950_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_951_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_952_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_953_bot__nat__0_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_954_field__lbound__gt__zero,axiom,
    ! [D1: real,D2: real] :
      ( ( ord_less_real @ zero_zero_real @ D1 )
     => ( ( ord_less_real @ zero_zero_real @ D2 )
       => ? [E: real] :
            ( ( ord_less_real @ zero_zero_real @ E )
            & ( ord_less_real @ E @ D1 )
            & ( ord_less_real @ E @ D2 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_955_field__lbound__gt__zero,axiom,
    ! [D1: rat,D2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ D1 )
     => ( ( ord_less_rat @ zero_zero_rat @ D2 )
       => ? [E: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ E )
            & ( ord_less_rat @ E @ D1 )
            & ( ord_less_rat @ E @ D2 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_956_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_957_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_958_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_959_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_960_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).

% of_nat_less_0_iff
thf(fact_961_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_962_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).

% of_nat_less_0_iff
thf(fact_963_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_964_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ zero_z3403309356797280102nteger ) ).

% of_nat_less_0_iff
thf(fact_965_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_966_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_967_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_968_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_969_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_970_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_971_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_972_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_973_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_974_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_975_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_976_pos2,axiom,
    ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).

% pos2
thf(fact_977_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ? [N2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N2 )
          & ( K
            = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_978_pos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ~ ! [N2: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N2 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% pos_int_cases
thf(fact_979_word__unat__power,axiom,
    ! [N: nat] :
      ( ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N )
      = ( semiri8819519690708144855l_num1 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% word_unat_power
thf(fact_980_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri8819519690708144855l_num1 @ K )
       != zero_z3563351764282998399l_num1 )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_981_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_982_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_983_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_984_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_985_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri4939895301339042750nteger @ K )
       != zero_z3403309356797280102nteger )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_986_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_987_realpow__pos__nth__unique,axiom,
    ! [N: nat,A2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A2 )
       => ? [X3: real] :
            ( ( ord_less_real @ zero_zero_real @ X3 )
            & ( ( power_power_real @ X3 @ N )
              = A2 )
            & ! [Y5: real] :
                ( ( ( ord_less_real @ zero_zero_real @ Y5 )
                  & ( ( power_power_real @ Y5 @ N )
                    = A2 ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% realpow_pos_nth_unique
thf(fact_988_realpow__pos__nth,axiom,
    ! [N: nat,A2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A2 )
       => ? [R2: real] :
            ( ( ord_less_real @ zero_zero_real @ R2 )
            & ( ( power_power_real @ R2 @ N )
              = A2 ) ) ) ) ).

% realpow_pos_nth
thf(fact_989_p2__eq__1,axiom,
    ! [N: nat] :
      ( ( ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N )
        = one_on7727431528512463931l_num1 )
      = ( N = zero_zero_nat ) ) ).

% p2_eq_1
thf(fact_990_not__one__less__zero,axiom,
    ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).

% not_one_less_zero
thf(fact_991_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_992_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_993_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_994_word__gt__0__no,axiom,
    ! [Y: num] :
      ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ ( numera7442385471795722001l_num1 @ Y ) )
      = ( zero_z3563351764282998399l_num1
       != ( numera7442385471795722001l_num1 @ Y ) ) ) ).

% word_gt_0_no
thf(fact_995_word__less__1,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ X @ one_on7727431528512463931l_num1 )
      = ( X = zero_z3563351764282998399l_num1 ) ) ).

% word_less_1
thf(fact_996_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_997_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_998_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_999_Abs__fnat__hom__0,axiom,
    ( zero_z3563351764282998399l_num1
    = ( semiri8819519690708144855l_num1 @ zero_zero_nat ) ) ).

% Abs_fnat_hom_0
thf(fact_1000_boolean__algebra__cancel_Osup1,axiom,
    ! [A: set_nat,K: set_nat,A2: set_nat,B2: set_nat] :
      ( ( A
        = ( sup_sup_set_nat @ K @ A2 ) )
     => ( ( sup_sup_set_nat @ A @ B2 )
        = ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_1001_boolean__algebra__cancel_Osup1,axiom,
    ! [A: int,K: int,A2: int,B2: int] :
      ( ( A
        = ( sup_sup_int @ K @ A2 ) )
     => ( ( sup_sup_int @ A @ B2 )
        = ( sup_sup_int @ K @ ( sup_sup_int @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_1002_boolean__algebra__cancel_Osup1,axiom,
    ! [A: nat,K: nat,A2: nat,B2: nat] :
      ( ( A
        = ( sup_sup_nat @ K @ A2 ) )
     => ( ( sup_sup_nat @ A @ B2 )
        = ( sup_sup_nat @ K @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_1003_boolean__algebra__cancel_Osup1,axiom,
    ! [A: extended_enat,K: extended_enat,A2: extended_enat,B2: extended_enat] :
      ( ( A
        = ( sup_su3973961784419623482d_enat @ K @ A2 ) )
     => ( ( sup_su3973961784419623482d_enat @ A @ B2 )
        = ( sup_su3973961784419623482d_enat @ K @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_1004_boolean__algebra__cancel_Osup1,axiom,
    ! [A: assn,K: assn,A2: assn,B2: assn] :
      ( ( A
        = ( sup_sup_assn @ K @ A2 ) )
     => ( ( sup_sup_assn @ A @ B2 )
        = ( sup_sup_assn @ K @ ( sup_sup_assn @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_1005_boolean__algebra__cancel_Osup1,axiom,
    ! [A: set_complex,K: set_complex,A2: set_complex,B2: set_complex] :
      ( ( A
        = ( sup_sup_set_complex @ K @ A2 ) )
     => ( ( sup_sup_set_complex @ A @ B2 )
        = ( sup_sup_set_complex @ K @ ( sup_sup_set_complex @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_1006_boolean__algebra__cancel_Osup1,axiom,
    ! [A: set_int,K: set_int,A2: set_int,B2: set_int] :
      ( ( A
        = ( sup_sup_set_int @ K @ A2 ) )
     => ( ( sup_sup_set_int @ A @ B2 )
        = ( sup_sup_set_int @ K @ ( sup_sup_set_int @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_1007_boolean__algebra__cancel_Osup1,axiom,
    ! [A: set_real,K: set_real,A2: set_real,B2: set_real] :
      ( ( A
        = ( sup_sup_set_real @ K @ A2 ) )
     => ( ( sup_sup_set_real @ A @ B2 )
        = ( sup_sup_set_real @ K @ ( sup_sup_set_real @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_1008_boolean__algebra__cancel_Osup2,axiom,
    ! [B: set_nat,K: set_nat,B2: set_nat,A2: set_nat] :
      ( ( B
        = ( sup_sup_set_nat @ K @ B2 ) )
     => ( ( sup_sup_set_nat @ A2 @ B )
        = ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_1009_boolean__algebra__cancel_Osup2,axiom,
    ! [B: int,K: int,B2: int,A2: int] :
      ( ( B
        = ( sup_sup_int @ K @ B2 ) )
     => ( ( sup_sup_int @ A2 @ B )
        = ( sup_sup_int @ K @ ( sup_sup_int @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_1010_boolean__algebra__cancel_Osup2,axiom,
    ! [B: nat,K: nat,B2: nat,A2: nat] :
      ( ( B
        = ( sup_sup_nat @ K @ B2 ) )
     => ( ( sup_sup_nat @ A2 @ B )
        = ( sup_sup_nat @ K @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_1011_boolean__algebra__cancel_Osup2,axiom,
    ! [B: extended_enat,K: extended_enat,B2: extended_enat,A2: extended_enat] :
      ( ( B
        = ( sup_su3973961784419623482d_enat @ K @ B2 ) )
     => ( ( sup_su3973961784419623482d_enat @ A2 @ B )
        = ( sup_su3973961784419623482d_enat @ K @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_1012_boolean__algebra__cancel_Osup2,axiom,
    ! [B: assn,K: assn,B2: assn,A2: assn] :
      ( ( B
        = ( sup_sup_assn @ K @ B2 ) )
     => ( ( sup_sup_assn @ A2 @ B )
        = ( sup_sup_assn @ K @ ( sup_sup_assn @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_1013_boolean__algebra__cancel_Osup2,axiom,
    ! [B: set_complex,K: set_complex,B2: set_complex,A2: set_complex] :
      ( ( B
        = ( sup_sup_set_complex @ K @ B2 ) )
     => ( ( sup_sup_set_complex @ A2 @ B )
        = ( sup_sup_set_complex @ K @ ( sup_sup_set_complex @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_1014_boolean__algebra__cancel_Osup2,axiom,
    ! [B: set_int,K: set_int,B2: set_int,A2: set_int] :
      ( ( B
        = ( sup_sup_set_int @ K @ B2 ) )
     => ( ( sup_sup_set_int @ A2 @ B )
        = ( sup_sup_set_int @ K @ ( sup_sup_set_int @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_1015_boolean__algebra__cancel_Osup2,axiom,
    ! [B: set_real,K: set_real,B2: set_real,A2: set_real] :
      ( ( B
        = ( sup_sup_set_real @ K @ B2 ) )
     => ( ( sup_sup_set_real @ A2 @ B )
        = ( sup_sup_set_real @ K @ ( sup_sup_set_real @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_1016_less__int__code_I1_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_int_code(1)
thf(fact_1017_int__int__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M = N ) ) ).

% int_int_eq
thf(fact_1018_zero__neq__one,axiom,
    zero_z3563351764282998399l_num1 != one_on7727431528512463931l_num1 ).

% zero_neq_one
thf(fact_1019_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
thf(fact_1020_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
thf(fact_1021_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_1022_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_1023_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ bot_bot_assn )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_1024_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_complex] :
      ( ( sup_sup_set_complex @ X @ bot_bot_set_complex )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_1025_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_real] :
      ( ( sup_sup_set_real @ X @ bot_bot_set_real )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_1026_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ X @ bot_bot_set_o )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_1027_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_1028_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_int] :
      ( ( sup_sup_set_int @ X @ bot_bot_set_int )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_1029_zero__less__one,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% zero_less_one
thf(fact_1030_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_1031_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_1032_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_1033_false__rule,axiom,
    ! [C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] : ( hoare_1429296392585015714_VEBTi @ bot_bot_assn @ C @ Q ) ).

% false_rule
thf(fact_1034_false__rule,axiom,
    ! [C: heap_Time_Heap_o,Q: $o > assn] : ( hoare_hoare_triple_o @ bot_bot_assn @ C @ Q ) ).

% false_rule
thf(fact_1035_the__elem__eq,axiom,
    ! [X: real] :
      ( ( the_elem_real @ ( insert_real @ X @ bot_bot_set_real ) )
      = X ) ).

% the_elem_eq
thf(fact_1036_the__elem__eq,axiom,
    ! [X: $o] :
      ( ( the_elem_o @ ( insert_o @ X @ bot_bot_set_o ) )
      = X ) ).

% the_elem_eq
thf(fact_1037_the__elem__eq,axiom,
    ! [X: nat] :
      ( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
      = X ) ).

% the_elem_eq
thf(fact_1038_the__elem__eq,axiom,
    ! [X: int] :
      ( ( the_elem_int @ ( insert_int @ X @ bot_bot_set_int ) )
      = X ) ).

% the_elem_eq
thf(fact_1039_Collect__empty__eq__bot,axiom,
    ! [P: complex > $o] :
      ( ( ( collect_complex @ P )
        = bot_bot_set_complex )
      = ( P = bot_bot_complex_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1040_Collect__empty__eq__bot,axiom,
    ! [P: product_prod_int_int > $o] :
      ( ( ( collec213857154873943460nt_int @ P )
        = bot_bo1796632182523588997nt_int )
      = ( P = bot_bo8147686125503663512_int_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1041_Collect__empty__eq__bot,axiom,
    ! [P: real > $o] :
      ( ( ( collect_real @ P )
        = bot_bot_set_real )
      = ( P = bot_bot_real_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1042_Collect__empty__eq__bot,axiom,
    ! [P: $o > $o] :
      ( ( ( collect_o @ P )
        = bot_bot_set_o )
      = ( P = bot_bot_o_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1043_Collect__empty__eq__bot,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( P = bot_bot_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1044_Collect__empty__eq__bot,axiom,
    ! [P: int > $o] :
      ( ( ( collect_int @ P )
        = bot_bot_set_int )
      = ( P = bot_bot_int_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1045_vebt__buildupi__rule__basic,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( hoare_1429296392585015714_VEBTi @ one_one_assn @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_Intf_vebt_assn @ N @ bot_bot_set_nat ) ) ) ).

% vebt_buildupi_rule_basic
thf(fact_1046_is__singletonI,axiom,
    ! [X: real] : ( is_singleton_real @ ( insert_real @ X @ bot_bot_set_real ) ) ).

% is_singletonI
thf(fact_1047_is__singletonI,axiom,
    ! [X: $o] : ( is_singleton_o @ ( insert_o @ X @ bot_bot_set_o ) ) ).

% is_singletonI
thf(fact_1048_is__singletonI,axiom,
    ! [X: nat] : ( is_singleton_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) ).

% is_singletonI
thf(fact_1049_is__singletonI,axiom,
    ! [X: int] : ( is_singleton_int @ ( insert_int @ X @ bot_bot_set_int ) ) ).

% is_singletonI
thf(fact_1050_forall__finite_I1_J,axiom,
    ! [P: nat > $o,I2: nat] :
      ( ( ord_less_nat @ I2 @ zero_zero_nat )
     => ( P @ I2 ) ) ).

% forall_finite(1)
thf(fact_1051_reals__Archimedean2,axiom,
    ! [X: rat] :
    ? [N2: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N2 ) ) ).

% reals_Archimedean2
thf(fact_1052_reals__Archimedean2,axiom,
    ! [X: real] :
    ? [N2: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).

% reals_Archimedean2
thf(fact_1053_bot_Oextremum__strict,axiom,
    ! [A2: set_real] :
      ~ ( ord_less_set_real @ A2 @ bot_bot_set_real ) ).

% bot.extremum_strict
thf(fact_1054_bot_Oextremum__strict,axiom,
    ! [A2: set_o] :
      ~ ( ord_less_set_o @ A2 @ bot_bot_set_o ) ).

% bot.extremum_strict
thf(fact_1055_bot_Oextremum__strict,axiom,
    ! [A2: set_nat] :
      ~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).

% bot.extremum_strict
thf(fact_1056_bot_Oextremum__strict,axiom,
    ! [A2: set_int] :
      ~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).

% bot.extremum_strict
thf(fact_1057_bot_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_1058_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_1059_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_1060_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_1061_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_1062_bot_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_1063_word__not__simps_I1_J,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ~ ( ord_le750835935415966154l_num1 @ X @ zero_z3563351764282998399l_num1 ) ).

% word_not_simps(1)
thf(fact_1064_word__coorder_Oextremum__strict,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ~ ( ord_le750835935415966154l_num1 @ A2 @ zero_z3563351764282998399l_num1 ) ).

% word_coorder.extremum_strict
thf(fact_1065_word__gt__0,axiom,
    ! [Y: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ Y )
      = ( zero_z3563351764282998399l_num1 != Y ) ) ).

% word_gt_0
thf(fact_1066_word__neq__0__conv,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( W != zero_z3563351764282998399l_num1 )
      = ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ W ) ) ).

% word_neq_0_conv
thf(fact_1067_word__gt__a__gt__0,axiom,
    ! [A2: word_N3645301735248828278l_num1,N: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ A2 @ N )
     => ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ N ) ) ).

% word_gt_a_gt_0
thf(fact_1068_word__greater__zero__iff,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ A2 )
      = ( A2 != zero_z3563351764282998399l_num1 ) ) ).

% word_greater_zero_iff
thf(fact_1069_is__singleton__the__elem,axiom,
    ( is_singleton_real
    = ( ^ [A3: set_real] :
          ( A3
          = ( insert_real @ ( the_elem_real @ A3 ) @ bot_bot_set_real ) ) ) ) ).

% is_singleton_the_elem
thf(fact_1070_is__singleton__the__elem,axiom,
    ( is_singleton_o
    = ( ^ [A3: set_o] :
          ( A3
          = ( insert_o @ ( the_elem_o @ A3 ) @ bot_bot_set_o ) ) ) ) ).

% is_singleton_the_elem
thf(fact_1071_is__singleton__the__elem,axiom,
    ( is_singleton_nat
    = ( ^ [A3: set_nat] :
          ( A3
          = ( insert_nat @ ( the_elem_nat @ A3 ) @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_the_elem
thf(fact_1072_is__singleton__the__elem,axiom,
    ( is_singleton_int
    = ( ^ [A3: set_int] :
          ( A3
          = ( insert_int @ ( the_elem_int @ A3 ) @ bot_bot_set_int ) ) ) ) ).

% is_singleton_the_elem
thf(fact_1073_is__singletonI_H,axiom,
    ! [A: set_complex] :
      ( ( A != bot_bot_set_complex )
     => ( ! [X3: complex,Y4: complex] :
            ( ( member_complex @ X3 @ A )
           => ( ( member_complex @ Y4 @ A )
             => ( X3 = Y4 ) ) )
       => ( is_singleton_complex @ A ) ) ) ).

% is_singletonI'
thf(fact_1074_is__singletonI_H,axiom,
    ! [A: set_real] :
      ( ( A != bot_bot_set_real )
     => ( ! [X3: real,Y4: real] :
            ( ( member_real @ X3 @ A )
           => ( ( member_real @ Y4 @ A )
             => ( X3 = Y4 ) ) )
       => ( is_singleton_real @ A ) ) ) ).

% is_singletonI'
thf(fact_1075_is__singletonI_H,axiom,
    ! [A: set_o] :
      ( ( A != bot_bot_set_o )
     => ( ! [X3: $o,Y4: $o] :
            ( ( member_o @ X3 @ A )
           => ( ( member_o @ Y4 @ A )
             => ( X3 = Y4 ) ) )
       => ( is_singleton_o @ A ) ) ) ).

% is_singletonI'
thf(fact_1076_is__singletonI_H,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
     => ( ! [X3: nat,Y4: nat] :
            ( ( member_nat @ X3 @ A )
           => ( ( member_nat @ Y4 @ A )
             => ( X3 = Y4 ) ) )
       => ( is_singleton_nat @ A ) ) ) ).

% is_singletonI'
thf(fact_1077_is__singletonI_H,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
     => ( ! [X3: int,Y4: int] :
            ( ( member_int @ X3 @ A )
           => ( ( member_int @ Y4 @ A )
             => ( X3 = Y4 ) ) )
       => ( is_singleton_int @ A ) ) ) ).

% is_singletonI'
thf(fact_1078_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_1079_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_1080_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_1081_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_1082_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_1083_order__less__imp__not__eq2,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_1084_order__less__imp__not__eq2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_1085_order__less__imp__not__eq2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_1086_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_1087_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_1088_order__less__imp__not__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1089_order__less__imp__not__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1090_order__less__imp__not__eq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1091_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1092_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1093_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_1094_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_1095_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_1096_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_1097_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_1098_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_1099_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_1100_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_1101_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_1102_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_1103_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_1104_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_1105_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_1106_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_1107_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_1108_order__less__subst2,axiom,
    ! [A2: real,B2: real,F: real > real,C: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_real @ ( F @ B2 ) @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1109_order__less__subst2,axiom,
    ! [A2: real,B2: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_rat @ ( F @ B2 ) @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1110_order__less__subst2,axiom,
    ! [A2: real,B2: real,F: real > num,C: num] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_num @ ( F @ B2 ) @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1111_order__less__subst2,axiom,
    ! [A2: real,B2: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1112_order__less__subst2,axiom,
    ! [A2: real,B2: real,F: real > int,C: int] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_int @ ( F @ B2 ) @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1113_order__less__subst2,axiom,
    ! [A2: rat,B2: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_real @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1114_order__less__subst2,axiom,
    ! [A2: rat,B2: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1115_order__less__subst2,axiom,
    ! [A2: rat,B2: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_num @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1116_order__less__subst2,axiom,
    ! [A2: rat,B2: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1117_order__less__subst2,axiom,
    ! [A2: rat,B2: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_int @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1118_order__less__subst1,axiom,
    ! [A2: real,F: real > real,B2: real,C: real] :
      ( ( ord_less_real @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1119_order__less__subst1,axiom,
    ! [A2: real,F: rat > real,B2: rat,C: rat] :
      ( ( ord_less_real @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1120_order__less__subst1,axiom,
    ! [A2: real,F: num > real,B2: num,C: num] :
      ( ( ord_less_real @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_num @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1121_order__less__subst1,axiom,
    ! [A2: real,F: nat > real,B2: nat,C: nat] :
      ( ( ord_less_real @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1122_order__less__subst1,axiom,
    ! [A2: real,F: int > real,B2: int,C: int] :
      ( ( ord_less_real @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_int @ B2 @ C )
       => ( ! [X3: int,Y4: int] :
              ( ( ord_less_int @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1123_order__less__subst1,axiom,
    ! [A2: rat,F: real > rat,B2: real,C: real] :
      ( ( ord_less_rat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1124_order__less__subst1,axiom,
    ! [A2: rat,F: rat > rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1125_order__less__subst1,axiom,
    ! [A2: rat,F: num > rat,B2: num,C: num] :
      ( ( ord_less_rat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_num @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1126_order__less__subst1,axiom,
    ! [A2: rat,F: nat > rat,B2: nat,C: nat] :
      ( ( ord_less_rat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1127_order__less__subst1,axiom,
    ! [A2: rat,F: int > rat,B2: int,C: int] :
      ( ( ord_less_rat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_int @ B2 @ C )
       => ( ! [X3: int,Y4: int] :
              ( ( ord_less_int @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1128_order__less__irrefl,axiom,
    ! [X: real] :
      ~ ( ord_less_real @ X @ X ) ).

% order_less_irrefl
thf(fact_1129_order__less__irrefl,axiom,
    ! [X: rat] :
      ~ ( ord_less_rat @ X @ X ) ).

% order_less_irrefl
thf(fact_1130_order__less__irrefl,axiom,
    ! [X: num] :
      ~ ( ord_less_num @ X @ X ) ).

% order_less_irrefl
thf(fact_1131_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_1132_order__less__irrefl,axiom,
    ! [X: int] :
      ~ ( ord_less_int @ X @ X ) ).

% order_less_irrefl
thf(fact_1133_ord__less__eq__subst,axiom,
    ! [A2: real,B2: real,F: real > real,C: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1134_ord__less__eq__subst,axiom,
    ! [A2: real,B2: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1135_ord__less__eq__subst,axiom,
    ! [A2: real,B2: real,F: real > num,C: num] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1136_ord__less__eq__subst,axiom,
    ! [A2: real,B2: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1137_ord__less__eq__subst,axiom,
    ! [A2: real,B2: real,F: real > int,C: int] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1138_ord__less__eq__subst,axiom,
    ! [A2: rat,B2: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1139_ord__less__eq__subst,axiom,
    ! [A2: rat,B2: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1140_ord__less__eq__subst,axiom,
    ! [A2: rat,B2: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1141_ord__less__eq__subst,axiom,
    ! [A2: rat,B2: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1142_ord__less__eq__subst,axiom,
    ! [A2: rat,B2: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1143_ord__eq__less__subst,axiom,
    ! [A2: real,F: real > real,B2: real,C: real] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1144_ord__eq__less__subst,axiom,
    ! [A2: rat,F: real > rat,B2: real,C: real] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1145_ord__eq__less__subst,axiom,
    ! [A2: num,F: real > num,B2: real,C: real] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1146_ord__eq__less__subst,axiom,
    ! [A2: nat,F: real > nat,B2: real,C: real] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1147_ord__eq__less__subst,axiom,
    ! [A2: int,F: real > int,B2: real,C: real] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1148_ord__eq__less__subst,axiom,
    ! [A2: real,F: rat > real,B2: rat,C: rat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1149_ord__eq__less__subst,axiom,
    ! [A2: rat,F: rat > rat,B2: rat,C: rat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1150_ord__eq__less__subst,axiom,
    ! [A2: num,F: rat > num,B2: rat,C: rat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1151_ord__eq__less__subst,axiom,
    ! [A2: nat,F: rat > nat,B2: rat,C: rat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1152_ord__eq__less__subst,axiom,
    ! [A2: int,F: rat > int,B2: rat,C: rat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1153_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_1154_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_1155_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_1156_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_1157_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_1158_order__less__asym_H,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ~ ( ord_less_real @ B2 @ A2 ) ) ).

% order_less_asym'
thf(fact_1159_order__less__asym_H,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ~ ( ord_less_rat @ B2 @ A2 ) ) ).

% order_less_asym'
thf(fact_1160_order__less__asym_H,axiom,
    ! [A2: num,B2: num] :
      ( ( ord_less_num @ A2 @ B2 )
     => ~ ( ord_less_num @ B2 @ A2 ) ) ).

% order_less_asym'
thf(fact_1161_order__less__asym_H,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ~ ( ord_less_nat @ B2 @ A2 ) ) ).

% order_less_asym'
thf(fact_1162_order__less__asym_H,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ~ ( ord_less_int @ B2 @ A2 ) ) ).

% order_less_asym'
thf(fact_1163_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_1164_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_1165_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_1166_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_1167_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_1168_order__less__asym,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_asym
thf(fact_1169_order__less__asym,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ~ ( ord_less_rat @ Y @ X ) ) ).

% order_less_asym
thf(fact_1170_order__less__asym,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ~ ( ord_less_num @ Y @ X ) ) ).

% order_less_asym
thf(fact_1171_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_1172_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_asym
thf(fact_1173_linorder__neqE,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ ( ord_less_real @ X @ Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_1174_linorder__neqE,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ ( ord_less_rat @ X @ Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_1175_linorder__neqE,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
     => ( ~ ( ord_less_num @ X @ Y )
       => ( ord_less_num @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_1176_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_1177_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_1178_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_1179_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_1180_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_1181_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_1182_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_1183_order_Ostrict__implies__not__eq,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_1184_order_Ostrict__implies__not__eq,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_1185_order_Ostrict__implies__not__eq,axiom,
    ! [A2: num,B2: num] :
      ( ( ord_less_num @ A2 @ B2 )
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_1186_order_Ostrict__implies__not__eq,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_1187_order_Ostrict__implies__not__eq,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_1188_dual__order_Ostrict__trans,axiom,
    ! [B2: real,A2: real,C: real] :
      ( ( ord_less_real @ B2 @ A2 )
     => ( ( ord_less_real @ C @ B2 )
       => ( ord_less_real @ C @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_1189_dual__order_Ostrict__trans,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ( ( ord_less_rat @ C @ B2 )
       => ( ord_less_rat @ C @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_1190_dual__order_Ostrict__trans,axiom,
    ! [B2: num,A2: num,C: num] :
      ( ( ord_less_num @ B2 @ A2 )
     => ( ( ord_less_num @ C @ B2 )
       => ( ord_less_num @ C @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_1191_dual__order_Ostrict__trans,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( ord_less_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_1192_dual__order_Ostrict__trans,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( ord_less_int @ C @ B2 )
       => ( ord_less_int @ C @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_1193_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_1194_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_1195_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_1196_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_1197_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_1198_order_Ostrict__trans,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_real @ B2 @ C )
       => ( ord_less_real @ A2 @ C ) ) ) ).

% order.strict_trans
thf(fact_1199_order_Ostrict__trans,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ord_less_rat @ A2 @ C ) ) ) ).

% order.strict_trans
thf(fact_1200_order_Ostrict__trans,axiom,
    ! [A2: num,B2: num,C: num] :
      ( ( ord_less_num @ A2 @ B2 )
     => ( ( ord_less_num @ B2 @ C )
       => ( ord_less_num @ A2 @ C ) ) ) ).

% order.strict_trans
thf(fact_1201_order_Ostrict__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% order.strict_trans
thf(fact_1202_order_Ostrict__trans,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ B2 @ C )
       => ( ord_less_int @ A2 @ C ) ) ) ).

% order.strict_trans
thf(fact_1203_linorder__less__wlog,axiom,
    ! [P: real > real > $o,A2: real,B2: real] :
      ( ! [A5: real,B6: real] :
          ( ( ord_less_real @ A5 @ B6 )
         => ( P @ A5 @ B6 ) )
     => ( ! [A5: real] : ( P @ A5 @ A5 )
       => ( ! [A5: real,B6: real] :
              ( ( P @ B6 @ A5 )
             => ( P @ A5 @ B6 ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_1204_linorder__less__wlog,axiom,
    ! [P: rat > rat > $o,A2: rat,B2: rat] :
      ( ! [A5: rat,B6: rat] :
          ( ( ord_less_rat @ A5 @ B6 )
         => ( P @ A5 @ B6 ) )
     => ( ! [A5: rat] : ( P @ A5 @ A5 )
       => ( ! [A5: rat,B6: rat] :
              ( ( P @ B6 @ A5 )
             => ( P @ A5 @ B6 ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_1205_linorder__less__wlog,axiom,
    ! [P: num > num > $o,A2: num,B2: num] :
      ( ! [A5: num,B6: num] :
          ( ( ord_less_num @ A5 @ B6 )
         => ( P @ A5 @ B6 ) )
     => ( ! [A5: num] : ( P @ A5 @ A5 )
       => ( ! [A5: num,B6: num] :
              ( ( P @ B6 @ A5 )
             => ( P @ A5 @ B6 ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_1206_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A2: nat,B2: nat] :
      ( ! [A5: nat,B6: nat] :
          ( ( ord_less_nat @ A5 @ B6 )
         => ( P @ A5 @ B6 ) )
     => ( ! [A5: nat] : ( P @ A5 @ A5 )
       => ( ! [A5: nat,B6: nat] :
              ( ( P @ B6 @ A5 )
             => ( P @ A5 @ B6 ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_1207_linorder__less__wlog,axiom,
    ! [P: int > int > $o,A2: int,B2: int] :
      ( ! [A5: int,B6: int] :
          ( ( ord_less_int @ A5 @ B6 )
         => ( P @ A5 @ B6 ) )
     => ( ! [A5: int] : ( P @ A5 @ A5 )
       => ( ! [A5: int,B6: int] :
              ( ( P @ B6 @ A5 )
             => ( P @ A5 @ B6 ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_1208_exists__least__iff,axiom,
    ( ( ^ [P2: nat > $o] :
        ? [X4: nat] : ( P2 @ X4 ) )
    = ( ^ [P3: nat > $o] :
        ? [N4: nat] :
          ( ( P3 @ N4 )
          & ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N4 )
             => ~ ( P3 @ M3 ) ) ) ) ) ).

% exists_least_iff
thf(fact_1209_dual__order_Oirrefl,axiom,
    ! [A2: real] :
      ~ ( ord_less_real @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_1210_dual__order_Oirrefl,axiom,
    ! [A2: rat] :
      ~ ( ord_less_rat @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_1211_dual__order_Oirrefl,axiom,
    ! [A2: num] :
      ~ ( ord_less_num @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_1212_dual__order_Oirrefl,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_1213_dual__order_Oirrefl,axiom,
    ! [A2: int] :
      ~ ( ord_less_int @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_1214_dual__order_Oasym,axiom,
    ! [B2: real,A2: real] :
      ( ( ord_less_real @ B2 @ A2 )
     => ~ ( ord_less_real @ A2 @ B2 ) ) ).

% dual_order.asym
thf(fact_1215_dual__order_Oasym,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ~ ( ord_less_rat @ A2 @ B2 ) ) ).

% dual_order.asym
thf(fact_1216_dual__order_Oasym,axiom,
    ! [B2: num,A2: num] :
      ( ( ord_less_num @ B2 @ A2 )
     => ~ ( ord_less_num @ A2 @ B2 ) ) ).

% dual_order.asym
thf(fact_1217_dual__order_Oasym,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ~ ( ord_less_nat @ A2 @ B2 ) ) ).

% dual_order.asym
thf(fact_1218_dual__order_Oasym,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ~ ( ord_less_int @ A2 @ B2 ) ) ).

% dual_order.asym
thf(fact_1219_linorder__cases,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_1220_linorder__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_1221_linorder__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_num @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_1222_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_1223_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_1224_antisym__conv3,axiom,
    ! [Y: real,X: real] :
      ( ~ ( ord_less_real @ Y @ X )
     => ( ( ~ ( ord_less_real @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_1225_antisym__conv3,axiom,
    ! [Y: rat,X: rat] :
      ( ~ ( ord_less_rat @ Y @ X )
     => ( ( ~ ( ord_less_rat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_1226_antisym__conv3,axiom,
    ! [Y: num,X: num] :
      ( ~ ( ord_less_num @ Y @ X )
     => ( ( ~ ( ord_less_num @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_1227_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_1228_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_int @ Y @ X )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_1229_less__induct,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ! [X3: nat] :
          ( ! [Y5: nat] :
              ( ( ord_less_nat @ Y5 @ X3 )
             => ( P @ Y5 ) )
         => ( P @ X3 ) )
     => ( P @ A2 ) ) ).

% less_induct
thf(fact_1230_ord__less__eq__trans,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_real @ A2 @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1231_ord__less__eq__trans,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_rat @ A2 @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1232_ord__less__eq__trans,axiom,
    ! [A2: num,B2: num,C: num] :
      ( ( ord_less_num @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_num @ A2 @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1233_ord__less__eq__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1234_ord__less__eq__trans,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_int @ A2 @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_1235_ord__eq__less__trans,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( A2 = B2 )
     => ( ( ord_less_real @ B2 @ C )
       => ( ord_less_real @ A2 @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1236_ord__eq__less__trans,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( A2 = B2 )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ord_less_rat @ A2 @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1237_ord__eq__less__trans,axiom,
    ! [A2: num,B2: num,C: num] :
      ( ( A2 = B2 )
     => ( ( ord_less_num @ B2 @ C )
       => ( ord_less_num @ A2 @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1238_ord__eq__less__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( A2 = B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1239_ord__eq__less__trans,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( A2 = B2 )
     => ( ( ord_less_int @ B2 @ C )
       => ( ord_less_int @ A2 @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_1240_order_Oasym,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ~ ( ord_less_real @ B2 @ A2 ) ) ).

% order.asym
thf(fact_1241_order_Oasym,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ~ ( ord_less_rat @ B2 @ A2 ) ) ).

% order.asym
thf(fact_1242_order_Oasym,axiom,
    ! [A2: num,B2: num] :
      ( ( ord_less_num @ A2 @ B2 )
     => ~ ( ord_less_num @ B2 @ A2 ) ) ).

% order.asym
thf(fact_1243_order_Oasym,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ~ ( ord_less_nat @ B2 @ A2 ) ) ).

% order.asym
thf(fact_1244_order_Oasym,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ~ ( ord_less_int @ B2 @ A2 ) ) ).

% order.asym
thf(fact_1245_less__imp__neq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_1246_less__imp__neq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_1247_less__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_1248_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_1249_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_1250_dense,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ? [Z3: real] :
          ( ( ord_less_real @ X @ Z3 )
          & ( ord_less_real @ Z3 @ Y ) ) ) ).

% dense
thf(fact_1251_dense,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ? [Z3: rat] :
          ( ( ord_less_rat @ X @ Z3 )
          & ( ord_less_rat @ Z3 @ Y ) ) ) ).

% dense
thf(fact_1252_gt__ex,axiom,
    ! [X: real] :
    ? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).

% gt_ex
thf(fact_1253_gt__ex,axiom,
    ! [X: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X @ X_1 ) ).

% gt_ex
thf(fact_1254_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).

% gt_ex
thf(fact_1255_gt__ex,axiom,
    ! [X: int] :
    ? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).

% gt_ex
thf(fact_1256_lt__ex,axiom,
    ! [X: real] :
    ? [Y4: real] : ( ord_less_real @ Y4 @ X ) ).

% lt_ex
thf(fact_1257_lt__ex,axiom,
    ! [X: rat] :
    ? [Y4: rat] : ( ord_less_rat @ Y4 @ X ) ).

% lt_ex
thf(fact_1258_lt__ex,axiom,
    ! [X: int] :
    ? [Y4: int] : ( ord_less_int @ Y4 @ X ) ).

% lt_ex
thf(fact_1259_split__rule,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,R: vEBT_VEBTi > assn,Q: assn] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ C @ R )
     => ( ( hoare_1429296392585015714_VEBTi @ Q @ C @ R )
       => ( hoare_1429296392585015714_VEBTi @ ( sup_sup_assn @ P @ Q ) @ C @ R ) ) ) ).

% split_rule
thf(fact_1260_split__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_o,R: $o > assn,Q: assn] :
      ( ( hoare_hoare_triple_o @ P @ C @ R )
     => ( ( hoare_hoare_triple_o @ Q @ C @ R )
       => ( hoare_hoare_triple_o @ ( sup_sup_assn @ P @ Q ) @ C @ R ) ) ) ).

% split_rule
thf(fact_1261_if__rule,axiom,
    ! [B2: $o,P: assn,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,G: heap_T8145700208782473153_VEBTi] :
      ( ( B2
       => ( hoare_1429296392585015714_VEBTi @ P @ F @ Q ) )
     => ( ( ~ B2
         => ( hoare_1429296392585015714_VEBTi @ P @ G @ Q ) )
       => ( hoare_1429296392585015714_VEBTi @ P @ ( if_Hea8453224502484754311_VEBTi @ B2 @ F @ G ) @ Q ) ) ) ).

% if_rule
thf(fact_1262_if__rule,axiom,
    ! [B2: $o,P: assn,F: heap_Time_Heap_o,Q: $o > assn,G: heap_Time_Heap_o] :
      ( ( B2
       => ( hoare_hoare_triple_o @ P @ F @ Q ) )
     => ( ( ~ B2
         => ( hoare_hoare_triple_o @ P @ G @ Q ) )
       => ( hoare_hoare_triple_o @ P @ ( if_Heap_Time_Heap_o @ B2 @ F @ G ) @ Q ) ) ) ).

% if_rule
thf(fact_1263_is__singleton__def,axiom,
    ( is_singleton_real
    = ( ^ [A3: set_real] :
        ? [X2: real] :
          ( A3
          = ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ).

% is_singleton_def
thf(fact_1264_is__singleton__def,axiom,
    ( is_singleton_o
    = ( ^ [A3: set_o] :
        ? [X2: $o] :
          ( A3
          = ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).

% is_singleton_def
thf(fact_1265_is__singleton__def,axiom,
    ( is_singleton_nat
    = ( ^ [A3: set_nat] :
        ? [X2: nat] :
          ( A3
          = ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_def
thf(fact_1266_is__singleton__def,axiom,
    ( is_singleton_int
    = ( ^ [A3: set_int] :
        ? [X2: int] :
          ( A3
          = ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ).

% is_singleton_def
thf(fact_1267_is__singletonE,axiom,
    ! [A: set_real] :
      ( ( is_singleton_real @ A )
     => ~ ! [X3: real] :
            ( A
           != ( insert_real @ X3 @ bot_bot_set_real ) ) ) ).

% is_singletonE
thf(fact_1268_is__singletonE,axiom,
    ! [A: set_o] :
      ( ( is_singleton_o @ A )
     => ~ ! [X3: $o] :
            ( A
           != ( insert_o @ X3 @ bot_bot_set_o ) ) ) ).

% is_singletonE
thf(fact_1269_is__singletonE,axiom,
    ! [A: set_nat] :
      ( ( is_singleton_nat @ A )
     => ~ ! [X3: nat] :
            ( A
           != ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).

% is_singletonE
thf(fact_1270_is__singletonE,axiom,
    ! [A: set_int] :
      ( ( is_singleton_int @ A )
     => ~ ! [X3: int] :
            ( A
           != ( insert_int @ X3 @ bot_bot_set_int ) ) ) ).

% is_singletonE
thf(fact_1271_assn__basic__inequalities_I3_J,axiom,
    bot_bot_assn != one_one_assn ).

% assn_basic_inequalities(3)
thf(fact_1272_uint__2p,axiom,
    ! [N: nat] :
      ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N ) )
     => ( ( semiri7338730514057886004m1_int @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N ) )
        = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% uint_2p
thf(fact_1273_vebt__heap__rules_I1_J,axiom,
    ! [N: nat] : ( hoare_1429296392585015714_VEBTi @ ( pure_assn @ ( ord_less_nat @ zero_zero_nat @ N ) ) @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_Intf_vebt_assn @ N @ bot_bot_set_nat ) ) ).

% vebt_heap_rules(1)
thf(fact_1274_dbl__simps_I4_J,axiom,
    ( ( neg_nu7865238048354675525l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
    = ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_1275_dbl__simps_I4_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_1276_dbl__simps_I4_J,axiom,
    ( ( neg_nu5314729912787363643uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) )
    = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_1277_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_1278_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_1279_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_1280_log2__of__power__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log2_of_power_less
thf(fact_1281_shiftl__1,axiom,
    ! [N: nat] :
      ( ( bit_Sh7051673377389942294nteger @ one_one_Code_integer @ N )
      = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ).

% shiftl_1
thf(fact_1282_shiftl__1,axiom,
    ! [N: nat] :
      ( ( bit_Sh9074413540854191407l_num1 @ one_on7727431528512463931l_num1 @ N )
      = ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N ) ) ).

% shiftl_1
thf(fact_1283_shiftl__1,axiom,
    ! [N: nat] :
      ( ( bit_Sh3963086678839698405tl_int @ one_one_int @ N )
      = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% shiftl_1
thf(fact_1284_shiftl__1,axiom,
    ! [N: nat] :
      ( ( bit_Sh3965577149348748681tl_nat @ one_one_nat @ N )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% shiftl_1
thf(fact_1285_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_1286_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_1287_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_1288_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_1289_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_1290_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_1291_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_1292_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_1293_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_1294_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8115118780965096967l_num1 @ zero_z3563351764282998399l_num1 )
    = one_on7727431528512463931l_num1 ) ).

% dbl_inc_simps(2)
thf(fact_1295_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
    = one_one_real ) ).

% dbl_inc_simps(2)
thf(fact_1296_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
    = one_one_rat ) ).

% dbl_inc_simps(2)
thf(fact_1297_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
    = one_one_int ) ).

% dbl_inc_simps(2)
thf(fact_1298_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_1299_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_1300_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_1301_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_1302_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_1303_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_1304_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_1305_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_1306_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_1307_order__refl,axiom,
    ! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).

% order_refl
thf(fact_1308_order__refl,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).

% order_refl
thf(fact_1309_order__refl,axiom,
    ! [X: num] : ( ord_less_eq_num @ X @ X ) ).

% order_refl
thf(fact_1310_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_1311_order__refl,axiom,
    ! [X: int] : ( ord_less_eq_int @ X @ X ) ).

% order_refl
thf(fact_1312_dual__order_Orefl,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_1313_dual__order_Orefl,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_1314_dual__order_Orefl,axiom,
    ! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_1315_dual__order_Orefl,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_1316_dual__order_Orefl,axiom,
    ! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_1317_semiring__norm_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).

% semiring_norm(6)
thf(fact_1318_semiring__norm_I71_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(71)
thf(fact_1319_semiring__norm_I68_J,axiom,
    ! [N: num] : ( ord_less_eq_num @ one @ N ) ).

% semiring_norm(68)
thf(fact_1320_add__left__cancel,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ( plus_plus_real @ A2 @ B2 )
        = ( plus_plus_real @ A2 @ C ) )
      = ( B2 = C ) ) ).

% add_left_cancel
thf(fact_1321_add__left__cancel,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ( plus_plus_rat @ A2 @ B2 )
        = ( plus_plus_rat @ A2 @ C ) )
      = ( B2 = C ) ) ).

% add_left_cancel
thf(fact_1322_add__left__cancel,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ( plus_plus_nat @ A2 @ B2 )
        = ( plus_plus_nat @ A2 @ C ) )
      = ( B2 = C ) ) ).

% add_left_cancel
thf(fact_1323_add__left__cancel,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ( plus_plus_int @ A2 @ B2 )
        = ( plus_plus_int @ A2 @ C ) )
      = ( B2 = C ) ) ).

% add_left_cancel
thf(fact_1324_add__right__cancel,axiom,
    ! [B2: real,A2: real,C: real] :
      ( ( ( plus_plus_real @ B2 @ A2 )
        = ( plus_plus_real @ C @ A2 ) )
      = ( B2 = C ) ) ).

% add_right_cancel
thf(fact_1325_add__right__cancel,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( ( plus_plus_rat @ B2 @ A2 )
        = ( plus_plus_rat @ C @ A2 ) )
      = ( B2 = C ) ) ).

% add_right_cancel
thf(fact_1326_add__right__cancel,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ( plus_plus_nat @ B2 @ A2 )
        = ( plus_plus_nat @ C @ A2 ) )
      = ( B2 = C ) ) ).

% add_right_cancel
thf(fact_1327_add__right__cancel,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ( plus_plus_int @ B2 @ A2 )
        = ( plus_plus_int @ C @ A2 ) )
      = ( B2 = C ) ) ).

% add_right_cancel
thf(fact_1328_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_1329_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_1330_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).

% bot_nat_0.extremum
thf(fact_1331_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_1332_add_Oinverse__inverse,axiom,
    ! [A2: complex] :
      ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A2 ) )
      = A2 ) ).

% add.inverse_inverse
thf(fact_1333_add_Oinverse__inverse,axiom,
    ! [A2: uint32] :
      ( ( uminus_uminus_uint32 @ ( uminus_uminus_uint32 @ A2 ) )
      = A2 ) ).

% add.inverse_inverse
thf(fact_1334_add_Oinverse__inverse,axiom,
    ! [A2: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ A2 ) )
      = A2 ) ).

% add.inverse_inverse
thf(fact_1335_add_Oinverse__inverse,axiom,
    ! [A2: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A2 ) )
      = A2 ) ).

% add.inverse_inverse
thf(fact_1336_add_Oinverse__inverse,axiom,
    ! [A2: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A2 ) )
      = A2 ) ).

% add.inverse_inverse
thf(fact_1337_neg__equal__iff__equal,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ( uminus1482373934393186551omplex @ A2 )
        = ( uminus1482373934393186551omplex @ B2 ) )
      = ( A2 = B2 ) ) ).

% neg_equal_iff_equal
thf(fact_1338_neg__equal__iff__equal,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( ( uminus_uminus_uint32 @ A2 )
        = ( uminus_uminus_uint32 @ B2 ) )
      = ( A2 = B2 ) ) ).

% neg_equal_iff_equal
thf(fact_1339_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_1340_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_1341_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_1342_verit__minus__simplify_I4_J,axiom,
    ! [B2: complex] :
      ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ B2 ) )
      = B2 ) ).

% verit_minus_simplify(4)
thf(fact_1343_verit__minus__simplify_I4_J,axiom,
    ! [B2: uint32] :
      ( ( uminus_uminus_uint32 @ ( uminus_uminus_uint32 @ B2 ) )
      = B2 ) ).

% verit_minus_simplify(4)
thf(fact_1344_verit__minus__simplify_I4_J,axiom,
    ! [B2: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ B2 ) )
      = B2 ) ).

% verit_minus_simplify(4)
thf(fact_1345_verit__minus__simplify_I4_J,axiom,
    ! [B2: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ B2 ) )
      = B2 ) ).

% verit_minus_simplify(4)
thf(fact_1346_verit__minus__simplify_I4_J,axiom,
    ! [B2: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ B2 ) )
      = B2 ) ).

% verit_minus_simplify(4)
thf(fact_1347_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_1348_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_1349_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_1350_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_1351_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_1352_empty__subsetI,axiom,
    ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).

% empty_subsetI
thf(fact_1353_empty__subsetI,axiom,
    ! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).

% empty_subsetI
thf(fact_1354_empty__subsetI,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).

% empty_subsetI
thf(fact_1355_empty__subsetI,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% empty_subsetI
thf(fact_1356_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_1357_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_1358_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_1359_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_1360_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_1361_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_1362_Un__subset__iff,axiom,
    ! [A: set_complex,B: set_complex,C2: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( sup_sup_set_complex @ A @ B ) @ C2 )
      = ( ( ord_le211207098394363844omplex @ A @ C2 )
        & ( ord_le211207098394363844omplex @ B @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_1363_Un__subset__iff,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ A @ B ) @ C2 )
      = ( ( ord_less_eq_set_int @ A @ C2 )
        & ( ord_less_eq_set_int @ B @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_1364_Un__subset__iff,axiom,
    ! [A: set_real,B: set_real,C2: set_real] :
      ( ( ord_less_eq_set_real @ ( sup_sup_set_real @ A @ B ) @ C2 )
      = ( ( ord_less_eq_set_real @ A @ C2 )
        & ( ord_less_eq_set_real @ B @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_1365_Un__subset__iff,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
      = ( ( ord_less_eq_set_nat @ A @ C2 )
        & ( ord_less_eq_set_nat @ B @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_1366_negative__zle,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zle
thf(fact_1367_max__word__less__eq__iff,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ W )
      = ( W
        = ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ) ).

% max_word_less_eq_iff
thf(fact_1368_word__minus__one__le,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ X )
      = ( X
        = ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ) ).

% word_minus_one_le
thf(fact_1369_max__word__max,axiom,
    ! [N: word_N3645301735248828278l_num1] : ( ord_le3335648743751981014l_num1 @ N @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ).

% max_word_max
thf(fact_1370_word__n1__ge,axiom,
    ! [Y: word_N3645301735248828278l_num1] : ( ord_le3335648743751981014l_num1 @ Y @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ).

% word_n1_ge
thf(fact_1371_word__order_Oextremum__unique,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ A2 )
      = ( A2
        = ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ) ).

% word_order.extremum_unique
thf(fact_1372_word__order_Oextremum,axiom,
    ! [A2: word_N3645301735248828278l_num1] : ( ord_le3335648743751981014l_num1 @ A2 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ).

% word_order.extremum
thf(fact_1373_word__coorder_Oextremum__unique,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ A2 @ zero_z3563351764282998399l_num1 )
      = ( A2 = zero_z3563351764282998399l_num1 ) ) ).

% word_coorder.extremum_unique
thf(fact_1374_word__le__0__iff,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ X @ zero_z3563351764282998399l_num1 )
      = ( X = zero_z3563351764282998399l_num1 ) ) ).

% word_le_0_iff
thf(fact_1375_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_1376_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_1377_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_1378_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_1379_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_1380_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_1381_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_1382_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_1383_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_1384_add__0,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ zero_z3563351764282998399l_num1 @ A2 )
      = A2 ) ).

% add_0
thf(fact_1385_add__0,axiom,
    ! [A2: real] :
      ( ( plus_plus_real @ zero_zero_real @ A2 )
      = A2 ) ).

% add_0
thf(fact_1386_add__0,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A2 )
      = A2 ) ).

% add_0
thf(fact_1387_add__0,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A2 )
      = A2 ) ).

% add_0
thf(fact_1388_add__0,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ zero_zero_int @ A2 )
      = A2 ) ).

% add_0
thf(fact_1389_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_1390_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_1391_add__cancel__right__right,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( A2
        = ( plus_p361126936061061375l_num1 @ A2 @ B2 ) )
      = ( B2 = zero_z3563351764282998399l_num1 ) ) ).

% add_cancel_right_right
thf(fact_1392_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_1393_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_1394_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_1395_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_1396_add__cancel__right__left,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( A2
        = ( plus_p361126936061061375l_num1 @ B2 @ A2 ) )
      = ( B2 = zero_z3563351764282998399l_num1 ) ) ).

% add_cancel_right_left
thf(fact_1397_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_1398_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_1399_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_1400_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_1401_add__cancel__left__right,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( ( plus_p361126936061061375l_num1 @ A2 @ B2 )
        = A2 )
      = ( B2 = zero_z3563351764282998399l_num1 ) ) ).

% add_cancel_left_right
thf(fact_1402_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_1403_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_1404_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_1405_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_1406_add__cancel__left__left,axiom,
    ! [B2: word_N3645301735248828278l_num1,A2: word_N3645301735248828278l_num1] :
      ( ( ( plus_p361126936061061375l_num1 @ B2 @ A2 )
        = A2 )
      = ( B2 = zero_z3563351764282998399l_num1 ) ) ).

% add_cancel_left_left
thf(fact_1407_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_1408_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_1409_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_1410_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_1411_double__zero__sym,axiom,
    ! [A2: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A2 @ A2 ) )
      = ( A2 = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_1412_double__zero__sym,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A2 @ A2 ) )
      = ( A2 = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_1413_double__zero__sym,axiom,
    ! [A2: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A2 @ A2 ) )
      = ( A2 = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_1414_add_Oright__neutral,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ A2 @ zero_z3563351764282998399l_num1 )
      = A2 ) ).

% add.right_neutral
thf(fact_1415_add_Oright__neutral,axiom,
    ! [A2: real] :
      ( ( plus_plus_real @ A2 @ zero_zero_real )
      = A2 ) ).

% add.right_neutral
thf(fact_1416_add_Oright__neutral,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ A2 @ zero_zero_rat )
      = A2 ) ).

% add.right_neutral
thf(fact_1417_add_Oright__neutral,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% add.right_neutral
thf(fact_1418_add_Oright__neutral,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% add.right_neutral
thf(fact_1419_add__le__cancel__left,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B2 ) )
      = ( ord_less_eq_real @ A2 @ B2 ) ) ).

% add_le_cancel_left
thf(fact_1420_add__le__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B2 ) )
      = ( ord_less_eq_rat @ A2 @ B2 ) ) ).

% add_le_cancel_left
thf(fact_1421_add__le__cancel__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
      = ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% add_le_cancel_left
thf(fact_1422_add__le__cancel__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
      = ( ord_less_eq_int @ A2 @ B2 ) ) ).

% add_le_cancel_left
thf(fact_1423_add__le__cancel__right,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ C ) )
      = ( ord_less_eq_real @ A2 @ B2 ) ) ).

% add_le_cancel_right
thf(fact_1424_add__le__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B2 @ C ) )
      = ( ord_less_eq_rat @ A2 @ B2 ) ) ).

% add_le_cancel_right
thf(fact_1425_add__le__cancel__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
      = ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% add_le_cancel_right
thf(fact_1426_add__le__cancel__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
      = ( ord_less_eq_int @ A2 @ B2 ) ) ).

% add_le_cancel_right
thf(fact_1427_add__less__cancel__left,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B2 ) )
      = ( ord_less_real @ A2 @ B2 ) ) ).

% add_less_cancel_left
thf(fact_1428_add__less__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B2 ) )
      = ( ord_less_rat @ A2 @ B2 ) ) ).

% add_less_cancel_left
thf(fact_1429_add__less__cancel__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
      = ( ord_less_nat @ A2 @ B2 ) ) ).

% add_less_cancel_left
thf(fact_1430_add__less__cancel__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
      = ( ord_less_int @ A2 @ B2 ) ) ).

% add_less_cancel_left
thf(fact_1431_add__less__cancel__right,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ C ) )
      = ( ord_less_real @ A2 @ B2 ) ) ).

% add_less_cancel_right
thf(fact_1432_add__less__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B2 @ C ) )
      = ( ord_less_rat @ A2 @ B2 ) ) ).

% add_less_cancel_right
thf(fact_1433_add__less__cancel__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
      = ( ord_less_nat @ A2 @ B2 ) ) ).

% add_less_cancel_right
thf(fact_1434_add__less__cancel__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
      = ( ord_less_int @ A2 @ B2 ) ) ).

% add_less_cancel_right
thf(fact_1435_add__numeral__left,axiom,
    ! [V: num,W: num,Z: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ ( numera7442385471795722001l_num1 @ V ) @ ( plus_p361126936061061375l_num1 @ ( numera7442385471795722001l_num1 @ W ) @ Z ) )
      = ( plus_p361126936061061375l_num1 @ ( numera7442385471795722001l_num1 @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_1436_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_1437_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_1438_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_1439_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_1440_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_p361126936061061375l_num1 @ ( numera7442385471795722001l_num1 @ M ) @ ( numera7442385471795722001l_num1 @ N ) )
      = ( numera7442385471795722001l_num1 @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_1441_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_1442_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_1443_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_1444_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_1445_semiring__norm_I2_J,axiom,
    ( ( plus_plus_num @ one @ one )
    = ( bit0 @ one ) ) ).

% semiring_norm(2)
thf(fact_1446_semiring__norm_I69_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).

% semiring_norm(69)
thf(fact_1447_neg__equal__zero,axiom,
    ! [A2: real] :
      ( ( ( uminus_uminus_real @ A2 )
        = A2 )
      = ( A2 = zero_zero_real ) ) ).

% neg_equal_zero
thf(fact_1448_neg__equal__zero,axiom,
    ! [A2: rat] :
      ( ( ( uminus_uminus_rat @ A2 )
        = A2 )
      = ( A2 = zero_zero_rat ) ) ).

% neg_equal_zero
thf(fact_1449_neg__equal__zero,axiom,
    ! [A2: int] :
      ( ( ( uminus_uminus_int @ A2 )
        = A2 )
      = ( A2 = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_1450_equal__neg__zero,axiom,
    ! [A2: real] :
      ( ( A2
        = ( uminus_uminus_real @ A2 ) )
      = ( A2 = zero_zero_real ) ) ).

% equal_neg_zero
thf(fact_1451_equal__neg__zero,axiom,
    ! [A2: rat] :
      ( ( A2
        = ( uminus_uminus_rat @ A2 ) )
      = ( A2 = zero_zero_rat ) ) ).

% equal_neg_zero
thf(fact_1452_equal__neg__zero,axiom,
    ! [A2: int] :
      ( ( A2
        = ( uminus_uminus_int @ A2 ) )
      = ( A2 = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_1453_neg__equal__0__iff__equal,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( ( uminus8244633308260627903l_num1 @ A2 )
        = zero_z3563351764282998399l_num1 )
      = ( A2 = zero_z3563351764282998399l_num1 ) ) ).

% neg_equal_0_iff_equal
thf(fact_1454_neg__equal__0__iff__equal,axiom,
    ! [A2: complex] :
      ( ( ( uminus1482373934393186551omplex @ A2 )
        = zero_zero_complex )
      = ( A2 = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
thf(fact_1455_neg__equal__0__iff__equal,axiom,
    ! [A2: uint32] :
      ( ( ( uminus_uminus_uint32 @ A2 )
        = zero_zero_uint32 )
      = ( A2 = zero_zero_uint32 ) ) ).

% neg_equal_0_iff_equal
thf(fact_1456_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_1457_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_1458_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_1459_neg__0__equal__iff__equal,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( zero_z3563351764282998399l_num1
        = ( uminus8244633308260627903l_num1 @ A2 ) )
      = ( zero_z3563351764282998399l_num1 = A2 ) ) ).

% neg_0_equal_iff_equal
thf(fact_1460_neg__0__equal__iff__equal,axiom,
    ! [A2: complex] :
      ( ( zero_zero_complex
        = ( uminus1482373934393186551omplex @ A2 ) )
      = ( zero_zero_complex = A2 ) ) ).

% neg_0_equal_iff_equal
thf(fact_1461_neg__0__equal__iff__equal,axiom,
    ! [A2: uint32] :
      ( ( zero_zero_uint32
        = ( uminus_uminus_uint32 @ A2 ) )
      = ( zero_zero_uint32 = A2 ) ) ).

% neg_0_equal_iff_equal
thf(fact_1462_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_1463_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_1464_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_1465_add_Oinverse__neutral,axiom,
    ( ( uminus8244633308260627903l_num1 @ zero_z3563351764282998399l_num1 )
    = zero_z3563351764282998399l_num1 ) ).

% add.inverse_neutral
thf(fact_1466_add_Oinverse__neutral,axiom,
    ( ( uminus1482373934393186551omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% add.inverse_neutral
thf(fact_1467_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_uint32 @ zero_zero_uint32 )
    = zero_zero_uint32 ) ).

% add.inverse_neutral
thf(fact_1468_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_real @ zero_zero_real )
    = zero_zero_real ) ).

% add.inverse_neutral
thf(fact_1469_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% add.inverse_neutral
thf(fact_1470_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_1471_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_1472_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_1473_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_1474_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_1475_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_1476_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_1477_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_1478_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( M = N ) ) ).

% neg_numeral_eq_iff
thf(fact_1479_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( M = N ) ) ).

% neg_numeral_eq_iff
thf(fact_1480_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( M = N ) ) ).

% neg_numeral_eq_iff
thf(fact_1481_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( M = N ) ) ).

% neg_numeral_eq_iff
thf(fact_1482_add__minus__cancel,axiom,
    ! [A2: complex,B2: complex] :
      ( ( plus_plus_complex @ A2 @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ B2 ) )
      = B2 ) ).

% add_minus_cancel
thf(fact_1483_add__minus__cancel,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( plus_plus_uint32 @ A2 @ ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ A2 ) @ B2 ) )
      = B2 ) ).

% add_minus_cancel
thf(fact_1484_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_1485_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_1486_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_1487_minus__add__cancel,axiom,
    ! [A2: complex,B2: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( plus_plus_complex @ A2 @ B2 ) )
      = B2 ) ).

% minus_add_cancel
thf(fact_1488_minus__add__cancel,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ A2 ) @ ( plus_plus_uint32 @ A2 @ B2 ) )
      = B2 ) ).

% minus_add_cancel
thf(fact_1489_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_1490_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_1491_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_1492_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_1493_minus__add__distrib,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( uminus_uminus_uint32 @ ( plus_plus_uint32 @ A2 @ B2 ) )
      = ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ A2 ) @ ( uminus_uminus_uint32 @ B2 ) ) ) ).

% minus_add_distrib
thf(fact_1494_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_1495_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_1496_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_1497_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_1498_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_1499_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_1500_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_1501_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_1502_le__sup__iff,axiom,
    ! [X: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( sup_su3973961784419623482d_enat @ X @ Y ) @ Z )
      = ( ( ord_le2932123472753598470d_enat @ X @ Z )
        & ( ord_le2932123472753598470d_enat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_1503_le__sup__iff,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_eq_assn @ ( sup_sup_assn @ X @ Y ) @ Z )
      = ( ( ord_less_eq_assn @ X @ Z )
        & ( ord_less_eq_assn @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_1504_le__sup__iff,axiom,
    ! [X: set_complex,Y: set_complex,Z: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( sup_sup_set_complex @ X @ Y ) @ Z )
      = ( ( ord_le211207098394363844omplex @ X @ Z )
        & ( ord_le211207098394363844omplex @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_1505_le__sup__iff,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ X @ Y ) @ Z )
      = ( ( ord_less_eq_set_int @ X @ Z )
        & ( ord_less_eq_set_int @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_1506_le__sup__iff,axiom,
    ! [X: set_real,Y: set_real,Z: set_real] :
      ( ( ord_less_eq_set_real @ ( sup_sup_set_real @ X @ Y ) @ Z )
      = ( ( ord_less_eq_set_real @ X @ Z )
        & ( ord_less_eq_set_real @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_1507_le__sup__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
      = ( ( ord_less_eq_set_nat @ X @ Z )
        & ( ord_less_eq_set_nat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_1508_le__sup__iff,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ X @ Y ) @ Z )
      = ( ( ord_less_eq_rat @ X @ Z )
        & ( ord_less_eq_rat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_1509_le__sup__iff,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
      = ( ( ord_less_eq_nat @ X @ Z )
        & ( ord_less_eq_nat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_1510_le__sup__iff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ X @ Y ) @ Z )
      = ( ( ord_less_eq_int @ X @ Z )
        & ( ord_less_eq_int @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_1511_sup_Obounded__iff,axiom,
    ! [B2: extended_enat,C: extended_enat,A2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( sup_su3973961784419623482d_enat @ B2 @ C ) @ A2 )
      = ( ( ord_le2932123472753598470d_enat @ B2 @ A2 )
        & ( ord_le2932123472753598470d_enat @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_1512_sup_Obounded__iff,axiom,
    ! [B2: assn,C: assn,A2: assn] :
      ( ( ord_less_eq_assn @ ( sup_sup_assn @ B2 @ C ) @ A2 )
      = ( ( ord_less_eq_assn @ B2 @ A2 )
        & ( ord_less_eq_assn @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_1513_sup_Obounded__iff,axiom,
    ! [B2: set_complex,C: set_complex,A2: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( sup_sup_set_complex @ B2 @ C ) @ A2 )
      = ( ( ord_le211207098394363844omplex @ B2 @ A2 )
        & ( ord_le211207098394363844omplex @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_1514_sup_Obounded__iff,axiom,
    ! [B2: set_int,C: set_int,A2: set_int] :
      ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ B2 @ C ) @ A2 )
      = ( ( ord_less_eq_set_int @ B2 @ A2 )
        & ( ord_less_eq_set_int @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_1515_sup_Obounded__iff,axiom,
    ! [B2: set_real,C: set_real,A2: set_real] :
      ( ( ord_less_eq_set_real @ ( sup_sup_set_real @ B2 @ C ) @ A2 )
      = ( ( ord_less_eq_set_real @ B2 @ A2 )
        & ( ord_less_eq_set_real @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_1516_sup_Obounded__iff,axiom,
    ! [B2: set_nat,C: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 )
      = ( ( ord_less_eq_set_nat @ B2 @ A2 )
        & ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_1517_sup_Obounded__iff,axiom,
    ! [B2: rat,C: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ B2 @ C ) @ A2 )
      = ( ( ord_less_eq_rat @ B2 @ A2 )
        & ( ord_less_eq_rat @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_1518_sup_Obounded__iff,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
      = ( ( ord_less_eq_nat @ B2 @ A2 )
        & ( ord_less_eq_nat @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_1519_sup_Obounded__iff,axiom,
    ! [B2: int,C: int,A2: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C ) @ A2 )
      = ( ( ord_less_eq_int @ B2 @ A2 )
        & ( ord_less_eq_int @ C @ A2 ) ) ) ).

% sup.bounded_iff
thf(fact_1520_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_1521_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% of_nat_add
thf(fact_1522_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_add
thf(fact_1523_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% of_nat_add
thf(fact_1524_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_add
thf(fact_1525_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4939895301339042750nteger @ ( plus_plus_nat @ M @ N ) )
      = ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% of_nat_add
thf(fact_1526_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).

% of_nat_add
thf(fact_1527_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_1528_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_1529_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_1530_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_1531_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_1532_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_1533_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_1534_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_1535_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_1536_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_1537_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_1538_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_1539_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_1540_Word_Oof__nat__unat,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( semiri1314217659103216013at_int @ ( semiri7341220984566936280m1_nat @ W ) )
      = ( semiri7338730514057886004m1_int @ W ) ) ).

% Word.of_nat_unat
thf(fact_1541_negative__eq__positive,axiom,
    ! [N: nat,M: nat] :
      ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
        = ( semiri1314217659103216013at_int @ M ) )
      = ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
thf(fact_1542_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_1543_uint__nonnegative,axiom,
    ! [W: word_N3645301735248828278l_num1] : ( ord_less_eq_int @ zero_zero_int @ ( semiri7338730514057886004m1_int @ W ) ) ).

% uint_nonnegative
thf(fact_1544_uint__le__0__iff,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( ord_less_eq_int @ ( semiri7338730514057886004m1_int @ X ) @ zero_zero_int )
      = ( ( semiri7338730514057886004m1_int @ X )
        = zero_zero_int ) ) ).

% uint_le_0_iff
thf(fact_1545_uint__ge__0,axiom,
    ! [X: word_N3645301735248828278l_num1] : ( ord_less_eq_int @ zero_zero_int @ ( semiri7338730514057886004m1_int @ X ) ) ).

% uint_ge_0
thf(fact_1546_word__le__no,axiom,
    ! [A2: num,B2: num] :
      ( ( ord_le3335648743751981014l_num1 @ ( numera7442385471795722001l_num1 @ A2 ) @ ( numera7442385471795722001l_num1 @ B2 ) )
      = ( ord_less_eq_int @ ( semiri7338730514057886004m1_int @ ( numera7442385471795722001l_num1 @ A2 ) ) @ ( semiri7338730514057886004m1_int @ ( numera7442385471795722001l_num1 @ B2 ) ) ) ) ).

% word_le_no
thf(fact_1547_shiftl__0,axiom,
    ! [N: nat] :
      ( ( bit_Sh9074413540854191407l_num1 @ zero_z3563351764282998399l_num1 @ N )
      = zero_z3563351764282998399l_num1 ) ).

% shiftl_0
thf(fact_1548_shiftl__0,axiom,
    ! [N: nat] :
      ( ( bit_Sh3963086678839698405tl_int @ zero_zero_int @ N )
      = zero_zero_int ) ).

% shiftl_0
thf(fact_1549_shiftl__0,axiom,
    ! [N: nat] :
      ( ( bit_Sh3965577149348748681tl_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% shiftl_0
thf(fact_1550_pure__true,axiom,
    ( ( pure_assn @ $true )
    = one_one_assn ) ).

% pure_true
thf(fact_1551_pure__assn__eq__emp__iff,axiom,
    ! [P: $o] :
      ( ( ( pure_assn @ P )
        = one_one_assn )
      = P ) ).

% pure_assn_eq_emp_iff
thf(fact_1552_shiftl__of__0,axiom,
    ! [A2: nat] :
      ( ( bit_Sh3965577149348748681tl_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% shiftl_of_0
thf(fact_1553_pure__false,axiom,
    ( ( pure_assn @ $false )
    = bot_bot_assn ) ).

% pure_false
thf(fact_1554_pure__assn__eq__false__iff,axiom,
    ! [P: $o] :
      ( ( ( pure_assn @ P )
        = bot_bot_assn )
      = ~ P ) ).

% pure_assn_eq_false_iff
thf(fact_1555_enat__ord__number_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).

% enat_ord_number(1)
thf(fact_1556_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_1557_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_1558_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_1559_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_1560_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_1561_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_1562_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_1563_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_1564_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_1565_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_1566_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_1567_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_1568_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_1569_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_1570_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_1571_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_1572_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_1573_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_1574_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_1575_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_1576_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_1577_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_1578_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_1579_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_1580_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_1581_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_1582_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_1583_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_1584_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_1585_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_1586_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_1587_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_1588_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_1589_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_1590_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_1591_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_1592_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_1593_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_1594_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_1595_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_1596_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_1597_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_1598_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_1599_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_1600_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_1601_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_1602_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_1603_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_1604_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_1605_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_1606_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_1607_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_1608_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_1609_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_1610_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_1611_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_1612_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_1613_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_1614_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_1615_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_1616_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_1617_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_1618_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_1619_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_1620_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_1621_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_1622_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_1623_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_1624_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( ord_less_eq_num @ N @ M ) ) ).

% neg_numeral_le_iff
thf(fact_1625_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( ord_less_eq_num @ N @ M ) ) ).

% neg_numeral_le_iff
thf(fact_1626_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( ord_less_eq_num @ N @ M ) ) ).

% neg_numeral_le_iff
thf(fact_1627_add_Oright__inverse,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ A2 @ ( uminus8244633308260627903l_num1 @ A2 ) )
      = zero_z3563351764282998399l_num1 ) ).

% add.right_inverse
thf(fact_1628_add_Oright__inverse,axiom,
    ! [A2: complex] :
      ( ( plus_plus_complex @ A2 @ ( uminus1482373934393186551omplex @ A2 ) )
      = zero_zero_complex ) ).

% add.right_inverse
thf(fact_1629_add_Oright__inverse,axiom,
    ! [A2: uint32] :
      ( ( plus_plus_uint32 @ A2 @ ( uminus_uminus_uint32 @ A2 ) )
      = zero_zero_uint32 ) ).

% add.right_inverse
thf(fact_1630_add_Oright__inverse,axiom,
    ! [A2: real] :
      ( ( plus_plus_real @ A2 @ ( uminus_uminus_real @ A2 ) )
      = zero_zero_real ) ).

% add.right_inverse
thf(fact_1631_add_Oright__inverse,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ A2 @ ( uminus_uminus_rat @ A2 ) )
      = zero_zero_rat ) ).

% add.right_inverse
thf(fact_1632_add_Oright__inverse,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ A2 @ ( uminus_uminus_int @ A2 ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_1633_ab__left__minus,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ ( uminus8244633308260627903l_num1 @ A2 ) @ A2 )
      = zero_z3563351764282998399l_num1 ) ).

% ab_left_minus
thf(fact_1634_ab__left__minus,axiom,
    ! [A2: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ A2 )
      = zero_zero_complex ) ).

% ab_left_minus
thf(fact_1635_ab__left__minus,axiom,
    ! [A2: uint32] :
      ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ A2 ) @ A2 )
      = zero_zero_uint32 ) ).

% ab_left_minus
thf(fact_1636_ab__left__minus,axiom,
    ! [A2: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ A2 )
      = zero_zero_real ) ).

% ab_left_minus
thf(fact_1637_ab__left__minus,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
      = zero_zero_rat ) ).

% ab_left_minus
thf(fact_1638_ab__left__minus,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_1639_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ V ) ) @ ( plus_p361126936061061375l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ W ) ) @ Y ) )
      = ( plus_p361126936061061375l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_1640_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_1641_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: uint32] :
      ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) @ ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ W ) ) @ Y ) )
      = ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_1642_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_1643_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_1644_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_1645_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_p361126936061061375l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ M ) ) @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ N ) ) )
      = ( uminus8244633308260627903l_num1 @ ( plus_p361126936061061375l_num1 @ ( numera7442385471795722001l_num1 @ M ) @ ( numera7442385471795722001l_num1 @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_1646_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_1647_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ M ) ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ N ) ) )
      = ( uminus_uminus_uint32 @ ( plus_plus_uint32 @ ( numera9087168376688890119uint32 @ M ) @ ( numera9087168376688890119uint32 @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_1648_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_1649_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_1650_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_1651_unsigned__0,axiom,
    ( ( semiri1312839663145358974l_num1 @ zero_z3563351764282998399l_num1 )
    = zero_z3563351764282998399l_num1 ) ).

% unsigned_0
thf(fact_1652_unsigned__0,axiom,
    ( ( semiri46416754965307273481_real @ zero_z3563351764282998399l_num1 )
    = zero_zero_real ) ).

% unsigned_0
thf(fact_1653_unsigned__0,axiom,
    ( ( semiri6706090924480440544m1_rat @ zero_z3563351764282998399l_num1 )
    = zero_zero_rat ) ).

% unsigned_0
thf(fact_1654_unsigned__0,axiom,
    ( ( semiri7341220984566936280m1_nat @ zero_z3563351764282998399l_num1 )
    = zero_zero_nat ) ).

% unsigned_0
thf(fact_1655_unsigned__0,axiom,
    ( ( semiri7338730514057886004m1_int @ zero_z3563351764282998399l_num1 )
    = zero_zero_int ) ).

% unsigned_0
thf(fact_1656_unsigned__1,axiom,
    ( ( semiri1312839663145358974l_num1 @ one_on7727431528512463931l_num1 )
    = one_on7727431528512463931l_num1 ) ).

% unsigned_1
thf(fact_1657_unsigned__1,axiom,
    ( ( semiri46416754965307273481_real @ one_on7727431528512463931l_num1 )
    = one_one_real ) ).

% unsigned_1
thf(fact_1658_unsigned__1,axiom,
    ( ( semiri6706090924480440544m1_rat @ one_on7727431528512463931l_num1 )
    = one_one_rat ) ).

% unsigned_1
thf(fact_1659_unsigned__1,axiom,
    ( ( semiri7341220984566936280m1_nat @ one_on7727431528512463931l_num1 )
    = one_one_nat ) ).

% unsigned_1
thf(fact_1660_unsigned__1,axiom,
    ( ( semiri7338730514057886004m1_int @ one_on7727431528512463931l_num1 )
    = one_one_int ) ).

% unsigned_1
thf(fact_1661_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M: nat] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).

% numeral_le_real_of_nat_iff
thf(fact_1662_uint__lt__0,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ~ ( ord_less_int @ ( semiri7338730514057886004m1_int @ X ) @ zero_zero_int ) ).

% uint_lt_0
thf(fact_1663_log__one,axiom,
    ! [A2: real] :
      ( ( log @ A2 @ one_one_real )
      = zero_zero_real ) ).

% log_one
thf(fact_1664_norm__pre__pure__iff__sng,axiom,
    ! [B2: $o,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( hoare_1429296392585015714_VEBTi @ ( pure_assn @ B2 ) @ F @ Q )
      = ( B2
       => ( hoare_1429296392585015714_VEBTi @ one_one_assn @ F @ Q ) ) ) ).

% norm_pre_pure_iff_sng
thf(fact_1665_norm__pre__pure__iff__sng,axiom,
    ! [B2: $o,F: heap_Time_Heap_o,Q: $o > assn] :
      ( ( hoare_hoare_triple_o @ ( pure_assn @ B2 ) @ F @ Q )
      = ( B2
       => ( hoare_hoare_triple_o @ one_one_assn @ F @ Q ) ) ) ).

% norm_pre_pure_iff_sng
thf(fact_1666_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu7865238048354675525l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ K ) ) )
      = ( uminus8244633308260627903l_num1 @ ( neg_nu7865238048354675525l_num1 @ ( numera7442385471795722001l_num1 @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_1667_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_1668_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu5314729912787363643uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ K ) ) )
      = ( uminus_uminus_uint32 @ ( neg_nu5314729912787363643uint32 @ ( numera9087168376688890119uint32 @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_1669_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_1670_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_1671_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_1672_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu8115118780965096967l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
    = ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ).

% dbl_inc_simps(4)
thf(fact_1673_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_inc_simps(4)
thf(fact_1674_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu4269007558841261821uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) )
    = ( uminus_uminus_uint32 @ one_one_uint32 ) ) ).

% dbl_inc_simps(4)
thf(fact_1675_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_inc_simps(4)
thf(fact_1676_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_inc_simps(4)
thf(fact_1677_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_inc_simps(4)
thf(fact_1678_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_1679_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_1680_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_1681_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_1682_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_p361126936061061375l_num1 @ ( numera7442385471795722001l_num1 @ N ) @ one_on7727431528512463931l_num1 )
      = ( numera7442385471795722001l_num1 @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_1683_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
      = ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_1684_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
      = ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_1685_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_1686_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
      = ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_1687_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ ( numera7442385471795722001l_num1 @ N ) )
      = ( numera7442385471795722001l_num1 @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_1688_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_1689_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_1690_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_1691_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_1692_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_1693_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_1694_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_1695_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_1696_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_1697_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_p361126936061061375l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ one_on7727431528512463931l_num1 )
    = zero_z3563351764282998399l_num1 ) ).

% add_neg_numeral_special(8)
thf(fact_1698_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_1699_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ one_one_uint32 )
    = zero_zero_uint32 ) ).

% add_neg_numeral_special(8)
thf(fact_1700_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_1701_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_1702_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_1703_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
    = zero_z3563351764282998399l_num1 ) ).

% add_neg_numeral_special(7)
thf(fact_1704_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_1705_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_plus_uint32 @ one_one_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) )
    = zero_zero_uint32 ) ).

% add_neg_numeral_special(7)
thf(fact_1706_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_1707_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_1708_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_1709_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1482373934393186551omplex @ one_one_complex )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_1710_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_real @ one_one_real )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_1711_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_rat @ one_one_rat )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_1712_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_int @ one_one_int )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_1713_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_1714_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_1715_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_1716_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_1717_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_1718_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_1719_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_1720_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_1721_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_1722_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_1723_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_1724_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_1725_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_1726_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_1727_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_1728_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_1729_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_1730_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_1731_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_1732_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_1733_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_1734_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_1735_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_1736_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_1737_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_1738_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_1739_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_1740_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_1741_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_1742_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_1743_word__less__no,axiom,
    ! [A2: num,B2: num] :
      ( ( ord_le750835935415966154l_num1 @ ( numera7442385471795722001l_num1 @ A2 ) @ ( numera7442385471795722001l_num1 @ B2 ) )
      = ( ord_less_int @ ( semiri7338730514057886004m1_int @ ( numera7442385471795722001l_num1 @ A2 ) ) @ ( semiri7338730514057886004m1_int @ ( numera7442385471795722001l_num1 @ B2 ) ) ) ) ).

% word_less_no
thf(fact_1744_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( ord_less_num @ N @ M ) ) ).

% neg_numeral_less_iff
thf(fact_1745_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( ord_less_num @ N @ M ) ) ).

% neg_numeral_less_iff
thf(fact_1746_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( ord_less_num @ N @ M ) ) ).

% neg_numeral_less_iff
thf(fact_1747_power__decreasing__iff,axiom,
    ! [B2: code_integer,M: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
     => ( ( ord_le6747313008572928689nteger @ B2 @ one_one_Code_integer )
       => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B2 @ M ) @ ( power_8256067586552552935nteger @ B2 @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_1748_power__decreasing__iff,axiom,
    ! [B2: real,M: nat,N: 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 @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_1749_power__decreasing__iff,axiom,
    ! [B2: rat,M: nat,N: 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 @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_1750_power__decreasing__iff,axiom,
    ! [B2: nat,M: nat,N: 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 @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_1751_power__decreasing__iff,axiom,
    ! [B2: int,M: nat,N: 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 @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_1752_one__add__one,axiom,
    ( ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ one_on7727431528512463931l_num1 )
    = ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_1753_one__add__one,axiom,
    ( ( plus_plus_real @ one_one_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_1754_one__add__one,axiom,
    ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_1755_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_1756_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_1757_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_1758_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_1759_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_1760_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_1761_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_1762_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_1763_power__mono__iff,axiom,
    ! [A2: real,B2: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ B2 @ N ) )
            = ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ) ).

% power_mono_iff
thf(fact_1764_power__mono__iff,axiom,
    ! [A2: code_integer,B2: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ ( power_8256067586552552935nteger @ B2 @ N ) )
            = ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ) ) ) ).

% power_mono_iff
thf(fact_1765_power__mono__iff,axiom,
    ! [A2: rat,B2: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ B2 @ N ) )
            = ( ord_less_eq_rat @ A2 @ B2 ) ) ) ) ) ).

% power_mono_iff
thf(fact_1766_power__mono__iff,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B2 @ N ) )
            = ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ) ).

% power_mono_iff
thf(fact_1767_power__mono__iff,axiom,
    ! [A2: int,B2: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B2 @ N ) )
            = ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ) ).

% power_mono_iff
thf(fact_1768_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_1769_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_1770_power2__minus,axiom,
    ! [A2: uint32] :
      ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ A2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_uint32 @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_1771_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_1772_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_1773_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_1774_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_p361126936061061375l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
    = ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_1775_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_1776_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ ( uminus_uminus_uint32 @ one_one_uint32 ) )
    = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_1777_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_1778_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_1779_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_1780_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_1781_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_1782_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_1783_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_1784_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_1785_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_1786_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_1787_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_1788_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_1789_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_1790_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_1791_overflow__plus__one__self,axiom,
    ! [P4: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ P4 ) @ P4 )
      = ( P4
        = ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ) ).

% overflow_plus_one_self
thf(fact_1792_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_1793_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_1794_plus__1__less,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ ( plus_p361126936061061375l_num1 @ X @ one_on7727431528512463931l_num1 ) @ X )
      = ( X
        = ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ) ).

% plus_1_less
thf(fact_1795_uint__add__ge0,axiom,
    ! [Xa: word_N3645301735248828278l_num1,X: word_N3645301735248828278l_num1] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( semiri7338730514057886004m1_int @ Xa ) @ ( semiri7338730514057886004m1_int @ X ) ) ) ).

% uint_add_ge0
thf(fact_1796_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_1797_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_1798_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_1799_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_1800_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_1801_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_1802_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_1803_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_1804_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_nat,Z2: set_nat] : Y3 = Z2 )
    = ( ^ [X2: set_nat,Y2: set_nat] :
          ( ( ord_less_eq_set_nat @ X2 @ Y2 )
          & ( ord_less_eq_set_nat @ Y2 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1805_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: rat,Z2: rat] : Y3 = Z2 )
    = ( ^ [X2: rat,Y2: rat] :
          ( ( ord_less_eq_rat @ X2 @ Y2 )
          & ( ord_less_eq_rat @ Y2 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1806_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: num,Z2: num] : Y3 = Z2 )
    = ( ^ [X2: num,Y2: num] :
          ( ( ord_less_eq_num @ X2 @ Y2 )
          & ( ord_less_eq_num @ Y2 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1807_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: nat,Z2: nat] : Y3 = Z2 )
    = ( ^ [X2: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y2 )
          & ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1808_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: int,Z2: int] : Y3 = Z2 )
    = ( ^ [X2: int,Y2: int] :
          ( ( ord_less_eq_int @ X2 @ Y2 )
          & ( ord_less_eq_int @ Y2 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1809_ord__eq__le__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C )
       => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1810_ord__eq__le__trans,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ord_less_eq_rat @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1811_ord__eq__le__trans,axiom,
    ! [A2: num,B2: num,C: num] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ord_less_eq_num @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1812_ord__eq__le__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1813_ord__eq__le__trans,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ord_less_eq_int @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1814_ord__le__eq__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1815_ord__le__eq__trans,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_rat @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1816_ord__le__eq__trans,axiom,
    ! [A2: num,B2: num,C: num] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_num @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1817_ord__le__eq__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1818_ord__le__eq__trans,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_int @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1819_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_1820_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_1821_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_1822_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_1823_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_1824_order_Otrans,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C )
       => ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).

% order.trans
thf(fact_1825_order_Otrans,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ord_less_eq_rat @ A2 @ C ) ) ) ).

% order.trans
thf(fact_1826_order_Otrans,axiom,
    ! [A2: num,B2: num,C: num] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ord_less_eq_num @ A2 @ C ) ) ) ).

% order.trans
thf(fact_1827_order_Otrans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% order.trans
thf(fact_1828_order_Otrans,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ord_less_eq_int @ A2 @ C ) ) ) ).

% order.trans
thf(fact_1829_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_1830_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_1831_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_1832_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_1833_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_1834_linorder__wlog,axiom,
    ! [P: rat > rat > $o,A2: rat,B2: rat] :
      ( ! [A5: rat,B6: rat] :
          ( ( ord_less_eq_rat @ A5 @ B6 )
         => ( P @ A5 @ B6 ) )
     => ( ! [A5: rat,B6: rat] :
            ( ( P @ B6 @ A5 )
           => ( P @ A5 @ B6 ) )
       => ( P @ A2 @ B2 ) ) ) ).

% linorder_wlog
thf(fact_1835_linorder__wlog,axiom,
    ! [P: num > num > $o,A2: num,B2: num] :
      ( ! [A5: num,B6: num] :
          ( ( ord_less_eq_num @ A5 @ B6 )
         => ( P @ A5 @ B6 ) )
     => ( ! [A5: num,B6: num] :
            ( ( P @ B6 @ A5 )
           => ( P @ A5 @ B6 ) )
       => ( P @ A2 @ B2 ) ) ) ).

% linorder_wlog
thf(fact_1836_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A2: nat,B2: nat] :
      ( ! [A5: nat,B6: nat] :
          ( ( ord_less_eq_nat @ A5 @ B6 )
         => ( P @ A5 @ B6 ) )
     => ( ! [A5: nat,B6: nat] :
            ( ( P @ B6 @ A5 )
           => ( P @ A5 @ B6 ) )
       => ( P @ A2 @ B2 ) ) ) ).

% linorder_wlog
thf(fact_1837_linorder__wlog,axiom,
    ! [P: int > int > $o,A2: int,B2: int] :
      ( ! [A5: int,B6: int] :
          ( ( ord_less_eq_int @ A5 @ B6 )
         => ( P @ A5 @ B6 ) )
     => ( ! [A5: int,B6: int] :
            ( ( P @ B6 @ A5 )
           => ( P @ A5 @ B6 ) )
       => ( P @ A2 @ B2 ) ) ) ).

% linorder_wlog
thf(fact_1838_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: set_nat,Z2: set_nat] : Y3 = Z2 )
    = ( ^ [A4: set_nat,B4: set_nat] :
          ( ( ord_less_eq_set_nat @ B4 @ A4 )
          & ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1839_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: rat,Z2: rat] : Y3 = Z2 )
    = ( ^ [A4: rat,B4: rat] :
          ( ( ord_less_eq_rat @ B4 @ A4 )
          & ( ord_less_eq_rat @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1840_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: num,Z2: num] : Y3 = Z2 )
    = ( ^ [A4: num,B4: num] :
          ( ( ord_less_eq_num @ B4 @ A4 )
          & ( ord_less_eq_num @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1841_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: nat,Z2: nat] : Y3 = Z2 )
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A4 )
          & ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1842_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y3: int,Z2: int] : Y3 = Z2 )
    = ( ^ [A4: int,B4: int] :
          ( ( ord_less_eq_int @ B4 @ A4 )
          & ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1843_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_1844_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_1845_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_1846_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_1847_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_1848_dual__order_Otrans,axiom,
    ! [B2: set_nat,A2: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( ord_less_eq_set_nat @ C @ B2 )
       => ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_1849_dual__order_Otrans,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ( ord_less_eq_rat @ C @ B2 )
       => ( ord_less_eq_rat @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_1850_dual__order_Otrans,axiom,
    ! [B2: num,A2: num,C: num] :
      ( ( ord_less_eq_num @ B2 @ A2 )
     => ( ( ord_less_eq_num @ C @ B2 )
       => ( ord_less_eq_num @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_1851_dual__order_Otrans,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_eq_nat @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_1852_dual__order_Otrans,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( ord_less_eq_int @ C @ B2 )
       => ( ord_less_eq_int @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_1853_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_1854_antisym,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_1855_antisym,axiom,
    ! [A2: num,B2: num] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ord_less_eq_num @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_1856_antisym,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_1857_antisym,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_1858_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: set_nat,Z2: set_nat] : Y3 = Z2 )
    = ( ^ [A4: set_nat,B4: set_nat] :
          ( ( ord_less_eq_set_nat @ A4 @ B4 )
          & ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1859_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: rat,Z2: rat] : Y3 = Z2 )
    = ( ^ [A4: rat,B4: rat] :
          ( ( ord_less_eq_rat @ A4 @ B4 )
          & ( ord_less_eq_rat @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1860_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: num,Z2: num] : Y3 = Z2 )
    = ( ^ [A4: num,B4: num] :
          ( ( ord_less_eq_num @ A4 @ B4 )
          & ( ord_less_eq_num @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1861_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: nat,Z2: nat] : Y3 = Z2 )
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
          & ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1862_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y3: int,Z2: int] : Y3 = Z2 )
    = ( ^ [A4: int,B4: int] :
          ( ( ord_less_eq_int @ A4 @ B4 )
          & ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1863_order__subst1,axiom,
    ! [A2: rat,F: rat > rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1864_order__subst1,axiom,
    ! [A2: rat,F: num > rat,B2: num,C: num] :
      ( ( ord_less_eq_rat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1865_order__subst1,axiom,
    ! [A2: rat,F: nat > rat,B2: nat,C: nat] :
      ( ( ord_less_eq_rat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1866_order__subst1,axiom,
    ! [A2: rat,F: int > rat,B2: int,C: int] :
      ( ( ord_less_eq_rat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ! [X3: int,Y4: int] :
              ( ( ord_less_eq_int @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1867_order__subst1,axiom,
    ! [A2: num,F: rat > num,B2: rat,C: rat] :
      ( ( ord_less_eq_num @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1868_order__subst1,axiom,
    ! [A2: num,F: num > num,B2: num,C: num] :
      ( ( ord_less_eq_num @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1869_order__subst1,axiom,
    ! [A2: num,F: nat > num,B2: nat,C: nat] :
      ( ( ord_less_eq_num @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1870_order__subst1,axiom,
    ! [A2: num,F: int > num,B2: int,C: int] :
      ( ( ord_less_eq_num @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ! [X3: int,Y4: int] :
              ( ( ord_less_eq_int @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1871_order__subst1,axiom,
    ! [A2: nat,F: rat > nat,B2: rat,C: rat] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1872_order__subst1,axiom,
    ! [A2: nat,F: num > nat,B2: num,C: num] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1873_order__subst2,axiom,
    ! [A2: rat,B2: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1874_order__subst2,axiom,
    ! [A2: rat,B2: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1875_order__subst2,axiom,
    ! [A2: rat,B2: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1876_order__subst2,axiom,
    ! [A2: rat,B2: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1877_order__subst2,axiom,
    ! [A2: num,B2: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1878_order__subst2,axiom,
    ! [A2: num,B2: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1879_order__subst2,axiom,
    ! [A2: num,B2: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1880_order__subst2,axiom,
    ! [A2: num,B2: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1881_order__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1882_order__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1883_order__eq__refl,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X = Y )
     => ( ord_less_eq_set_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_1884_order__eq__refl,axiom,
    ! [X: rat,Y: rat] :
      ( ( X = Y )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_1885_order__eq__refl,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
     => ( ord_less_eq_num @ X @ Y ) ) ).

% order_eq_refl
thf(fact_1886_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_1887_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_eq_refl
thf(fact_1888_linorder__linear,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
      | ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_linear
thf(fact_1889_linorder__linear,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
      | ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_linear
thf(fact_1890_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_1891_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_linear
thf(fact_1892_ord__eq__le__subst,axiom,
    ! [A2: rat,F: rat > rat,B2: rat,C: rat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1893_ord__eq__le__subst,axiom,
    ! [A2: num,F: rat > num,B2: rat,C: rat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1894_ord__eq__le__subst,axiom,
    ! [A2: nat,F: rat > nat,B2: rat,C: rat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1895_ord__eq__le__subst,axiom,
    ! [A2: int,F: rat > int,B2: rat,C: rat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1896_ord__eq__le__subst,axiom,
    ! [A2: rat,F: num > rat,B2: num,C: num] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1897_ord__eq__le__subst,axiom,
    ! [A2: num,F: num > num,B2: num,C: num] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1898_ord__eq__le__subst,axiom,
    ! [A2: nat,F: num > nat,B2: num,C: num] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1899_ord__eq__le__subst,axiom,
    ! [A2: int,F: num > int,B2: num,C: num] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1900_ord__eq__le__subst,axiom,
    ! [A2: rat,F: nat > rat,B2: nat,C: nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1901_ord__eq__le__subst,axiom,
    ! [A2: num,F: nat > num,B2: nat,C: nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1902_ord__le__eq__subst,axiom,
    ! [A2: rat,B2: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1903_ord__le__eq__subst,axiom,
    ! [A2: rat,B2: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1904_ord__le__eq__subst,axiom,
    ! [A2: rat,B2: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1905_ord__le__eq__subst,axiom,
    ! [A2: rat,B2: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1906_ord__le__eq__subst,axiom,
    ! [A2: num,B2: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1907_ord__le__eq__subst,axiom,
    ! [A2: num,B2: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1908_ord__le__eq__subst,axiom,
    ! [A2: num,B2: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1909_ord__le__eq__subst,axiom,
    ! [A2: num,B2: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1910_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1911_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1912_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_1913_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_1914_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_1915_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_1916_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_1917_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_1918_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_1919_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_1920_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_1921_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_real @ I @ K )
        = ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_1922_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_rat @ I @ K )
        = ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_1923_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_nat @ I @ K )
        = ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_1924_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_int @ I @ K )
        = ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_1925_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1926_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1927_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1928_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1929_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1930_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1931_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1932_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1933_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1934_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1935_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1936_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1937_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_1938_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_1939_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_1940_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_1941_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_1942_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_1943_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_1944_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_1945_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_1946_group__cancel_Oneg1,axiom,
    ! [A: uint32,K: uint32,A2: uint32] :
      ( ( A
        = ( plus_plus_uint32 @ K @ A2 ) )
     => ( ( uminus_uminus_uint32 @ A )
        = ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ K ) @ ( uminus_uminus_uint32 @ A2 ) ) ) ) ).

% group_cancel.neg1
thf(fact_1947_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_1948_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_1949_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_1950_add_Oassoc,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A2 @ B2 ) @ C )
      = ( plus_plus_real @ A2 @ ( plus_plus_real @ B2 @ C ) ) ) ).

% add.assoc
thf(fact_1951_add_Oassoc,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C )
      = ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B2 @ C ) ) ) ).

% add.assoc
thf(fact_1952_add_Oassoc,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
      = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add.assoc
thf(fact_1953_add_Oassoc,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
      = ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).

% add.assoc
thf(fact_1954_add_Oleft__cancel,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ( plus_plus_real @ A2 @ B2 )
        = ( plus_plus_real @ A2 @ C ) )
      = ( B2 = C ) ) ).

% add.left_cancel
thf(fact_1955_add_Oleft__cancel,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ( plus_plus_rat @ A2 @ B2 )
        = ( plus_plus_rat @ A2 @ C ) )
      = ( B2 = C ) ) ).

% add.left_cancel
thf(fact_1956_add_Oleft__cancel,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ( plus_plus_int @ A2 @ B2 )
        = ( plus_plus_int @ A2 @ C ) )
      = ( B2 = C ) ) ).

% add.left_cancel
thf(fact_1957_add_Oright__cancel,axiom,
    ! [B2: real,A2: real,C: real] :
      ( ( ( plus_plus_real @ B2 @ A2 )
        = ( plus_plus_real @ C @ A2 ) )
      = ( B2 = C ) ) ).

% add.right_cancel
thf(fact_1958_add_Oright__cancel,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( ( plus_plus_rat @ B2 @ A2 )
        = ( plus_plus_rat @ C @ A2 ) )
      = ( B2 = C ) ) ).

% add.right_cancel
thf(fact_1959_add_Oright__cancel,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ( plus_plus_int @ B2 @ A2 )
        = ( plus_plus_int @ C @ A2 ) )
      = ( B2 = C ) ) ).

% add.right_cancel
thf(fact_1960_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_real
    = ( ^ [A4: real,B4: real] : ( plus_plus_real @ B4 @ A4 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_1961_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A4: rat,B4: rat] : ( plus_plus_rat @ B4 @ A4 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_1962_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_1963_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A4: int,B4: int] : ( plus_plus_int @ B4 @ A4 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_1964_equation__minus__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( A2
        = ( uminus1482373934393186551omplex @ B2 ) )
      = ( B2
        = ( uminus1482373934393186551omplex @ A2 ) ) ) ).

% equation_minus_iff
thf(fact_1965_equation__minus__iff,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( A2
        = ( uminus_uminus_uint32 @ B2 ) )
      = ( B2
        = ( uminus_uminus_uint32 @ A2 ) ) ) ).

% equation_minus_iff
thf(fact_1966_equation__minus__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( A2
        = ( uminus_uminus_real @ B2 ) )
      = ( B2
        = ( uminus_uminus_real @ A2 ) ) ) ).

% equation_minus_iff
thf(fact_1967_equation__minus__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2
        = ( uminus_uminus_rat @ B2 ) )
      = ( B2
        = ( uminus_uminus_rat @ A2 ) ) ) ).

% equation_minus_iff
thf(fact_1968_equation__minus__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( A2
        = ( uminus_uminus_int @ B2 ) )
      = ( B2
        = ( uminus_uminus_int @ A2 ) ) ) ).

% equation_minus_iff
thf(fact_1969_minus__equation__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ( uminus1482373934393186551omplex @ A2 )
        = B2 )
      = ( ( uminus1482373934393186551omplex @ B2 )
        = A2 ) ) ).

% minus_equation_iff
thf(fact_1970_minus__equation__iff,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( ( uminus_uminus_uint32 @ A2 )
        = B2 )
      = ( ( uminus_uminus_uint32 @ B2 )
        = A2 ) ) ).

% minus_equation_iff
thf(fact_1971_minus__equation__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ( uminus_uminus_real @ A2 )
        = B2 )
      = ( ( uminus_uminus_real @ B2 )
        = A2 ) ) ).

% minus_equation_iff
thf(fact_1972_minus__equation__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( uminus_uminus_rat @ A2 )
        = B2 )
      = ( ( uminus_uminus_rat @ B2 )
        = A2 ) ) ).

% minus_equation_iff
thf(fact_1973_minus__equation__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ( uminus_uminus_int @ A2 )
        = B2 )
      = ( ( uminus_uminus_int @ B2 )
        = A2 ) ) ).

% minus_equation_iff
thf(fact_1974_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: real,A2: real,C: real] :
      ( ( plus_plus_real @ B2 @ ( plus_plus_real @ A2 @ C ) )
      = ( plus_plus_real @ A2 @ ( plus_plus_real @ B2 @ C ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_1975_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( plus_plus_rat @ B2 @ ( plus_plus_rat @ A2 @ C ) )
      = ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B2 @ C ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_1976_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) )
      = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_1977_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( plus_plus_int @ B2 @ ( plus_plus_int @ A2 @ C ) )
      = ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_1978_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_1979_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_1980_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_1981_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_1982_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_1983_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_1984_add__mono,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).

% add_mono
thf(fact_1985_add__mono,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).

% add_mono
thf(fact_1986_add__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).

% add_mono
thf(fact_1987_add__mono,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).

% add_mono
thf(fact_1988_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_1989_add_Oinverse__distrib__swap,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( uminus_uminus_uint32 @ ( plus_plus_uint32 @ A2 @ B2 ) )
      = ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ B2 ) @ ( uminus_uminus_uint32 @ A2 ) ) ) ).

% add.inverse_distrib_swap
thf(fact_1990_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_1991_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_1992_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_1993_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_1994_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_1995_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_1996_add__left__imp__eq,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ( plus_plus_real @ A2 @ B2 )
        = ( plus_plus_real @ A2 @ C ) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
thf(fact_1997_add__left__imp__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ( plus_plus_rat @ A2 @ B2 )
        = ( plus_plus_rat @ A2 @ C ) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
thf(fact_1998_add__left__imp__eq,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ( plus_plus_nat @ A2 @ B2 )
        = ( plus_plus_nat @ A2 @ C ) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
thf(fact_1999_add__left__imp__eq,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ( plus_plus_int @ A2 @ B2 )
        = ( plus_plus_int @ A2 @ C ) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
thf(fact_2000_add__right__imp__eq,axiom,
    ! [B2: real,A2: real,C: real] :
      ( ( ( plus_plus_real @ B2 @ A2 )
        = ( plus_plus_real @ C @ A2 ) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
thf(fact_2001_add__right__imp__eq,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( ( plus_plus_rat @ B2 @ A2 )
        = ( plus_plus_rat @ C @ A2 ) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
thf(fact_2002_add__right__imp__eq,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ( plus_plus_nat @ B2 @ A2 )
        = ( plus_plus_nat @ C @ A2 ) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
thf(fact_2003_add__right__imp__eq,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ( plus_plus_int @ B2 @ A2 )
        = ( plus_plus_int @ C @ A2 ) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
thf(fact_2004_add__left__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ord_less_eq_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B2 ) ) ) ).

% add_left_mono
thf(fact_2005_add__left__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B2 ) ) ) ).

% add_left_mono
thf(fact_2006_add__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).

% add_left_mono
thf(fact_2007_add__left__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) ) ) ).

% add_left_mono
thf(fact_2008_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_2009_add__right__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ C ) ) ) ).

% add_right_mono
thf(fact_2010_add__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B2 @ C ) ) ) ).

% add_right_mono
thf(fact_2011_add__right__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add_right_mono
thf(fact_2012_add__right__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).

% add_right_mono
thf(fact_2013_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
        ? [C5: nat] :
          ( B4
          = ( plus_plus_nat @ A4 @ C5 ) ) ) ) ).

% le_iff_add
thf(fact_2014_add__le__imp__le__left,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B2 ) )
     => ( ord_less_eq_real @ A2 @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_2015_add__le__imp__le__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B2 ) )
     => ( ord_less_eq_rat @ A2 @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_2016_add__le__imp__le__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
     => ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_2017_add__le__imp__le__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
     => ( ord_less_eq_int @ A2 @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_2018_add__le__imp__le__right,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ C ) )
     => ( ord_less_eq_real @ A2 @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_2019_add__le__imp__le__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B2 @ C ) )
     => ( ord_less_eq_rat @ A2 @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_2020_add__le__imp__le__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
     => ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_2021_add__le__imp__le__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
     => ( ord_less_eq_int @ A2 @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_2022_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_2023_is__num__normalize_I8_J,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( uminus_uminus_uint32 @ ( plus_plus_uint32 @ A2 @ B2 ) )
      = ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ B2 ) @ ( uminus_uminus_uint32 @ A2 ) ) ) ).

% is_num_normalize(8)
thf(fact_2024_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_2025_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_2026_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_2027_is__num__normalize_I1_J,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A2 @ B2 ) @ C )
      = ( plus_plus_real @ A2 @ ( plus_plus_real @ B2 @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_2028_is__num__normalize_I1_J,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C )
      = ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B2 @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_2029_is__num__normalize_I1_J,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
      = ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_2030_verit__comp__simplify1_I2_J,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).

% verit_comp_simplify1(2)
thf(fact_2031_verit__comp__simplify1_I2_J,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ A2 @ A2 ) ).

% verit_comp_simplify1(2)
thf(fact_2032_verit__comp__simplify1_I2_J,axiom,
    ! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).

% verit_comp_simplify1(2)
thf(fact_2033_verit__comp__simplify1_I2_J,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).

% verit_comp_simplify1(2)
thf(fact_2034_verit__comp__simplify1_I2_J,axiom,
    ! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).

% verit_comp_simplify1(2)
thf(fact_2035_verit__negate__coefficient_I3_J,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 = B2 )
     => ( ( uminus_uminus_real @ A2 )
        = ( uminus_uminus_real @ B2 ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_2036_verit__negate__coefficient_I3_J,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = B2 )
     => ( ( uminus_uminus_rat @ A2 )
        = ( uminus_uminus_rat @ B2 ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_2037_verit__negate__coefficient_I3_J,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = B2 )
     => ( ( uminus_uminus_int @ A2 )
        = ( uminus_uminus_int @ B2 ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_2038_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_2039_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_2040_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_2041_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_2042_unsigned__word__eqI,axiom,
    ! [V: word_N3645301735248828278l_num1,W: word_N3645301735248828278l_num1] :
      ( ( ( semiri7338730514057886004m1_int @ V )
        = ( semiri7338730514057886004m1_int @ W ) )
     => ( V = W ) ) ).

% unsigned_word_eqI
thf(fact_2043_word__eq__iff__unsigned,axiom,
    ( ( ^ [Y3: word_N3645301735248828278l_num1,Z2: word_N3645301735248828278l_num1] : Y3 = Z2 )
    = ( ^ [V2: word_N3645301735248828278l_num1,W2: word_N3645301735248828278l_num1] :
          ( ( semiri7338730514057886004m1_int @ V2 )
          = ( semiri7338730514057886004m1_int @ W2 ) ) ) ) ).

% word_eq_iff_unsigned
thf(fact_2044_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_2045_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_2046_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_2047_word__less__eq__iff__unsigned,axiom,
    ( ord_le3335648743751981014l_num1
    = ( ^ [A4: word_N3645301735248828278l_num1,B4: word_N3645301735248828278l_num1] : ( ord_less_eq_int @ ( semiri7338730514057886004m1_int @ A4 ) @ ( semiri7338730514057886004m1_int @ B4 ) ) ) ) ).

% word_less_eq_iff_unsigned
thf(fact_2048_uint__cong,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( X = Y )
     => ( ( semiri7338730514057886004m1_int @ X )
        = ( semiri7338730514057886004m1_int @ Y ) ) ) ).

% uint_cong
thf(fact_2049_uint__add__le,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] : ( ord_less_eq_int @ ( semiri7338730514057886004m1_int @ ( plus_p361126936061061375l_num1 @ X @ Y ) ) @ ( plus_plus_int @ ( semiri7338730514057886004m1_int @ X ) @ ( semiri7338730514057886004m1_int @ Y ) ) ) ).

% uint_add_le
thf(fact_2050_word__le__def,axiom,
    ( ord_le3335648743751981014l_num1
    = ( ^ [A4: word_N3645301735248828278l_num1,B4: word_N3645301735248828278l_num1] : ( ord_less_eq_int @ ( semiri7338730514057886004m1_int @ A4 ) @ ( semiri7338730514057886004m1_int @ B4 ) ) ) ) ).

% word_le_def
thf(fact_2051_word__uint__eqI,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( ( semiri7338730514057886004m1_int @ A2 )
        = ( semiri7338730514057886004m1_int @ B2 ) )
     => ( A2 = B2 ) ) ).

% word_uint_eqI
thf(fact_2052_uint__plus__simple,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ X @ ( plus_p361126936061061375l_num1 @ X @ Y ) )
     => ( ( semiri7338730514057886004m1_int @ ( plus_p361126936061061375l_num1 @ X @ Y ) )
        = ( plus_plus_int @ ( semiri7338730514057886004m1_int @ X ) @ ( semiri7338730514057886004m1_int @ Y ) ) ) ) ).

% uint_plus_simple
thf(fact_2053_word__uint__eq__iff,axiom,
    ( ( ^ [Y3: word_N3645301735248828278l_num1,Z2: word_N3645301735248828278l_num1] : Y3 = Z2 )
    = ( ^ [A4: word_N3645301735248828278l_num1,B4: word_N3645301735248828278l_num1] :
          ( ( semiri7338730514057886004m1_int @ A4 )
          = ( semiri7338730514057886004m1_int @ B4 ) ) ) ) ).

% word_uint_eq_iff
thf(fact_2054_uint__plus__simple__iff,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ X @ ( plus_p361126936061061375l_num1 @ X @ Y ) )
      = ( ( semiri7338730514057886004m1_int @ ( plus_p361126936061061375l_num1 @ X @ Y ) )
        = ( plus_plus_int @ ( semiri7338730514057886004m1_int @ X ) @ ( semiri7338730514057886004m1_int @ Y ) ) ) ) ).

% uint_plus_simple_iff
thf(fact_2055_add__eq__0__iff,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( ( plus_p361126936061061375l_num1 @ A2 @ B2 )
        = zero_z3563351764282998399l_num1 )
      = ( B2
        = ( uminus8244633308260627903l_num1 @ A2 ) ) ) ).

% add_eq_0_iff
thf(fact_2056_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_2057_add__eq__0__iff,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( ( plus_plus_uint32 @ A2 @ B2 )
        = zero_zero_uint32 )
      = ( B2
        = ( uminus_uminus_uint32 @ A2 ) ) ) ).

% add_eq_0_iff
thf(fact_2058_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_2059_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_2060_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_2061_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ ( uminus8244633308260627903l_num1 @ A2 ) @ A2 )
      = zero_z3563351764282998399l_num1 ) ).

% ab_group_add_class.ab_left_minus
thf(fact_2062_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_2063_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: uint32] :
      ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ A2 ) @ A2 )
      = zero_zero_uint32 ) ).

% ab_group_add_class.ab_left_minus
thf(fact_2064_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_2065_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_2066_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_2067_add_Oinverse__unique,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( ( plus_p361126936061061375l_num1 @ A2 @ B2 )
        = zero_z3563351764282998399l_num1 )
     => ( ( uminus8244633308260627903l_num1 @ A2 )
        = B2 ) ) ).

% add.inverse_unique
thf(fact_2068_add_Oinverse__unique,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ( plus_plus_complex @ A2 @ B2 )
        = zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ A2 )
        = B2 ) ) ).

% add.inverse_unique
thf(fact_2069_add_Oinverse__unique,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( ( plus_plus_uint32 @ A2 @ B2 )
        = zero_zero_uint32 )
     => ( ( uminus_uminus_uint32 @ A2 )
        = B2 ) ) ).

% add.inverse_unique
thf(fact_2070_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_2071_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_2072_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_2073_eq__neg__iff__add__eq__0,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( A2
        = ( uminus8244633308260627903l_num1 @ B2 ) )
      = ( ( plus_p361126936061061375l_num1 @ A2 @ B2 )
        = zero_z3563351764282998399l_num1 ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_2074_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_2075_eq__neg__iff__add__eq__0,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( A2
        = ( uminus_uminus_uint32 @ B2 ) )
      = ( ( plus_plus_uint32 @ A2 @ B2 )
        = zero_zero_uint32 ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_2076_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_2077_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_2078_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_2079_neg__eq__iff__add__eq__0,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( ( uminus8244633308260627903l_num1 @ A2 )
        = B2 )
      = ( ( plus_p361126936061061375l_num1 @ A2 @ B2 )
        = zero_z3563351764282998399l_num1 ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_2080_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_2081_neg__eq__iff__add__eq__0,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( ( uminus_uminus_uint32 @ A2 )
        = B2 )
      = ( ( plus_plus_uint32 @ A2 @ B2 )
        = zero_zero_uint32 ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_2082_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_2083_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_2084_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_2085_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_2086_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_2087_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_2088_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_2089_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_2090_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_2091_add__decreasing,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ A2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ C @ B2 )
       => ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_2092_add__decreasing,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ C @ B2 )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_2093_add__decreasing,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_2094_add__decreasing,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ C @ B2 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_2095_add__increasing,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).

% add_increasing
thf(fact_2096_add__increasing,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ord_less_eq_rat @ B2 @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).

% add_increasing
thf(fact_2097_add__increasing,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_increasing
thf(fact_2098_add__increasing,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).

% add_increasing
thf(fact_2099_add__decreasing2,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( ord_less_eq_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A2 @ B2 )
       => ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_2100_add__decreasing2,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A2 @ B2 )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_2101_add__decreasing2,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_2102_add__decreasing2,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ A2 @ B2 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_2103_add__increasing2,axiom,
    ! [C: real,B2: real,A2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ B2 @ A2 )
       => ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).

% add_increasing2
thf(fact_2104_add__increasing2,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ B2 @ A2 )
       => ( ord_less_eq_rat @ B2 @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).

% add_increasing2
thf(fact_2105_add__increasing2,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B2 @ A2 )
       => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_increasing2
thf(fact_2106_add__increasing2,axiom,
    ! [C: int,B2: int,A2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ B2 @ A2 )
       => ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).

% add_increasing2
thf(fact_2107_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_2108_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_2109_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_2110_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_2111_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_2112_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_2113_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_2114_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_2115_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_2116_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_2117_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_2118_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_2119_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_2120_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_2121_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_2122_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_2123_add__less__le__mono,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_2124_add__less__le__mono,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_2125_add__less__le__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_2126_add__less__le__mono,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_2127_add__le__less__mono,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_2128_add__le__less__mono,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_2129_add__le__less__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_2130_add__le__less__mono,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_2131_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2132_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2133_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2134_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2135_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2136_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2137_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2138_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2139_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_2140_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_2141_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_2142_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_2143_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_2144_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_2145_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_2146_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_2147_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_2148_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_2149_unsigned__greater__eq,axiom,
    ! [W: word_N3645301735248828278l_num1] : ( ord_less_eq_int @ zero_zero_int @ ( semiri7338730514057886004m1_int @ W ) ) ).

% unsigned_greater_eq
thf(fact_2150_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_2151_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_2152_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_2153_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_2154_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_2155_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_2156_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M4: nat,N2: nat] :
          ( ( ord_less_nat @ M4 @ N2 )
         => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_2157_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).

% of_nat_mono
thf(fact_2158_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ I ) @ ( semiri4939895301339042750nteger @ J ) ) ) ).

% of_nat_mono
thf(fact_2159_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).

% of_nat_mono
thf(fact_2160_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).

% of_nat_mono
thf(fact_2161_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).

% of_nat_mono
thf(fact_2162_int__ge__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_eq_int @ K @ I )
     => ( ( P @ K )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ K @ I3 )
             => ( ( P @ I3 )
               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_ge_induct
thf(fact_2163_word__order_Oextremum__uniqueI,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ A2 )
     => ( A2
        = ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ) ).

% word_order.extremum_uniqueI
thf(fact_2164_zadd__int__left,axiom,
    ! [M: nat,N: nat,Z: int] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).

% zadd_int_left
thf(fact_2165_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_2166_int__plus,axiom,
    ! [N: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% int_plus
thf(fact_2167_zle__iff__zadd,axiom,
    ( ord_less_eq_int
    = ( ^ [W2: int,Z4: int] :
        ? [N4: nat] :
          ( Z4
          = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).

% zle_iff_zadd
thf(fact_2168_neq__0__no__wrap,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ X @ ( plus_p361126936061061375l_num1 @ X @ Y ) )
     => ( ( X != zero_z3563351764282998399l_num1 )
       => ( ( plus_p361126936061061375l_num1 @ X @ Y )
         != zero_z3563351764282998399l_num1 ) ) ) ).

% neq_0_no_wrap
thf(fact_2169_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% zle_int
thf(fact_2170_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_2171_word__le__make__less,axiom,
    ! [Y: word_N3645301735248828278l_num1,X: word_N3645301735248828278l_num1] :
      ( ( Y
       != ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
     => ( ( ord_le3335648743751981014l_num1 @ X @ Y )
        = ( ord_le750835935415966154l_num1 @ X @ ( plus_p361126936061061375l_num1 @ Y @ one_on7727431528512463931l_num1 ) ) ) ) ).

% word_le_make_less
thf(fact_2172_word__Suc__leq,axiom,
    ! [K: word_N3645301735248828278l_num1,X: word_N3645301735248828278l_num1] :
      ( ( K
       != ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
     => ( ( ord_le750835935415966154l_num1 @ X @ ( plus_p361126936061061375l_num1 @ K @ one_on7727431528512463931l_num1 ) )
        = ( ord_le3335648743751981014l_num1 @ X @ K ) ) ) ).

% word_Suc_leq
thf(fact_2173_word__Suc__le,axiom,
    ! [X: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1] :
      ( ( X
       != ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
     => ( ( ord_le3335648743751981014l_num1 @ ( plus_p361126936061061375l_num1 @ X @ one_on7727431528512463931l_num1 ) @ K )
        = ( ord_le750835935415966154l_num1 @ X @ K ) ) ) ).

% word_Suc_le
thf(fact_2174_dbl__inc__def,axiom,
    ( neg_nu8115118780965096967l_num1
    = ( ^ [X2: word_N3645301735248828278l_num1] : ( plus_p361126936061061375l_num1 @ ( plus_p361126936061061375l_num1 @ X2 @ X2 ) @ one_on7727431528512463931l_num1 ) ) ) ).

% dbl_inc_def
thf(fact_2175_dbl__inc__def,axiom,
    ( neg_nu8295874005876285629c_real
    = ( ^ [X2: real] : ( plus_plus_real @ ( plus_plus_real @ X2 @ X2 ) @ one_one_real ) ) ) ).

% dbl_inc_def
thf(fact_2176_dbl__inc__def,axiom,
    ( neg_nu5219082963157363817nc_rat
    = ( ^ [X2: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).

% dbl_inc_def
thf(fact_2177_dbl__inc__def,axiom,
    ( neg_nu5851722552734809277nc_int
    = ( ^ [X2: int] : ( plus_plus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).

% dbl_inc_def
thf(fact_2178_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_leq_as_int
thf(fact_2179_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).

% neg_numeral_le_zero
thf(fact_2180_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).

% neg_numeral_le_zero
thf(fact_2181_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).

% neg_numeral_le_zero
thf(fact_2182_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_2183_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_2184_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_2185_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_2186_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_2187_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_2188_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_2189_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_2190_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_2191_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_2192_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_2193_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_2194_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_2195_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_2196_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_2197_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_2198_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_2199_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_2200_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_2201_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_2202_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_2203_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_2204_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_2205_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_2206_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_2207_add__strict__increasing,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ord_less_real @ B2 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_2208_add__strict__increasing,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_2209_add__strict__increasing,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_2210_add__strict__increasing,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_2211_add__strict__increasing2,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_real @ B2 @ C )
       => ( ord_less_real @ B2 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_2212_add__strict__increasing2,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_2213_add__strict__increasing2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_2214_add__strict__increasing2,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ B2 @ C )
       => ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_2215_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_2216_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_2217_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_2218_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_2219_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_2220_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_2221_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_2222_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_2223_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_2224_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_2225_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_2226_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_2227_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_2228_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_2229_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_2230_power__increasing,axiom,
    ! [N: nat,N3: nat,A2: real] :
      ( ( ord_less_eq_nat @ N @ N3 )
     => ( ( ord_less_eq_real @ one_one_real @ A2 )
       => ( ord_less_eq_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ A2 @ N3 ) ) ) ) ).

% power_increasing
thf(fact_2231_power__increasing,axiom,
    ! [N: nat,N3: nat,A2: code_integer] :
      ( ( ord_less_eq_nat @ N @ N3 )
     => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ) ).

% power_increasing
thf(fact_2232_power__increasing,axiom,
    ! [N: nat,N3: nat,A2: rat] :
      ( ( ord_less_eq_nat @ N @ N3 )
     => ( ( ord_less_eq_rat @ one_one_rat @ A2 )
       => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ A2 @ N3 ) ) ) ) ).

% power_increasing
thf(fact_2233_power__increasing,axiom,
    ! [N: nat,N3: nat,A2: nat] :
      ( ( ord_less_eq_nat @ N @ N3 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A2 )
       => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ A2 @ N3 ) ) ) ) ).

% power_increasing
thf(fact_2234_power__increasing,axiom,
    ! [N: nat,N3: nat,A2: int] :
      ( ( ord_less_eq_nat @ N @ N3 )
     => ( ( ord_less_eq_int @ one_one_int @ A2 )
       => ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ A2 @ N3 ) ) ) ) ).

% power_increasing
thf(fact_2235_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_2236_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_2237_int__zle__neg,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
      = ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
thf(fact_2238_nonpos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ~ ! [N2: nat] :
            ( K
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% nonpos_int_cases
thf(fact_2239_negative__zle__0,axiom,
    ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).

% negative_zle_0
thf(fact_2240_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_2241_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_2242_max__word__wrap,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( ( plus_p361126936061061375l_num1 @ X @ one_on7727431528512463931l_num1 )
        = zero_z3563351764282998399l_num1 )
     => ( X
        = ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ) ).

% max_word_wrap
thf(fact_2243_less__x__plus__1,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( X
       != ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
     => ( ( ord_le750835935415966154l_num1 @ Y @ ( plus_p361126936061061375l_num1 @ X @ one_on7727431528512463931l_num1 ) )
        = ( ( ord_le750835935415966154l_num1 @ Y @ X )
          | ( Y = X ) ) ) ) ).

% less_x_plus_1
thf(fact_2244_word__add__no__overflow,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ X @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
     => ( ord_le750835935415966154l_num1 @ X @ ( plus_p361126936061061375l_num1 @ X @ one_on7727431528512463931l_num1 ) ) ) ).

% word_add_no_overflow
thf(fact_2245_word__plus__one__nonzero,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ X @ ( plus_p361126936061061375l_num1 @ X @ Y ) )
     => ( ( Y != zero_z3563351764282998399l_num1 )
       => ( ( plus_p361126936061061375l_num1 @ X @ one_on7727431528512463931l_num1 )
         != zero_z3563351764282998399l_num1 ) ) ) ).

% word_plus_one_nonzero
thf(fact_2246_inc__i,axiom,
    ! [I: word_N3645301735248828278l_num1,M: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ one_on7727431528512463931l_num1 @ I )
     => ( ( ord_le750835935415966154l_num1 @ I @ M )
       => ( ( ord_le3335648743751981014l_num1 @ one_on7727431528512463931l_num1 @ ( plus_p361126936061061375l_num1 @ I @ one_on7727431528512463931l_num1 ) )
          & ( ord_le3335648743751981014l_num1 @ ( plus_p361126936061061375l_num1 @ I @ one_on7727431528512463931l_num1 ) @ M ) ) ) ) ).

% inc_i
thf(fact_2247_inc__le,axiom,
    ! [I: word_N3645301735248828278l_num1,M: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ I @ M )
     => ( ord_le3335648743751981014l_num1 @ ( plus_p361126936061061375l_num1 @ I @ one_on7727431528512463931l_num1 ) @ M ) ) ).

% inc_le
thf(fact_2248_word__le__plus__1,axiom,
    ! [Y: word_N3645301735248828278l_num1,N: word_N3645301735248828278l_num1,A2: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ Y @ ( plus_p361126936061061375l_num1 @ Y @ N ) )
     => ( ( ord_le750835935415966154l_num1 @ A2 @ N )
       => ( ord_le3335648743751981014l_num1 @ ( plus_p361126936061061375l_num1 @ Y @ A2 ) @ ( plus_p361126936061061375l_num1 @ ( plus_p361126936061061375l_num1 @ Y @ A2 ) @ one_on7727431528512463931l_num1 ) ) ) ) ).

% word_le_plus_1
thf(fact_2249_plus__one__helper,axiom,
    ! [X: word_N3645301735248828278l_num1,N: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ X @ ( plus_p361126936061061375l_num1 @ N @ one_on7727431528512463931l_num1 ) )
     => ( ord_le3335648743751981014l_num1 @ X @ N ) ) ).

% plus_one_helper
thf(fact_2250_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_2251_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_2252_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_2253_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_2254_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_2255_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_2256_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_2257_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_2258_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_2259_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
     != ( numera6690914467698888265omplex @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_2260_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
     != ( numeral_numeral_real @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_2261_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
     != ( numeral_numeral_rat @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_2262_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
     != ( numeral_numeral_int @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_2263_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numera6690914467698888265omplex @ M )
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_2264_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_real @ M )
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_2265_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_rat @ M )
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_2266_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_int @ M )
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_2267_verit__sum__simplify,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ A2 @ zero_z3563351764282998399l_num1 )
      = A2 ) ).

% verit_sum_simplify
thf(fact_2268_verit__sum__simplify,axiom,
    ! [A2: real] :
      ( ( plus_plus_real @ A2 @ zero_zero_real )
      = A2 ) ).

% verit_sum_simplify
thf(fact_2269_verit__sum__simplify,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ A2 @ zero_zero_rat )
      = A2 ) ).

% verit_sum_simplify
thf(fact_2270_verit__sum__simplify,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% verit_sum_simplify
thf(fact_2271_verit__sum__simplify,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% verit_sum_simplify
thf(fact_2272_add_Ogroup__left__neutral,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ zero_z3563351764282998399l_num1 @ A2 )
      = A2 ) ).

% add.group_left_neutral
thf(fact_2273_add_Ogroup__left__neutral,axiom,
    ! [A2: real] :
      ( ( plus_plus_real @ zero_zero_real @ A2 )
      = A2 ) ).

% add.group_left_neutral
thf(fact_2274_add_Ogroup__left__neutral,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A2 )
      = A2 ) ).

% add.group_left_neutral
thf(fact_2275_add_Ogroup__left__neutral,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ zero_zero_int @ A2 )
      = A2 ) ).

% add.group_left_neutral
thf(fact_2276_add_Ocomm__neutral,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ A2 @ zero_z3563351764282998399l_num1 )
      = A2 ) ).

% add.comm_neutral
thf(fact_2277_add_Ocomm__neutral,axiom,
    ! [A2: real] :
      ( ( plus_plus_real @ A2 @ zero_zero_real )
      = A2 ) ).

% add.comm_neutral
thf(fact_2278_add_Ocomm__neutral,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ A2 @ zero_zero_rat )
      = A2 ) ).

% add.comm_neutral
thf(fact_2279_add_Ocomm__neutral,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% add.comm_neutral
thf(fact_2280_add_Ocomm__neutral,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% add.comm_neutral
thf(fact_2281_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ zero_z3563351764282998399l_num1 @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_2282_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_2283_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_2284_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_2285_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_2286_one__neq__neg__one,axiom,
    ( one_one_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% one_neq_neg_one
thf(fact_2287_one__neq__neg__one,axiom,
    ( one_one_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% one_neq_neg_one
thf(fact_2288_one__neq__neg__one,axiom,
    ( one_one_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% one_neq_neg_one
thf(fact_2289_one__neq__neg__one,axiom,
    ( one_one_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% one_neq_neg_one
thf(fact_2290_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_2291_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_2292_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_2293_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_2294_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_2295_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_2296_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_2297_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_2298_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( K = L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_2299_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( K = L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_2300_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_2301_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( K = L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_2302_add__strict__mono,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).

% add_strict_mono
thf(fact_2303_add__strict__mono,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).

% add_strict_mono
thf(fact_2304_add__strict__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).

% add_strict_mono
thf(fact_2305_add__strict__mono,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).

% add_strict_mono
thf(fact_2306_add__strict__left__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ord_less_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_2307_add__strict__left__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_2308_add__strict__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_2309_add__strict__left__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_2310_add__strict__right__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_2311_add__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B2 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_2312_add__strict__right__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_2313_add__strict__right__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_2314_add__less__imp__less__left,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B2 ) )
     => ( ord_less_real @ A2 @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_2315_add__less__imp__less__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B2 ) )
     => ( ord_less_rat @ A2 @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_2316_add__less__imp__less__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
     => ( ord_less_nat @ A2 @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_2317_add__less__imp__less__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
     => ( ord_less_int @ A2 @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_2318_add__less__imp__less__right,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ C ) )
     => ( ord_less_real @ A2 @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_2319_add__less__imp__less__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B2 @ C ) )
     => ( ord_less_rat @ A2 @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_2320_add__less__imp__less__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
     => ( ord_less_nat @ A2 @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_2321_add__less__imp__less__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
     => ( ord_less_int @ A2 @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_2322_add__One__commute,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ N )
      = ( plus_plus_num @ N @ one ) ) ).

% add_One_commute
thf(fact_2323_le__numeral__extra_I3_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% le_numeral_extra(3)
thf(fact_2324_le__numeral__extra_I3_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).

% le_numeral_extra(3)
thf(fact_2325_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_2326_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_2327_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_2328_verit__comp__simplify1_I3_J,axiom,
    ! [B7: real,A6: real] :
      ( ( ~ ( ord_less_eq_real @ B7 @ A6 ) )
      = ( ord_less_real @ A6 @ B7 ) ) ).

% verit_comp_simplify1(3)
thf(fact_2329_verit__comp__simplify1_I3_J,axiom,
    ! [B7: rat,A6: rat] :
      ( ( ~ ( ord_less_eq_rat @ B7 @ A6 ) )
      = ( ord_less_rat @ A6 @ B7 ) ) ).

% verit_comp_simplify1(3)
thf(fact_2330_verit__comp__simplify1_I3_J,axiom,
    ! [B7: num,A6: num] :
      ( ( ~ ( ord_less_eq_num @ B7 @ A6 ) )
      = ( ord_less_num @ A6 @ B7 ) ) ).

% verit_comp_simplify1(3)
thf(fact_2331_verit__comp__simplify1_I3_J,axiom,
    ! [B7: nat,A6: nat] :
      ( ( ~ ( ord_less_eq_nat @ B7 @ A6 ) )
      = ( ord_less_nat @ A6 @ B7 ) ) ).

% verit_comp_simplify1(3)
thf(fact_2332_verit__comp__simplify1_I3_J,axiom,
    ! [B7: int,A6: int] :
      ( ( ~ ( ord_less_eq_int @ B7 @ A6 ) )
      = ( ord_less_int @ A6 @ B7 ) ) ).

% verit_comp_simplify1(3)
thf(fact_2333_leD,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ Y @ X )
     => ~ ( ord_less_real @ X @ Y ) ) ).

% leD
thf(fact_2334_leD,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ~ ( ord_less_set_nat @ X @ Y ) ) ).

% leD
thf(fact_2335_leD,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ~ ( ord_less_rat @ X @ Y ) ) ).

% leD
thf(fact_2336_leD,axiom,
    ! [Y: num,X: num] :
      ( ( ord_less_eq_num @ Y @ X )
     => ~ ( ord_less_num @ X @ Y ) ) ).

% leD
thf(fact_2337_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_2338_leD,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ~ ( ord_less_int @ X @ Y ) ) ).

% leD
thf(fact_2339_leI,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ord_less_eq_real @ Y @ X ) ) ).

% leI
thf(fact_2340_leI,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ord_less_eq_rat @ Y @ X ) ) ).

% leI
thf(fact_2341_leI,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ord_less_eq_num @ Y @ X ) ) ).

% leI
thf(fact_2342_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_2343_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% leI
thf(fact_2344_nless__le,axiom,
    ! [A2: real,B2: real] :
      ( ( ~ ( ord_less_real @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_real @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_2345_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_2346_nless__le,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ~ ( ord_less_rat @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_rat @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_2347_nless__le,axiom,
    ! [A2: num,B2: num] :
      ( ( ~ ( ord_less_num @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_num @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_2348_nless__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ~ ( ord_less_nat @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_nat @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_2349_nless__le,axiom,
    ! [A2: int,B2: int] :
      ( ( ~ ( ord_less_int @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_int @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_2350_antisym__conv1,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ( ord_less_eq_real @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2351_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_2352_antisym__conv1,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2353_antisym__conv1,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ( ord_less_eq_num @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2354_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2355_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2356_antisym__conv2,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ~ ( ord_less_real @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2357_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_2358_antisym__conv2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ~ ( ord_less_rat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2359_antisym__conv2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ~ ( ord_less_num @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2360_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2361_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2362_dense__ge,axiom,
    ! [Z: real,Y: real] :
      ( ! [X3: real] :
          ( ( ord_less_real @ Z @ X3 )
         => ( ord_less_eq_real @ Y @ X3 ) )
     => ( ord_less_eq_real @ Y @ Z ) ) ).

% dense_ge
thf(fact_2363_dense__ge,axiom,
    ! [Z: rat,Y: rat] :
      ( ! [X3: rat] :
          ( ( ord_less_rat @ Z @ X3 )
         => ( ord_less_eq_rat @ Y @ X3 ) )
     => ( ord_less_eq_rat @ Y @ Z ) ) ).

% dense_ge
thf(fact_2364_dense__le,axiom,
    ! [Y: real,Z: real] :
      ( ! [X3: real] :
          ( ( ord_less_real @ X3 @ Y )
         => ( ord_less_eq_real @ X3 @ Z ) )
     => ( ord_less_eq_real @ Y @ Z ) ) ).

% dense_le
thf(fact_2365_dense__le,axiom,
    ! [Y: rat,Z: rat] :
      ( ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Y )
         => ( ord_less_eq_rat @ X3 @ Z ) )
     => ( ord_less_eq_rat @ Y @ Z ) ) ).

% dense_le
thf(fact_2366_less__le__not__le,axiom,
    ( ord_less_real
    = ( ^ [X2: real,Y2: real] :
          ( ( ord_less_eq_real @ X2 @ Y2 )
          & ~ ( ord_less_eq_real @ Y2 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_2367_less__le__not__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X2: set_nat,Y2: set_nat] :
          ( ( ord_less_eq_set_nat @ X2 @ Y2 )
          & ~ ( ord_less_eq_set_nat @ Y2 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_2368_less__le__not__le,axiom,
    ( ord_less_rat
    = ( ^ [X2: rat,Y2: rat] :
          ( ( ord_less_eq_rat @ X2 @ Y2 )
          & ~ ( ord_less_eq_rat @ Y2 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_2369_less__le__not__le,axiom,
    ( ord_less_num
    = ( ^ [X2: num,Y2: num] :
          ( ( ord_less_eq_num @ X2 @ Y2 )
          & ~ ( ord_less_eq_num @ Y2 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_2370_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X2: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y2 )
          & ~ ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_2371_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X2: int,Y2: int] :
          ( ( ord_less_eq_int @ X2 @ Y2 )
          & ~ ( ord_less_eq_int @ Y2 @ X2 ) ) ) ) ).

% less_le_not_le
thf(fact_2372_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_2373_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_2374_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_2375_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_2376_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_2377_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [A4: real,B4: real] :
          ( ( ord_less_real @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2378_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A4: set_nat,B4: set_nat] :
          ( ( ord_less_set_nat @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2379_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [A4: rat,B4: rat] :
          ( ( ord_less_rat @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2380_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [A4: num,B4: num] :
          ( ( ord_less_num @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2381_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_nat @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2382_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A4: int,B4: int] :
          ( ( ord_less_int @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2383_order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [A4: real,B4: real] :
          ( ( ord_less_eq_real @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2384_order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [A4: set_nat,B4: set_nat] :
          ( ( ord_less_eq_set_nat @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2385_order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [A4: rat,B4: rat] :
          ( ( ord_less_eq_rat @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2386_order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [A4: num,B4: num] :
          ( ( ord_less_eq_num @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2387_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2388_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A4: int,B4: int] :
          ( ( ord_less_eq_int @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2389_order_Ostrict__trans1,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_real @ B2 @ C )
       => ( ord_less_real @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_2390_order_Ostrict__trans1,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_set_nat @ B2 @ C )
       => ( ord_less_set_nat @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_2391_order_Ostrict__trans1,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ord_less_rat @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_2392_order_Ostrict__trans1,axiom,
    ! [A2: num,B2: num,C: num] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ord_less_num @ B2 @ C )
       => ( ord_less_num @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_2393_order_Ostrict__trans1,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_2394_order_Ostrict__trans1,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_int @ B2 @ C )
       => ( ord_less_int @ A2 @ C ) ) ) ).

% order.strict_trans1
thf(fact_2395_order_Ostrict__trans2,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ord_less_real @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_2396_order_Ostrict__trans2,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C )
       => ( ord_less_set_nat @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_2397_order_Ostrict__trans2,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ord_less_rat @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_2398_order_Ostrict__trans2,axiom,
    ! [A2: num,B2: num,C: num] :
      ( ( ord_less_num @ A2 @ B2 )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ord_less_num @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_2399_order_Ostrict__trans2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_2400_order_Ostrict__trans2,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ord_less_int @ A2 @ C ) ) ) ).

% order.strict_trans2
thf(fact_2401_order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [A4: real,B4: real] :
          ( ( ord_less_eq_real @ A4 @ B4 )
          & ~ ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_2402_order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [A4: set_nat,B4: set_nat] :
          ( ( ord_less_eq_set_nat @ A4 @ B4 )
          & ~ ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_2403_order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [A4: rat,B4: rat] :
          ( ( ord_less_eq_rat @ A4 @ B4 )
          & ~ ( ord_less_eq_rat @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_2404_order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [A4: num,B4: num] :
          ( ( ord_less_eq_num @ A4 @ B4 )
          & ~ ( ord_less_eq_num @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_2405_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A4 @ B4 )
          & ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_2406_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A4: int,B4: int] :
          ( ( ord_less_eq_int @ A4 @ B4 )
          & ~ ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_2407_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_2408_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_2409_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_2410_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_2411_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [B4: real,A4: real] :
          ( ( ord_less_real @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_2412_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B4: set_nat,A4: set_nat] :
          ( ( ord_less_set_nat @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_2413_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A4: rat] :
          ( ( ord_less_rat @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_2414_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [B4: num,A4: num] :
          ( ( ord_less_num @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_2415_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_less_nat @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_2416_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A4: int] :
          ( ( ord_less_int @ B4 @ A4 )
          | ( A4 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_2417_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [B4: real,A4: real] :
          ( ( ord_less_eq_real @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_2418_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [B4: set_nat,A4: set_nat] :
          ( ( ord_less_eq_set_nat @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_2419_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [B4: rat,A4: rat] :
          ( ( ord_less_eq_rat @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_2420_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [B4: num,A4: num] :
          ( ( ord_less_eq_num @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_2421_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_2422_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B4: int,A4: int] :
          ( ( ord_less_eq_int @ B4 @ A4 )
          & ( A4 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_2423_dual__order_Ostrict__trans1,axiom,
    ! [B2: real,A2: real,C: real] :
      ( ( ord_less_eq_real @ B2 @ A2 )
     => ( ( ord_less_real @ C @ B2 )
       => ( ord_less_real @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_2424_dual__order_Ostrict__trans1,axiom,
    ! [B2: set_nat,A2: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( ord_less_set_nat @ C @ B2 )
       => ( ord_less_set_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_2425_dual__order_Ostrict__trans1,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ( ord_less_rat @ C @ B2 )
       => ( ord_less_rat @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_2426_dual__order_Ostrict__trans1,axiom,
    ! [B2: num,A2: num,C: num] :
      ( ( ord_less_eq_num @ B2 @ A2 )
     => ( ( ord_less_num @ C @ B2 )
       => ( ord_less_num @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_2427_dual__order_Ostrict__trans1,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_2428_dual__order_Ostrict__trans1,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( ord_less_int @ C @ B2 )
       => ( ord_less_int @ C @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_2429_dual__order_Ostrict__trans2,axiom,
    ! [B2: real,A2: real,C: real] :
      ( ( ord_less_real @ B2 @ A2 )
     => ( ( ord_less_eq_real @ C @ B2 )
       => ( ord_less_real @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_2430_dual__order_Ostrict__trans2,axiom,
    ! [B2: set_nat,A2: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ B2 @ A2 )
     => ( ( ord_less_eq_set_nat @ C @ B2 )
       => ( ord_less_set_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_2431_dual__order_Ostrict__trans2,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ( ( ord_less_eq_rat @ C @ B2 )
       => ( ord_less_rat @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_2432_dual__order_Ostrict__trans2,axiom,
    ! [B2: num,A2: num,C: num] :
      ( ( ord_less_num @ B2 @ A2 )
     => ( ( ord_less_eq_num @ C @ B2 )
       => ( ord_less_num @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_2433_dual__order_Ostrict__trans2,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_2434_dual__order_Ostrict__trans2,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( ord_less_eq_int @ C @ B2 )
       => ( ord_less_int @ C @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_2435_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [B4: real,A4: real] :
          ( ( ord_less_eq_real @ B4 @ A4 )
          & ~ ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_2436_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [B4: set_nat,A4: set_nat] :
          ( ( ord_less_eq_set_nat @ B4 @ A4 )
          & ~ ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_2437_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [B4: rat,A4: rat] :
          ( ( ord_less_eq_rat @ B4 @ A4 )
          & ~ ( ord_less_eq_rat @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_2438_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [B4: num,A4: num] :
          ( ( ord_less_eq_num @ B4 @ A4 )
          & ~ ( ord_less_eq_num @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_2439_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A4 )
          & ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_2440_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B4: int,A4: int] :
          ( ( ord_less_eq_int @ B4 @ A4 )
          & ~ ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_2441_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_2442_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_2443_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_2444_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_2445_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_2446_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_2447_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_2448_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_2449_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_2450_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_2451_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_2452_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_2453_order__le__less,axiom,
    ( ord_less_eq_real
    = ( ^ [X2: real,Y2: real] :
          ( ( ord_less_real @ X2 @ Y2 )
          | ( X2 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_2454_order__le__less,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X2: set_nat,Y2: set_nat] :
          ( ( ord_less_set_nat @ X2 @ Y2 )
          | ( X2 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_2455_order__le__less,axiom,
    ( ord_less_eq_rat
    = ( ^ [X2: rat,Y2: rat] :
          ( ( ord_less_rat @ X2 @ Y2 )
          | ( X2 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_2456_order__le__less,axiom,
    ( ord_less_eq_num
    = ( ^ [X2: num,Y2: num] :
          ( ( ord_less_num @ X2 @ Y2 )
          | ( X2 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_2457_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X2: nat,Y2: nat] :
          ( ( ord_less_nat @ X2 @ Y2 )
          | ( X2 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_2458_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X2: int,Y2: int] :
          ( ( ord_less_int @ X2 @ Y2 )
          | ( X2 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_2459_order__less__le,axiom,
    ( ord_less_real
    = ( ^ [X2: real,Y2: real] :
          ( ( ord_less_eq_real @ X2 @ Y2 )
          & ( X2 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_2460_order__less__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X2: set_nat,Y2: set_nat] :
          ( ( ord_less_eq_set_nat @ X2 @ Y2 )
          & ( X2 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_2461_order__less__le,axiom,
    ( ord_less_rat
    = ( ^ [X2: rat,Y2: rat] :
          ( ( ord_less_eq_rat @ X2 @ Y2 )
          & ( X2 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_2462_order__less__le,axiom,
    ( ord_less_num
    = ( ^ [X2: num,Y2: num] :
          ( ( ord_less_eq_num @ X2 @ Y2 )
          & ( X2 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_2463_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X2: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y2 )
          & ( X2 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_2464_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X2: int,Y2: int] :
          ( ( ord_less_eq_int @ X2 @ Y2 )
          & ( X2 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_2465_linorder__not__le,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_eq_real @ X @ Y ) )
      = ( ord_less_real @ Y @ X ) ) ).

% linorder_not_le
thf(fact_2466_linorder__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_eq_rat @ X @ Y ) )
      = ( ord_less_rat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_2467_linorder__not__le,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_eq_num @ X @ Y ) )
      = ( ord_less_num @ Y @ X ) ) ).

% linorder_not_le
thf(fact_2468_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_2469_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_eq_int @ X @ Y ) )
      = ( ord_less_int @ Y @ X ) ) ).

% linorder_not_le
thf(fact_2470_linorder__not__less,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_real @ X @ Y ) )
      = ( ord_less_eq_real @ Y @ X ) ) ).

% linorder_not_less
thf(fact_2471_linorder__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_rat @ X @ Y ) )
      = ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_2472_linorder__not__less,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_num @ X @ Y ) )
      = ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_not_less
thf(fact_2473_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_2474_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_not_less
thf(fact_2475_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_2476_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_2477_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_2478_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_2479_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_2480_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_2481_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_2482_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_2483_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_2484_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_2485_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_2486_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_2487_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_2488_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_2489_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_2490_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_2491_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_2492_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_2493_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_2494_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_2495_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_2496_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_2497_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_2498_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_2499_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_2500_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_2501_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_2502_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_2503_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_2504_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_2505_order__le__less__subst1,axiom,
    ! [A2: real,F: real > real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2506_order__le__less__subst1,axiom,
    ! [A2: real,F: rat > real,B2: rat,C: rat] :
      ( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2507_order__le__less__subst1,axiom,
    ! [A2: real,F: num > real,B2: num,C: num] :
      ( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_num @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2508_order__le__less__subst1,axiom,
    ! [A2: real,F: nat > real,B2: nat,C: nat] :
      ( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2509_order__le__less__subst1,axiom,
    ! [A2: real,F: int > real,B2: int,C: int] :
      ( ( ord_less_eq_real @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_int @ B2 @ C )
       => ( ! [X3: int,Y4: int] :
              ( ( ord_less_int @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2510_order__le__less__subst1,axiom,
    ! [A2: rat,F: real > rat,B2: real,C: real] :
      ( ( ord_less_eq_rat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2511_order__le__less__subst1,axiom,
    ! [A2: rat,F: rat > rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2512_order__le__less__subst1,axiom,
    ! [A2: rat,F: num > rat,B2: num,C: num] :
      ( ( ord_less_eq_rat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_num @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2513_order__le__less__subst1,axiom,
    ! [A2: rat,F: nat > rat,B2: nat,C: nat] :
      ( ( ord_less_eq_rat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2514_order__le__less__subst1,axiom,
    ! [A2: rat,F: int > rat,B2: int,C: int] :
      ( ( ord_less_eq_rat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_int @ B2 @ C )
       => ( ! [X3: int,Y4: int] :
              ( ( ord_less_int @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2515_order__le__less__subst2,axiom,
    ! [A2: rat,B2: rat,F: rat > real,C: real] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_real @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_2516_order__le__less__subst2,axiom,
    ! [A2: rat,B2: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_2517_order__le__less__subst2,axiom,
    ! [A2: rat,B2: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_num @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_2518_order__le__less__subst2,axiom,
    ! [A2: rat,B2: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_2519_order__le__less__subst2,axiom,
    ! [A2: rat,B2: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_int @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_2520_order__le__less__subst2,axiom,
    ! [A2: num,B2: num,F: num > real,C: real] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ord_less_real @ ( F @ B2 ) @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_2521_order__le__less__subst2,axiom,
    ! [A2: num,B2: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ord_less_rat @ ( F @ B2 ) @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_2522_order__le__less__subst2,axiom,
    ! [A2: num,B2: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ord_less_num @ ( F @ B2 ) @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_2523_order__le__less__subst2,axiom,
    ! [A2: num,B2: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_2524_order__le__less__subst2,axiom,
    ! [A2: num,B2: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ord_less_int @ ( F @ B2 ) @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_2525_order__less__le__subst1,axiom,
    ! [A2: real,F: rat > real,B2: rat,C: rat] :
      ( ( ord_less_real @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2526_order__less__le__subst1,axiom,
    ! [A2: rat,F: rat > rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2527_order__less__le__subst1,axiom,
    ! [A2: num,F: rat > num,B2: rat,C: rat] :
      ( ( ord_less_num @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2528_order__less__le__subst1,axiom,
    ! [A2: nat,F: rat > nat,B2: rat,C: rat] :
      ( ( ord_less_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2529_order__less__le__subst1,axiom,
    ! [A2: int,F: rat > int,B2: rat,C: rat] :
      ( ( ord_less_int @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2530_order__less__le__subst1,axiom,
    ! [A2: real,F: num > real,B2: num,C: num] :
      ( ( ord_less_real @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2531_order__less__le__subst1,axiom,
    ! [A2: rat,F: num > rat,B2: num,C: num] :
      ( ( ord_less_rat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2532_order__less__le__subst1,axiom,
    ! [A2: num,F: num > num,B2: num,C: num] :
      ( ( ord_less_num @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2533_order__less__le__subst1,axiom,
    ! [A2: nat,F: num > nat,B2: num,C: num] :
      ( ( ord_less_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2534_order__less__le__subst1,axiom,
    ! [A2: int,F: num > int,B2: num,C: num] :
      ( ( ord_less_int @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_eq_num @ X3 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2535_order__less__le__subst2,axiom,
    ! [A2: real,B2: real,F: real > real,C: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_2536_order__less__le__subst2,axiom,
    ! [A2: rat,B2: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_2537_order__less__le__subst2,axiom,
    ! [A2: num,B2: num,F: num > real,C: real] :
      ( ( ord_less_num @ A2 @ B2 )
     => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_num @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_2538_order__less__le__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > real,C: real] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_2539_order__less__le__subst2,axiom,
    ! [A2: int,B2: int,F: int > real,C: real] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
       => ( ! [X3: int,Y4: int] :
              ( ( ord_less_int @ X3 @ Y4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_2540_order__less__le__subst2,axiom,
    ! [A2: real,B2: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
       => ( ! [X3: real,Y4: real] :
              ( ( ord_less_real @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_2541_order__less__le__subst2,axiom,
    ! [A2: rat,B2: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
       => ( ! [X3: rat,Y4: rat] :
              ( ( ord_less_rat @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_2542_order__less__le__subst2,axiom,
    ! [A2: num,B2: num,F: num > rat,C: rat] :
      ( ( ord_less_num @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
       => ( ! [X3: num,Y4: num] :
              ( ( ord_less_num @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_2543_order__less__le__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > rat,C: rat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_2544_order__less__le__subst2,axiom,
    ! [A2: int,B2: int,F: int > rat,C: rat] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
       => ( ! [X3: int,Y4: int] :
              ( ( ord_less_int @ X3 @ Y4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_2545_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_2546_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_2547_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_2548_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_2549_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_2550_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_2551_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_2552_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_2553_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_2554_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_2555_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_2556_le__num__One__iff,axiom,
    ! [X: num] :
      ( ( ord_less_eq_num @ X @ one )
      = ( X = one ) ) ).

% le_num_One_iff
thf(fact_2557_le__numeral__extra_I4_J,axiom,
    ord_less_eq_real @ one_one_real @ one_one_real ).

% le_numeral_extra(4)
thf(fact_2558_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_2559_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_2560_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_2561_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_2562_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_2563_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_2564_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_2565_bot_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
     => ( A2 = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_2566_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_2567_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_2568_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_2569_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_2570_bot_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
      = ( A2 = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_2571_bot_Oextremum,axiom,
    ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).

% bot.extremum
thf(fact_2572_bot_Oextremum,axiom,
    ! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).

% bot.extremum
thf(fact_2573_bot_Oextremum,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).

% bot.extremum
thf(fact_2574_bot_Oextremum,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).

% bot.extremum
thf(fact_2575_bot_Oextremum,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).

% bot.extremum
thf(fact_2576_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_2577_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_2578_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
     => ( ord_less_nat @ I @ K ) ) ).

% add_lessD1
thf(fact_2579_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ K @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_2580_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_2581_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_2582_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_less_mono1
thf(fact_2583_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_less_add1
thf(fact_2584_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_less_add2
thf(fact_2585_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M @ L )
          = ( plus_plus_nat @ K @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% less_add_eq_less
thf(fact_2586_uminus__int__code_I1_J,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% uminus_int_code(1)
thf(fact_2587_plus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( plus_plus_int @ K @ zero_zero_int )
      = K ) ).

% plus_int_code(1)
thf(fact_2588_plus__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( plus_plus_int @ zero_zero_int @ L )
      = L ) ).

% plus_int_code(2)
thf(fact_2589_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_2590_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_2591_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_2592_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_2593_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N4: nat] :
          ( ( ord_less_eq_nat @ M3 @ N4 )
          & ( M3 != N4 ) ) ) ) ).

% nat_less_le
thf(fact_2594_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_imp_le_nat
thf(fact_2595_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N4: nat] :
          ( ( ord_less_nat @ M3 @ N4 )
          | ( M3 = N4 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_2596_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_nat @ M @ N )
        | ( M = N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_or_eq_imp_le
thf(fact_2597_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( M != N )
       => ( ord_less_nat @ M @ N ) ) ) ).

% le_neq_implies_less
thf(fact_2598_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( ( ord_less_nat @ I3 @ J2 )
         => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_2599_real__arch__simple,axiom,
    ! [X: real] :
    ? [N2: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).

% real_arch_simple
thf(fact_2600_real__arch__simple,axiom,
    ! [X: rat] :
    ? [N2: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N2 ) ) ).

% real_arch_simple
thf(fact_2601_inf__sup__ord_I4_J,axiom,
    ! [Y: extended_enat,X: extended_enat] : ( ord_le2932123472753598470d_enat @ Y @ ( sup_su3973961784419623482d_enat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_2602_inf__sup__ord_I4_J,axiom,
    ! [Y: assn,X: assn] : ( ord_less_eq_assn @ Y @ ( sup_sup_assn @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_2603_inf__sup__ord_I4_J,axiom,
    ! [Y: set_complex,X: set_complex] : ( ord_le211207098394363844omplex @ Y @ ( sup_sup_set_complex @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_2604_inf__sup__ord_I4_J,axiom,
    ! [Y: set_int,X: set_int] : ( ord_less_eq_set_int @ Y @ ( sup_sup_set_int @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_2605_inf__sup__ord_I4_J,axiom,
    ! [Y: set_real,X: set_real] : ( ord_less_eq_set_real @ Y @ ( sup_sup_set_real @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_2606_inf__sup__ord_I4_J,axiom,
    ! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_2607_inf__sup__ord_I4_J,axiom,
    ! [Y: rat,X: rat] : ( ord_less_eq_rat @ Y @ ( sup_sup_rat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_2608_inf__sup__ord_I4_J,axiom,
    ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_2609_inf__sup__ord_I4_J,axiom,
    ! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_2610_inf__sup__ord_I3_J,axiom,
    ! [X: extended_enat,Y: extended_enat] : ( ord_le2932123472753598470d_enat @ X @ ( sup_su3973961784419623482d_enat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_2611_inf__sup__ord_I3_J,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ X @ ( sup_sup_assn @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_2612_inf__sup__ord_I3_J,axiom,
    ! [X: set_complex,Y: set_complex] : ( ord_le211207098394363844omplex @ X @ ( sup_sup_set_complex @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_2613_inf__sup__ord_I3_J,axiom,
    ! [X: set_int,Y: set_int] : ( ord_less_eq_set_int @ X @ ( sup_sup_set_int @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_2614_inf__sup__ord_I3_J,axiom,
    ! [X: set_real,Y: set_real] : ( ord_less_eq_set_real @ X @ ( sup_sup_set_real @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_2615_inf__sup__ord_I3_J,axiom,
    ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_2616_inf__sup__ord_I3_J,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ X @ ( sup_sup_rat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_2617_inf__sup__ord_I3_J,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_2618_inf__sup__ord_I3_J,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_2619_le__supE,axiom,
    ! [A2: extended_enat,B2: extended_enat,X: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) @ X )
     => ~ ( ( ord_le2932123472753598470d_enat @ A2 @ X )
         => ~ ( ord_le2932123472753598470d_enat @ B2 @ X ) ) ) ).

% le_supE
thf(fact_2620_le__supE,axiom,
    ! [A2: assn,B2: assn,X: assn] :
      ( ( ord_less_eq_assn @ ( sup_sup_assn @ A2 @ B2 ) @ X )
     => ~ ( ( ord_less_eq_assn @ A2 @ X )
         => ~ ( ord_less_eq_assn @ B2 @ X ) ) ) ).

% le_supE
thf(fact_2621_le__supE,axiom,
    ! [A2: set_complex,B2: set_complex,X: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( sup_sup_set_complex @ A2 @ B2 ) @ X )
     => ~ ( ( ord_le211207098394363844omplex @ A2 @ X )
         => ~ ( ord_le211207098394363844omplex @ B2 @ X ) ) ) ).

% le_supE
thf(fact_2622_le__supE,axiom,
    ! [A2: set_int,B2: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ A2 @ B2 ) @ X )
     => ~ ( ( ord_less_eq_set_int @ A2 @ X )
         => ~ ( ord_less_eq_set_int @ B2 @ X ) ) ) ).

% le_supE
thf(fact_2623_le__supE,axiom,
    ! [A2: set_real,B2: set_real,X: set_real] :
      ( ( ord_less_eq_set_real @ ( sup_sup_set_real @ A2 @ B2 ) @ X )
     => ~ ( ( ord_less_eq_set_real @ A2 @ X )
         => ~ ( ord_less_eq_set_real @ B2 @ X ) ) ) ).

% le_supE
thf(fact_2624_le__supE,axiom,
    ! [A2: set_nat,B2: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X )
     => ~ ( ( ord_less_eq_set_nat @ A2 @ X )
         => ~ ( ord_less_eq_set_nat @ B2 @ X ) ) ) ).

% le_supE
thf(fact_2625_le__supE,axiom,
    ! [A2: rat,B2: rat,X: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ A2 @ B2 ) @ X )
     => ~ ( ( ord_less_eq_rat @ A2 @ X )
         => ~ ( ord_less_eq_rat @ B2 @ X ) ) ) ).

% le_supE
thf(fact_2626_le__supE,axiom,
    ! [A2: nat,B2: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X )
     => ~ ( ( ord_less_eq_nat @ A2 @ X )
         => ~ ( ord_less_eq_nat @ B2 @ X ) ) ) ).

% le_supE
thf(fact_2627_le__supE,axiom,
    ! [A2: int,B2: int,X: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B2 ) @ X )
     => ~ ( ( ord_less_eq_int @ A2 @ X )
         => ~ ( ord_less_eq_int @ B2 @ X ) ) ) ).

% le_supE
thf(fact_2628_le__supI,axiom,
    ! [A2: extended_enat,X: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ X )
     => ( ( ord_le2932123472753598470d_enat @ B2 @ X )
       => ( ord_le2932123472753598470d_enat @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_2629_le__supI,axiom,
    ! [A2: assn,X: assn,B2: assn] :
      ( ( ord_less_eq_assn @ A2 @ X )
     => ( ( ord_less_eq_assn @ B2 @ X )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_2630_le__supI,axiom,
    ! [A2: set_complex,X: set_complex,B2: set_complex] :
      ( ( ord_le211207098394363844omplex @ A2 @ X )
     => ( ( ord_le211207098394363844omplex @ B2 @ X )
       => ( ord_le211207098394363844omplex @ ( sup_sup_set_complex @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_2631_le__supI,axiom,
    ! [A2: set_int,X: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ X )
     => ( ( ord_less_eq_set_int @ B2 @ X )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_2632_le__supI,axiom,
    ! [A2: set_real,X: set_real,B2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ X )
     => ( ( ord_less_eq_set_real @ B2 @ X )
       => ( ord_less_eq_set_real @ ( sup_sup_set_real @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_2633_le__supI,axiom,
    ! [A2: set_nat,X: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ X )
     => ( ( ord_less_eq_set_nat @ B2 @ X )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_2634_le__supI,axiom,
    ! [A2: rat,X: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ X )
     => ( ( ord_less_eq_rat @ B2 @ X )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_2635_le__supI,axiom,
    ! [A2: nat,X: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ X )
     => ( ( ord_less_eq_nat @ B2 @ X )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_2636_le__supI,axiom,
    ! [A2: int,X: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ X )
     => ( ( ord_less_eq_int @ B2 @ X )
       => ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B2 ) @ X ) ) ) ).

% le_supI
thf(fact_2637_sup__ge1,axiom,
    ! [X: extended_enat,Y: extended_enat] : ( ord_le2932123472753598470d_enat @ X @ ( sup_su3973961784419623482d_enat @ X @ Y ) ) ).

% sup_ge1
thf(fact_2638_sup__ge1,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ X @ ( sup_sup_assn @ X @ Y ) ) ).

% sup_ge1
thf(fact_2639_sup__ge1,axiom,
    ! [X: set_complex,Y: set_complex] : ( ord_le211207098394363844omplex @ X @ ( sup_sup_set_complex @ X @ Y ) ) ).

% sup_ge1
thf(fact_2640_sup__ge1,axiom,
    ! [X: set_int,Y: set_int] : ( ord_less_eq_set_int @ X @ ( sup_sup_set_int @ X @ Y ) ) ).

% sup_ge1
thf(fact_2641_sup__ge1,axiom,
    ! [X: set_real,Y: set_real] : ( ord_less_eq_set_real @ X @ ( sup_sup_set_real @ X @ Y ) ) ).

% sup_ge1
thf(fact_2642_sup__ge1,axiom,
    ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_2643_sup__ge1,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ X @ ( sup_sup_rat @ X @ Y ) ) ).

% sup_ge1
thf(fact_2644_sup__ge1,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_2645_sup__ge1,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).

% sup_ge1
thf(fact_2646_sup__ge2,axiom,
    ! [Y: extended_enat,X: extended_enat] : ( ord_le2932123472753598470d_enat @ Y @ ( sup_su3973961784419623482d_enat @ X @ Y ) ) ).

% sup_ge2
thf(fact_2647_sup__ge2,axiom,
    ! [Y: assn,X: assn] : ( ord_less_eq_assn @ Y @ ( sup_sup_assn @ X @ Y ) ) ).

% sup_ge2
thf(fact_2648_sup__ge2,axiom,
    ! [Y: set_complex,X: set_complex] : ( ord_le211207098394363844omplex @ Y @ ( sup_sup_set_complex @ X @ Y ) ) ).

% sup_ge2
thf(fact_2649_sup__ge2,axiom,
    ! [Y: set_int,X: set_int] : ( ord_less_eq_set_int @ Y @ ( sup_sup_set_int @ X @ Y ) ) ).

% sup_ge2
thf(fact_2650_sup__ge2,axiom,
    ! [Y: set_real,X: set_real] : ( ord_less_eq_set_real @ Y @ ( sup_sup_set_real @ X @ Y ) ) ).

% sup_ge2
thf(fact_2651_sup__ge2,axiom,
    ! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_2652_sup__ge2,axiom,
    ! [Y: rat,X: rat] : ( ord_less_eq_rat @ Y @ ( sup_sup_rat @ X @ Y ) ) ).

% sup_ge2
thf(fact_2653_sup__ge2,axiom,
    ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_2654_sup__ge2,axiom,
    ! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).

% sup_ge2
thf(fact_2655_le__supI1,axiom,
    ! [X: extended_enat,A2: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X @ A2 )
     => ( ord_le2932123472753598470d_enat @ X @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_2656_le__supI1,axiom,
    ! [X: assn,A2: assn,B2: assn] :
      ( ( ord_less_eq_assn @ X @ A2 )
     => ( ord_less_eq_assn @ X @ ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_2657_le__supI1,axiom,
    ! [X: set_complex,A2: set_complex,B2: set_complex] :
      ( ( ord_le211207098394363844omplex @ X @ A2 )
     => ( ord_le211207098394363844omplex @ X @ ( sup_sup_set_complex @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_2658_le__supI1,axiom,
    ! [X: set_int,A2: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ X @ A2 )
     => ( ord_less_eq_set_int @ X @ ( sup_sup_set_int @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_2659_le__supI1,axiom,
    ! [X: set_real,A2: set_real,B2: set_real] :
      ( ( ord_less_eq_set_real @ X @ A2 )
     => ( ord_less_eq_set_real @ X @ ( sup_sup_set_real @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_2660_le__supI1,axiom,
    ! [X: set_nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ A2 )
     => ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_2661_le__supI1,axiom,
    ! [X: rat,A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ X @ A2 )
     => ( ord_less_eq_rat @ X @ ( sup_sup_rat @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_2662_le__supI1,axiom,
    ! [X: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ X @ A2 )
     => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_2663_le__supI1,axiom,
    ! [X: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ X @ A2 )
     => ( ord_less_eq_int @ X @ ( sup_sup_int @ A2 @ B2 ) ) ) ).

% le_supI1
thf(fact_2664_le__supI2,axiom,
    ! [X: extended_enat,B2: extended_enat,A2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X @ B2 )
     => ( ord_le2932123472753598470d_enat @ X @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_2665_le__supI2,axiom,
    ! [X: assn,B2: assn,A2: assn] :
      ( ( ord_less_eq_assn @ X @ B2 )
     => ( ord_less_eq_assn @ X @ ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_2666_le__supI2,axiom,
    ! [X: set_complex,B2: set_complex,A2: set_complex] :
      ( ( ord_le211207098394363844omplex @ X @ B2 )
     => ( ord_le211207098394363844omplex @ X @ ( sup_sup_set_complex @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_2667_le__supI2,axiom,
    ! [X: set_int,B2: set_int,A2: set_int] :
      ( ( ord_less_eq_set_int @ X @ B2 )
     => ( ord_less_eq_set_int @ X @ ( sup_sup_set_int @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_2668_le__supI2,axiom,
    ! [X: set_real,B2: set_real,A2: set_real] :
      ( ( ord_less_eq_set_real @ X @ B2 )
     => ( ord_less_eq_set_real @ X @ ( sup_sup_set_real @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_2669_le__supI2,axiom,
    ! [X: set_nat,B2: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ B2 )
     => ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_2670_le__supI2,axiom,
    ! [X: rat,B2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ X @ B2 )
     => ( ord_less_eq_rat @ X @ ( sup_sup_rat @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_2671_le__supI2,axiom,
    ! [X: nat,B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ X @ B2 )
     => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_2672_le__supI2,axiom,
    ! [X: int,B2: int,A2: int] :
      ( ( ord_less_eq_int @ X @ B2 )
     => ( ord_less_eq_int @ X @ ( sup_sup_int @ A2 @ B2 ) ) ) ).

% le_supI2
thf(fact_2673_sup_Omono,axiom,
    ! [C: extended_enat,A2: extended_enat,D: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ C @ A2 )
     => ( ( ord_le2932123472753598470d_enat @ D @ B2 )
       => ( ord_le2932123472753598470d_enat @ ( sup_su3973961784419623482d_enat @ C @ D ) @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_2674_sup_Omono,axiom,
    ! [C: assn,A2: assn,D: assn,B2: assn] :
      ( ( ord_less_eq_assn @ C @ A2 )
     => ( ( ord_less_eq_assn @ D @ B2 )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ C @ D ) @ ( sup_sup_assn @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_2675_sup_Omono,axiom,
    ! [C: set_complex,A2: set_complex,D: set_complex,B2: set_complex] :
      ( ( ord_le211207098394363844omplex @ C @ A2 )
     => ( ( ord_le211207098394363844omplex @ D @ B2 )
       => ( ord_le211207098394363844omplex @ ( sup_sup_set_complex @ C @ D ) @ ( sup_sup_set_complex @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_2676_sup_Omono,axiom,
    ! [C: set_int,A2: set_int,D: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ C @ A2 )
     => ( ( ord_less_eq_set_int @ D @ B2 )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ C @ D ) @ ( sup_sup_set_int @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_2677_sup_Omono,axiom,
    ! [C: set_real,A2: set_real,D: set_real,B2: set_real] :
      ( ( ord_less_eq_set_real @ C @ A2 )
     => ( ( ord_less_eq_set_real @ D @ B2 )
       => ( ord_less_eq_set_real @ ( sup_sup_set_real @ C @ D ) @ ( sup_sup_set_real @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_2678_sup_Omono,axiom,
    ! [C: set_nat,A2: set_nat,D: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ C @ A2 )
     => ( ( ord_less_eq_set_nat @ D @ B2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D ) @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_2679_sup_Omono,axiom,
    ! [C: rat,A2: rat,D: rat,B2: rat] :
      ( ( ord_less_eq_rat @ C @ A2 )
     => ( ( ord_less_eq_rat @ D @ B2 )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ C @ D ) @ ( sup_sup_rat @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_2680_sup_Omono,axiom,
    ! [C: nat,A2: nat,D: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C @ A2 )
     => ( ( ord_less_eq_nat @ D @ B2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_2681_sup_Omono,axiom,
    ! [C: int,A2: int,D: int,B2: int] :
      ( ( ord_less_eq_int @ C @ A2 )
     => ( ( ord_less_eq_int @ D @ B2 )
       => ( ord_less_eq_int @ ( sup_sup_int @ C @ D ) @ ( sup_sup_int @ A2 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_2682_sup__mono,axiom,
    ! [A2: extended_enat,C: extended_enat,B2: extended_enat,D: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ C )
     => ( ( ord_le2932123472753598470d_enat @ B2 @ D )
       => ( ord_le2932123472753598470d_enat @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) @ ( sup_su3973961784419623482d_enat @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_2683_sup__mono,axiom,
    ! [A2: assn,C: assn,B2: assn,D: assn] :
      ( ( ord_less_eq_assn @ A2 @ C )
     => ( ( ord_less_eq_assn @ B2 @ D )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ A2 @ B2 ) @ ( sup_sup_assn @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_2684_sup__mono,axiom,
    ! [A2: set_complex,C: set_complex,B2: set_complex,D: set_complex] :
      ( ( ord_le211207098394363844omplex @ A2 @ C )
     => ( ( ord_le211207098394363844omplex @ B2 @ D )
       => ( ord_le211207098394363844omplex @ ( sup_sup_set_complex @ A2 @ B2 ) @ ( sup_sup_set_complex @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_2685_sup__mono,axiom,
    ! [A2: set_int,C: set_int,B2: set_int,D: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ C )
     => ( ( ord_less_eq_set_int @ B2 @ D )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A2 @ B2 ) @ ( sup_sup_set_int @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_2686_sup__mono,axiom,
    ! [A2: set_real,C: set_real,B2: set_real,D: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ C )
     => ( ( ord_less_eq_set_real @ B2 @ D )
       => ( ord_less_eq_set_real @ ( sup_sup_set_real @ A2 @ B2 ) @ ( sup_sup_set_real @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_2687_sup__mono,axiom,
    ! [A2: set_nat,C: set_nat,B2: set_nat,D: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ C )
     => ( ( ord_less_eq_set_nat @ B2 @ D )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_2688_sup__mono,axiom,
    ! [A2: rat,C: rat,B2: rat,D: rat] :
      ( ( ord_less_eq_rat @ A2 @ C )
     => ( ( ord_less_eq_rat @ B2 @ D )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ A2 @ B2 ) @ ( sup_sup_rat @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_2689_sup__mono,axiom,
    ! [A2: nat,C: nat,B2: nat,D: nat] :
      ( ( ord_less_eq_nat @ A2 @ C )
     => ( ( ord_less_eq_nat @ B2 @ D )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_2690_sup__mono,axiom,
    ! [A2: int,C: int,B2: int,D: int] :
      ( ( ord_less_eq_int @ A2 @ C )
     => ( ( ord_less_eq_int @ B2 @ D )
       => ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B2 ) @ ( sup_sup_int @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_2691_sup__least,axiom,
    ! [Y: extended_enat,X: extended_enat,Z: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Y @ X )
     => ( ( ord_le2932123472753598470d_enat @ Z @ X )
       => ( ord_le2932123472753598470d_enat @ ( sup_su3973961784419623482d_enat @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_2692_sup__least,axiom,
    ! [Y: assn,X: assn,Z: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ( ( ord_less_eq_assn @ Z @ X )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_2693_sup__least,axiom,
    ! [Y: set_complex,X: set_complex,Z: set_complex] :
      ( ( ord_le211207098394363844omplex @ Y @ X )
     => ( ( ord_le211207098394363844omplex @ Z @ X )
       => ( ord_le211207098394363844omplex @ ( sup_sup_set_complex @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_2694_sup__least,axiom,
    ! [Y: set_int,X: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X )
     => ( ( ord_less_eq_set_int @ Z @ X )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_2695_sup__least,axiom,
    ! [Y: set_real,X: set_real,Z: set_real] :
      ( ( ord_less_eq_set_real @ Y @ X )
     => ( ( ord_less_eq_set_real @ Z @ X )
       => ( ord_less_eq_set_real @ ( sup_sup_set_real @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_2696_sup__least,axiom,
    ! [Y: set_nat,X: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( ord_less_eq_set_nat @ Z @ X )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_2697_sup__least,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( ord_less_eq_rat @ Z @ X )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_2698_sup__least,axiom,
    ! [Y: nat,X: nat,Z: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ Z @ X )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_2699_sup__least,axiom,
    ! [Y: int,X: int,Z: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_less_eq_int @ Z @ X )
       => ( ord_less_eq_int @ ( sup_sup_int @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_2700_le__iff__sup,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [X2: extended_enat,Y2: extended_enat] :
          ( ( sup_su3973961784419623482d_enat @ X2 @ Y2 )
          = Y2 ) ) ) ).

% le_iff_sup
thf(fact_2701_le__iff__sup,axiom,
    ( ord_less_eq_assn
    = ( ^ [X2: assn,Y2: assn] :
          ( ( sup_sup_assn @ X2 @ Y2 )
          = Y2 ) ) ) ).

% le_iff_sup
thf(fact_2702_le__iff__sup,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [X2: set_complex,Y2: set_complex] :
          ( ( sup_sup_set_complex @ X2 @ Y2 )
          = Y2 ) ) ) ).

% le_iff_sup
thf(fact_2703_le__iff__sup,axiom,
    ( ord_less_eq_set_int
    = ( ^ [X2: set_int,Y2: set_int] :
          ( ( sup_sup_set_int @ X2 @ Y2 )
          = Y2 ) ) ) ).

% le_iff_sup
thf(fact_2704_le__iff__sup,axiom,
    ( ord_less_eq_set_real
    = ( ^ [X2: set_real,Y2: set_real] :
          ( ( sup_sup_set_real @ X2 @ Y2 )
          = Y2 ) ) ) ).

% le_iff_sup
thf(fact_2705_le__iff__sup,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X2: set_nat,Y2: set_nat] :
          ( ( sup_sup_set_nat @ X2 @ Y2 )
          = Y2 ) ) ) ).

% le_iff_sup
thf(fact_2706_le__iff__sup,axiom,
    ( ord_less_eq_rat
    = ( ^ [X2: rat,Y2: rat] :
          ( ( sup_sup_rat @ X2 @ Y2 )
          = Y2 ) ) ) ).

% le_iff_sup
thf(fact_2707_le__iff__sup,axiom,
    ( ord_less_eq_nat
    = ( ^ [X2: nat,Y2: nat] :
          ( ( sup_sup_nat @ X2 @ Y2 )
          = Y2 ) ) ) ).

% le_iff_sup
thf(fact_2708_le__iff__sup,axiom,
    ( ord_less_eq_int
    = ( ^ [X2: int,Y2: int] :
          ( ( sup_sup_int @ X2 @ Y2 )
          = Y2 ) ) ) ).

% le_iff_sup
thf(fact_2709_sup_OorderE,axiom,
    ! [B2: extended_enat,A2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B2 @ A2 )
     => ( A2
        = ( sup_su3973961784419623482d_enat @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_2710_sup_OorderE,axiom,
    ! [B2: assn,A2: assn] :
      ( ( ord_less_eq_assn @ B2 @ A2 )
     => ( A2
        = ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_2711_sup_OorderE,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( A2
        = ( sup_sup_set_complex @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_2712_sup_OorderE,axiom,
    ! [B2: set_int,A2: set_int] :
      ( ( ord_less_eq_set_int @ B2 @ A2 )
     => ( A2
        = ( sup_sup_set_int @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_2713_sup_OorderE,axiom,
    ! [B2: set_real,A2: set_real] :
      ( ( ord_less_eq_set_real @ B2 @ A2 )
     => ( A2
        = ( sup_sup_set_real @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_2714_sup_OorderE,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( A2
        = ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_2715_sup_OorderE,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( A2
        = ( sup_sup_rat @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_2716_sup_OorderE,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( A2
        = ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_2717_sup_OorderE,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( A2
        = ( sup_sup_int @ A2 @ B2 ) ) ) ).

% sup.orderE
thf(fact_2718_sup_OorderI,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( A2
        = ( sup_su3973961784419623482d_enat @ A2 @ B2 ) )
     => ( ord_le2932123472753598470d_enat @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_2719_sup_OorderI,axiom,
    ! [A2: assn,B2: assn] :
      ( ( A2
        = ( sup_sup_assn @ A2 @ B2 ) )
     => ( ord_less_eq_assn @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_2720_sup_OorderI,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ( A2
        = ( sup_sup_set_complex @ A2 @ B2 ) )
     => ( ord_le211207098394363844omplex @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_2721_sup_OorderI,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( A2
        = ( sup_sup_set_int @ A2 @ B2 ) )
     => ( ord_less_eq_set_int @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_2722_sup_OorderI,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( A2
        = ( sup_sup_set_real @ A2 @ B2 ) )
     => ( ord_less_eq_set_real @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_2723_sup_OorderI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2
        = ( sup_sup_set_nat @ A2 @ B2 ) )
     => ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_2724_sup_OorderI,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2
        = ( sup_sup_rat @ A2 @ B2 ) )
     => ( ord_less_eq_rat @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_2725_sup_OorderI,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2
        = ( sup_sup_nat @ A2 @ B2 ) )
     => ( ord_less_eq_nat @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_2726_sup_OorderI,axiom,
    ! [A2: int,B2: int] :
      ( ( A2
        = ( sup_sup_int @ A2 @ B2 ) )
     => ( ord_less_eq_int @ B2 @ A2 ) ) ).

% sup.orderI
thf(fact_2727_sup__unique,axiom,
    ! [F: extended_enat > extended_enat > extended_enat,X: extended_enat,Y: extended_enat] :
      ( ! [X3: extended_enat,Y4: extended_enat] : ( ord_le2932123472753598470d_enat @ X3 @ ( F @ X3 @ Y4 ) )
     => ( ! [X3: extended_enat,Y4: extended_enat] : ( ord_le2932123472753598470d_enat @ Y4 @ ( F @ X3 @ Y4 ) )
       => ( ! [X3: extended_enat,Y4: extended_enat,Z3: extended_enat] :
              ( ( ord_le2932123472753598470d_enat @ Y4 @ X3 )
             => ( ( ord_le2932123472753598470d_enat @ Z3 @ X3 )
               => ( ord_le2932123472753598470d_enat @ ( F @ Y4 @ Z3 ) @ X3 ) ) )
         => ( ( sup_su3973961784419623482d_enat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_2728_sup__unique,axiom,
    ! [F: assn > assn > assn,X: assn,Y: assn] :
      ( ! [X3: assn,Y4: assn] : ( ord_less_eq_assn @ X3 @ ( F @ X3 @ Y4 ) )
     => ( ! [X3: assn,Y4: assn] : ( ord_less_eq_assn @ Y4 @ ( F @ X3 @ Y4 ) )
       => ( ! [X3: assn,Y4: assn,Z3: assn] :
              ( ( ord_less_eq_assn @ Y4 @ X3 )
             => ( ( ord_less_eq_assn @ Z3 @ X3 )
               => ( ord_less_eq_assn @ ( F @ Y4 @ Z3 ) @ X3 ) ) )
         => ( ( sup_sup_assn @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_2729_sup__unique,axiom,
    ! [F: set_complex > set_complex > set_complex,X: set_complex,Y: set_complex] :
      ( ! [X3: set_complex,Y4: set_complex] : ( ord_le211207098394363844omplex @ X3 @ ( F @ X3 @ Y4 ) )
     => ( ! [X3: set_complex,Y4: set_complex] : ( ord_le211207098394363844omplex @ Y4 @ ( F @ X3 @ Y4 ) )
       => ( ! [X3: set_complex,Y4: set_complex,Z3: set_complex] :
              ( ( ord_le211207098394363844omplex @ Y4 @ X3 )
             => ( ( ord_le211207098394363844omplex @ Z3 @ X3 )
               => ( ord_le211207098394363844omplex @ ( F @ Y4 @ Z3 ) @ X3 ) ) )
         => ( ( sup_sup_set_complex @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_2730_sup__unique,axiom,
    ! [F: set_int > set_int > set_int,X: set_int,Y: set_int] :
      ( ! [X3: set_int,Y4: set_int] : ( ord_less_eq_set_int @ X3 @ ( F @ X3 @ Y4 ) )
     => ( ! [X3: set_int,Y4: set_int] : ( ord_less_eq_set_int @ Y4 @ ( F @ X3 @ Y4 ) )
       => ( ! [X3: set_int,Y4: set_int,Z3: set_int] :
              ( ( ord_less_eq_set_int @ Y4 @ X3 )
             => ( ( ord_less_eq_set_int @ Z3 @ X3 )
               => ( ord_less_eq_set_int @ ( F @ Y4 @ Z3 ) @ X3 ) ) )
         => ( ( sup_sup_set_int @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_2731_sup__unique,axiom,
    ! [F: set_real > set_real > set_real,X: set_real,Y: set_real] :
      ( ! [X3: set_real,Y4: set_real] : ( ord_less_eq_set_real @ X3 @ ( F @ X3 @ Y4 ) )
     => ( ! [X3: set_real,Y4: set_real] : ( ord_less_eq_set_real @ Y4 @ ( F @ X3 @ Y4 ) )
       => ( ! [X3: set_real,Y4: set_real,Z3: set_real] :
              ( ( ord_less_eq_set_real @ Y4 @ X3 )
             => ( ( ord_less_eq_set_real @ Z3 @ X3 )
               => ( ord_less_eq_set_real @ ( F @ Y4 @ Z3 ) @ X3 ) ) )
         => ( ( sup_sup_set_real @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_2732_sup__unique,axiom,
    ! [F: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
      ( ! [X3: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( F @ X3 @ Y4 ) )
     => ( ! [X3: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ Y4 @ ( F @ X3 @ Y4 ) )
       => ( ! [X3: set_nat,Y4: set_nat,Z3: set_nat] :
              ( ( ord_less_eq_set_nat @ Y4 @ X3 )
             => ( ( ord_less_eq_set_nat @ Z3 @ X3 )
               => ( ord_less_eq_set_nat @ ( F @ Y4 @ Z3 ) @ X3 ) ) )
         => ( ( sup_sup_set_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_2733_sup__unique,axiom,
    ! [F: rat > rat > rat,X: rat,Y: rat] :
      ( ! [X3: rat,Y4: rat] : ( ord_less_eq_rat @ X3 @ ( F @ X3 @ Y4 ) )
     => ( ! [X3: rat,Y4: rat] : ( ord_less_eq_rat @ Y4 @ ( F @ X3 @ Y4 ) )
       => ( ! [X3: rat,Y4: rat,Z3: rat] :
              ( ( ord_less_eq_rat @ Y4 @ X3 )
             => ( ( ord_less_eq_rat @ Z3 @ X3 )
               => ( ord_less_eq_rat @ ( F @ Y4 @ Z3 ) @ X3 ) ) )
         => ( ( sup_sup_rat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_2734_sup__unique,axiom,
    ! [F: nat > nat > nat,X: nat,Y: nat] :
      ( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y4 ) )
     => ( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ ( F @ X3 @ Y4 ) )
       => ( ! [X3: nat,Y4: nat,Z3: nat] :
              ( ( ord_less_eq_nat @ Y4 @ X3 )
             => ( ( ord_less_eq_nat @ Z3 @ X3 )
               => ( ord_less_eq_nat @ ( F @ Y4 @ Z3 ) @ X3 ) ) )
         => ( ( sup_sup_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_2735_sup__unique,axiom,
    ! [F: int > int > int,X: int,Y: int] :
      ( ! [X3: int,Y4: int] : ( ord_less_eq_int @ X3 @ ( F @ X3 @ Y4 ) )
     => ( ! [X3: int,Y4: int] : ( ord_less_eq_int @ Y4 @ ( F @ X3 @ Y4 ) )
       => ( ! [X3: int,Y4: int,Z3: int] :
              ( ( ord_less_eq_int @ Y4 @ X3 )
             => ( ( ord_less_eq_int @ Z3 @ X3 )
               => ( ord_less_eq_int @ ( F @ Y4 @ Z3 ) @ X3 ) ) )
         => ( ( sup_sup_int @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_2736_sup_Oabsorb1,axiom,
    ! [B2: extended_enat,A2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B2 @ A2 )
     => ( ( sup_su3973961784419623482d_enat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_2737_sup_Oabsorb1,axiom,
    ! [B2: assn,A2: assn] :
      ( ( ord_less_eq_assn @ B2 @ A2 )
     => ( ( sup_sup_assn @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_2738_sup_Oabsorb1,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( sup_sup_set_complex @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_2739_sup_Oabsorb1,axiom,
    ! [B2: set_int,A2: set_int] :
      ( ( ord_less_eq_set_int @ B2 @ A2 )
     => ( ( sup_sup_set_int @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_2740_sup_Oabsorb1,axiom,
    ! [B2: set_real,A2: set_real] :
      ( ( ord_less_eq_set_real @ B2 @ A2 )
     => ( ( sup_sup_set_real @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_2741_sup_Oabsorb1,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( sup_sup_set_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_2742_sup_Oabsorb1,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ( sup_sup_rat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_2743_sup_Oabsorb1,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( sup_sup_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_2744_sup_Oabsorb1,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( sup_sup_int @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb1
thf(fact_2745_sup_Oabsorb2,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
     => ( ( sup_su3973961784419623482d_enat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_2746_sup_Oabsorb2,axiom,
    ! [A2: assn,B2: assn] :
      ( ( ord_less_eq_assn @ A2 @ B2 )
     => ( ( sup_sup_assn @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_2747_sup_Oabsorb2,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ( ord_le211207098394363844omplex @ A2 @ B2 )
     => ( ( sup_sup_set_complex @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_2748_sup_Oabsorb2,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ( sup_sup_set_int @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_2749_sup_Oabsorb2,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ B2 )
     => ( ( sup_sup_set_real @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_2750_sup_Oabsorb2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( sup_sup_set_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_2751_sup_Oabsorb2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( sup_sup_rat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_2752_sup_Oabsorb2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( sup_sup_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_2753_sup_Oabsorb2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( sup_sup_int @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_2754_sup__absorb1,axiom,
    ! [Y: extended_enat,X: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Y @ X )
     => ( ( sup_su3973961784419623482d_enat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_2755_sup__absorb1,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ( ( sup_sup_assn @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_2756_sup__absorb1,axiom,
    ! [Y: set_complex,X: set_complex] :
      ( ( ord_le211207098394363844omplex @ Y @ X )
     => ( ( sup_sup_set_complex @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_2757_sup__absorb1,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X )
     => ( ( sup_sup_set_int @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_2758_sup__absorb1,axiom,
    ! [Y: set_real,X: set_real] :
      ( ( ord_less_eq_set_real @ Y @ X )
     => ( ( sup_sup_set_real @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_2759_sup__absorb1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( sup_sup_set_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_2760_sup__absorb1,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( sup_sup_rat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_2761_sup__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( sup_sup_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_2762_sup__absorb1,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( sup_sup_int @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_2763_sup__absorb2,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X @ Y )
     => ( ( sup_su3973961784419623482d_enat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_2764_sup__absorb2,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( sup_sup_assn @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_2765_sup__absorb2,axiom,
    ! [X: set_complex,Y: set_complex] :
      ( ( ord_le211207098394363844omplex @ X @ Y )
     => ( ( sup_sup_set_complex @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_2766_sup__absorb2,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( sup_sup_set_int @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_2767_sup__absorb2,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( ord_less_eq_set_real @ X @ Y )
     => ( ( sup_sup_set_real @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_2768_sup__absorb2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( sup_sup_set_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_2769_sup__absorb2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( sup_sup_rat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_2770_sup__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( sup_sup_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_2771_sup__absorb2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( sup_sup_int @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_2772_sup_OboundedE,axiom,
    ! [B2: extended_enat,C: extended_enat,A2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( sup_su3973961784419623482d_enat @ B2 @ C ) @ A2 )
     => ~ ( ( ord_le2932123472753598470d_enat @ B2 @ A2 )
         => ~ ( ord_le2932123472753598470d_enat @ C @ A2 ) ) ) ).

% sup.boundedE
thf(fact_2773_sup_OboundedE,axiom,
    ! [B2: assn,C: assn,A2: assn] :
      ( ( ord_less_eq_assn @ ( sup_sup_assn @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_eq_assn @ B2 @ A2 )
         => ~ ( ord_less_eq_assn @ C @ A2 ) ) ) ).

% sup.boundedE
thf(fact_2774_sup_OboundedE,axiom,
    ! [B2: set_complex,C: set_complex,A2: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( sup_sup_set_complex @ B2 @ C ) @ A2 )
     => ~ ( ( ord_le211207098394363844omplex @ B2 @ A2 )
         => ~ ( ord_le211207098394363844omplex @ C @ A2 ) ) ) ).

% sup.boundedE
thf(fact_2775_sup_OboundedE,axiom,
    ! [B2: set_int,C: set_int,A2: set_int] :
      ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_eq_set_int @ B2 @ A2 )
         => ~ ( ord_less_eq_set_int @ C @ A2 ) ) ) ).

% sup.boundedE
thf(fact_2776_sup_OboundedE,axiom,
    ! [B2: set_real,C: set_real,A2: set_real] :
      ( ( ord_less_eq_set_real @ ( sup_sup_set_real @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_eq_set_real @ B2 @ A2 )
         => ~ ( ord_less_eq_set_real @ C @ A2 ) ) ) ).

% sup.boundedE
thf(fact_2777_sup_OboundedE,axiom,
    ! [B2: set_nat,C: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_eq_set_nat @ B2 @ A2 )
         => ~ ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).

% sup.boundedE
thf(fact_2778_sup_OboundedE,axiom,
    ! [B2: rat,C: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_eq_rat @ B2 @ A2 )
         => ~ ( ord_less_eq_rat @ C @ A2 ) ) ) ).

% sup.boundedE
thf(fact_2779_sup_OboundedE,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_eq_nat @ B2 @ A2 )
         => ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).

% sup.boundedE
thf(fact_2780_sup_OboundedE,axiom,
    ! [B2: int,C: int,A2: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_eq_int @ B2 @ A2 )
         => ~ ( ord_less_eq_int @ C @ A2 ) ) ) ).

% sup.boundedE
thf(fact_2781_sup_OboundedI,axiom,
    ! [B2: extended_enat,A2: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B2 @ A2 )
     => ( ( ord_le2932123472753598470d_enat @ C @ A2 )
       => ( ord_le2932123472753598470d_enat @ ( sup_su3973961784419623482d_enat @ B2 @ C ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_2782_sup_OboundedI,axiom,
    ! [B2: assn,A2: assn,C: assn] :
      ( ( ord_less_eq_assn @ B2 @ A2 )
     => ( ( ord_less_eq_assn @ C @ A2 )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ B2 @ C ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_2783_sup_OboundedI,axiom,
    ! [B2: set_complex,A2: set_complex,C: set_complex] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( ord_le211207098394363844omplex @ C @ A2 )
       => ( ord_le211207098394363844omplex @ ( sup_sup_set_complex @ B2 @ C ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_2784_sup_OboundedI,axiom,
    ! [B2: set_int,A2: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ B2 @ A2 )
     => ( ( ord_less_eq_set_int @ C @ A2 )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ B2 @ C ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_2785_sup_OboundedI,axiom,
    ! [B2: set_real,A2: set_real,C: set_real] :
      ( ( ord_less_eq_set_real @ B2 @ A2 )
     => ( ( ord_less_eq_set_real @ C @ A2 )
       => ( ord_less_eq_set_real @ ( sup_sup_set_real @ B2 @ C ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_2786_sup_OboundedI,axiom,
    ! [B2: set_nat,A2: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( ord_less_eq_set_nat @ C @ A2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_2787_sup_OboundedI,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ( ord_less_eq_rat @ C @ A2 )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ B2 @ C ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_2788_sup_OboundedI,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ C @ A2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_2789_sup_OboundedI,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( ord_less_eq_int @ C @ A2 )
       => ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C ) @ A2 ) ) ) ).

% sup.boundedI
thf(fact_2790_sup_Oorder__iff,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [B4: extended_enat,A4: extended_enat] :
          ( A4
          = ( sup_su3973961784419623482d_enat @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_2791_sup_Oorder__iff,axiom,
    ( ord_less_eq_assn
    = ( ^ [B4: assn,A4: assn] :
          ( A4
          = ( sup_sup_assn @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_2792_sup_Oorder__iff,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [B4: set_complex,A4: set_complex] :
          ( A4
          = ( sup_sup_set_complex @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_2793_sup_Oorder__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [B4: set_int,A4: set_int] :
          ( A4
          = ( sup_sup_set_int @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_2794_sup_Oorder__iff,axiom,
    ( ord_less_eq_set_real
    = ( ^ [B4: set_real,A4: set_real] :
          ( A4
          = ( sup_sup_set_real @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_2795_sup_Oorder__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B4: set_nat,A4: set_nat] :
          ( A4
          = ( sup_sup_set_nat @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_2796_sup_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A4: rat] :
          ( A4
          = ( sup_sup_rat @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_2797_sup_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( A4
          = ( sup_sup_nat @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_2798_sup_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A4: int] :
          ( A4
          = ( sup_sup_int @ A4 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_2799_sup_Ocobounded1,axiom,
    ! [A2: extended_enat,B2: extended_enat] : ( ord_le2932123472753598470d_enat @ A2 @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_2800_sup_Ocobounded1,axiom,
    ! [A2: assn,B2: assn] : ( ord_less_eq_assn @ A2 @ ( sup_sup_assn @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_2801_sup_Ocobounded1,axiom,
    ! [A2: set_complex,B2: set_complex] : ( ord_le211207098394363844omplex @ A2 @ ( sup_sup_set_complex @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_2802_sup_Ocobounded1,axiom,
    ! [A2: set_int,B2: set_int] : ( ord_less_eq_set_int @ A2 @ ( sup_sup_set_int @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_2803_sup_Ocobounded1,axiom,
    ! [A2: set_real,B2: set_real] : ( ord_less_eq_set_real @ A2 @ ( sup_sup_set_real @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_2804_sup_Ocobounded1,axiom,
    ! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_2805_sup_Ocobounded1,axiom,
    ! [A2: rat,B2: rat] : ( ord_less_eq_rat @ A2 @ ( sup_sup_rat @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_2806_sup_Ocobounded1,axiom,
    ! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_2807_sup_Ocobounded1,axiom,
    ! [A2: int,B2: int] : ( ord_less_eq_int @ A2 @ ( sup_sup_int @ A2 @ B2 ) ) ).

% sup.cobounded1
thf(fact_2808_sup_Ocobounded2,axiom,
    ! [B2: extended_enat,A2: extended_enat] : ( ord_le2932123472753598470d_enat @ B2 @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_2809_sup_Ocobounded2,axiom,
    ! [B2: assn,A2: assn] : ( ord_less_eq_assn @ B2 @ ( sup_sup_assn @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_2810_sup_Ocobounded2,axiom,
    ! [B2: set_complex,A2: set_complex] : ( ord_le211207098394363844omplex @ B2 @ ( sup_sup_set_complex @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_2811_sup_Ocobounded2,axiom,
    ! [B2: set_int,A2: set_int] : ( ord_less_eq_set_int @ B2 @ ( sup_sup_set_int @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_2812_sup_Ocobounded2,axiom,
    ! [B2: set_real,A2: set_real] : ( ord_less_eq_set_real @ B2 @ ( sup_sup_set_real @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_2813_sup_Ocobounded2,axiom,
    ! [B2: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_2814_sup_Ocobounded2,axiom,
    ! [B2: rat,A2: rat] : ( ord_less_eq_rat @ B2 @ ( sup_sup_rat @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_2815_sup_Ocobounded2,axiom,
    ! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_2816_sup_Ocobounded2,axiom,
    ! [B2: int,A2: int] : ( ord_less_eq_int @ B2 @ ( sup_sup_int @ A2 @ B2 ) ) ).

% sup.cobounded2
thf(fact_2817_sup_Oabsorb__iff1,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [B4: extended_enat,A4: extended_enat] :
          ( ( sup_su3973961784419623482d_enat @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_2818_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_assn
    = ( ^ [B4: assn,A4: assn] :
          ( ( sup_sup_assn @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_2819_sup_Oabsorb__iff1,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [B4: set_complex,A4: set_complex] :
          ( ( sup_sup_set_complex @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_2820_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_int
    = ( ^ [B4: set_int,A4: set_int] :
          ( ( sup_sup_set_int @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_2821_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_real
    = ( ^ [B4: set_real,A4: set_real] :
          ( ( sup_sup_set_real @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_2822_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B4: set_nat,A4: set_nat] :
          ( ( sup_sup_set_nat @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_2823_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A4: rat] :
          ( ( sup_sup_rat @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_2824_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A4: nat] :
          ( ( sup_sup_nat @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_2825_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A4: int] :
          ( ( sup_sup_int @ A4 @ B4 )
          = A4 ) ) ) ).

% sup.absorb_iff1
thf(fact_2826_sup_Oabsorb__iff2,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [A4: extended_enat,B4: extended_enat] :
          ( ( sup_su3973961784419623482d_enat @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_2827_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_assn
    = ( ^ [A4: assn,B4: assn] :
          ( ( sup_sup_assn @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_2828_sup_Oabsorb__iff2,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [A4: set_complex,B4: set_complex] :
          ( ( sup_sup_set_complex @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_2829_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A4: set_int,B4: set_int] :
          ( ( sup_sup_set_int @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_2830_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A4: set_real,B4: set_real] :
          ( ( sup_sup_set_real @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_2831_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A4: set_nat,B4: set_nat] :
          ( ( sup_sup_set_nat @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_2832_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A4: rat,B4: rat] :
          ( ( sup_sup_rat @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_2833_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( sup_sup_nat @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_2834_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A4: int,B4: int] :
          ( ( sup_sup_int @ A4 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_2835_sup_OcoboundedI1,axiom,
    ! [C: extended_enat,A2: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ C @ A2 )
     => ( ord_le2932123472753598470d_enat @ C @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_2836_sup_OcoboundedI1,axiom,
    ! [C: assn,A2: assn,B2: assn] :
      ( ( ord_less_eq_assn @ C @ A2 )
     => ( ord_less_eq_assn @ C @ ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_2837_sup_OcoboundedI1,axiom,
    ! [C: set_complex,A2: set_complex,B2: set_complex] :
      ( ( ord_le211207098394363844omplex @ C @ A2 )
     => ( ord_le211207098394363844omplex @ C @ ( sup_sup_set_complex @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_2838_sup_OcoboundedI1,axiom,
    ! [C: set_int,A2: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ C @ A2 )
     => ( ord_less_eq_set_int @ C @ ( sup_sup_set_int @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_2839_sup_OcoboundedI1,axiom,
    ! [C: set_real,A2: set_real,B2: set_real] :
      ( ( ord_less_eq_set_real @ C @ A2 )
     => ( ord_less_eq_set_real @ C @ ( sup_sup_set_real @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_2840_sup_OcoboundedI1,axiom,
    ! [C: set_nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ C @ A2 )
     => ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_2841_sup_OcoboundedI1,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ C @ A2 )
     => ( ord_less_eq_rat @ C @ ( sup_sup_rat @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_2842_sup_OcoboundedI1,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C @ A2 )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_2843_sup_OcoboundedI1,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ C @ A2 )
     => ( ord_less_eq_int @ C @ ( sup_sup_int @ A2 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_2844_sup_OcoboundedI2,axiom,
    ! [C: extended_enat,B2: extended_enat,A2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ C @ B2 )
     => ( ord_le2932123472753598470d_enat @ C @ ( sup_su3973961784419623482d_enat @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_2845_sup_OcoboundedI2,axiom,
    ! [C: assn,B2: assn,A2: assn] :
      ( ( ord_less_eq_assn @ C @ B2 )
     => ( ord_less_eq_assn @ C @ ( sup_sup_assn @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_2846_sup_OcoboundedI2,axiom,
    ! [C: set_complex,B2: set_complex,A2: set_complex] :
      ( ( ord_le211207098394363844omplex @ C @ B2 )
     => ( ord_le211207098394363844omplex @ C @ ( sup_sup_set_complex @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_2847_sup_OcoboundedI2,axiom,
    ! [C: set_int,B2: set_int,A2: set_int] :
      ( ( ord_less_eq_set_int @ C @ B2 )
     => ( ord_less_eq_set_int @ C @ ( sup_sup_set_int @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_2848_sup_OcoboundedI2,axiom,
    ! [C: set_real,B2: set_real,A2: set_real] :
      ( ( ord_less_eq_set_real @ C @ B2 )
     => ( ord_less_eq_set_real @ C @ ( sup_sup_set_real @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_2849_sup_OcoboundedI2,axiom,
    ! [C: set_nat,B2: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ C @ B2 )
     => ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_2850_sup_OcoboundedI2,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ C @ B2 )
     => ( ord_less_eq_rat @ C @ ( sup_sup_rat @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_2851_sup_OcoboundedI2,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ C @ B2 )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_2852_sup_OcoboundedI2,axiom,
    ! [C: int,B2: int,A2: int] :
      ( ( ord_less_eq_int @ C @ B2 )
     => ( ord_less_eq_int @ C @ ( sup_sup_int @ A2 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_2853_insert__mono,axiom,
    ! [C2: set_int,D3: set_int,A2: int] :
      ( ( ord_less_eq_set_int @ C2 @ D3 )
     => ( ord_less_eq_set_int @ ( insert_int @ A2 @ C2 ) @ ( insert_int @ A2 @ D3 ) ) ) ).

% insert_mono
thf(fact_2854_insert__mono,axiom,
    ! [C2: set_real,D3: set_real,A2: real] :
      ( ( ord_less_eq_set_real @ C2 @ D3 )
     => ( ord_less_eq_set_real @ ( insert_real @ A2 @ C2 ) @ ( insert_real @ A2 @ D3 ) ) ) ).

% insert_mono
thf(fact_2855_insert__mono,axiom,
    ! [C2: set_o,D3: set_o,A2: $o] :
      ( ( ord_less_eq_set_o @ C2 @ D3 )
     => ( ord_less_eq_set_o @ ( insert_o @ A2 @ C2 ) @ ( insert_o @ A2 @ D3 ) ) ) ).

% insert_mono
thf(fact_2856_insert__mono,axiom,
    ! [C2: set_nat,D3: set_nat,A2: nat] :
      ( ( ord_less_eq_set_nat @ C2 @ D3 )
     => ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ C2 ) @ ( insert_nat @ A2 @ D3 ) ) ) ).

% insert_mono
thf(fact_2857_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_2858_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_2859_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_2860_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_2861_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_2862_subset__insertI,axiom,
    ! [B: set_int,A2: int] : ( ord_less_eq_set_int @ B @ ( insert_int @ A2 @ B ) ) ).

% subset_insertI
thf(fact_2863_subset__insertI,axiom,
    ! [B: set_real,A2: real] : ( ord_less_eq_set_real @ B @ ( insert_real @ A2 @ B ) ) ).

% subset_insertI
thf(fact_2864_subset__insertI,axiom,
    ! [B: set_o,A2: $o] : ( ord_less_eq_set_o @ B @ ( insert_o @ A2 @ B ) ) ).

% subset_insertI
thf(fact_2865_subset__insertI,axiom,
    ! [B: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat @ A2 @ B ) ) ).

% subset_insertI
thf(fact_2866_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_2867_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_2868_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_2869_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_2870_less__eq__real__def,axiom,
    ( ord_less_eq_real
    = ( ^ [X2: real,Y2: real] :
          ( ( ord_less_real @ X2 @ Y2 )
          | ( X2 = Y2 ) ) ) ) ).

% less_eq_real_def
thf(fact_2871_less__eq__int__code_I1_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% less_eq_int_code(1)
thf(fact_2872_subset__code_I1_J,axiom,
    ! [Xs: list_real,B: set_real] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B )
      = ( ! [X2: real] :
            ( ( member_real @ X2 @ ( set_real2 @ Xs ) )
           => ( member_real @ X2 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_2873_subset__code_I1_J,axiom,
    ! [Xs: list_int,B: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B )
      = ( ! [X2: int] :
            ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
           => ( member_int @ X2 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_2874_subset__code_I1_J,axiom,
    ! [Xs: list_complex,B: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ B )
      = ( ! [X2: complex] :
            ( ( member_complex @ X2 @ ( set_complex2 @ Xs ) )
           => ( member_complex @ X2 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_2875_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
           => ( member_nat @ X2 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_2876_int__cases2,axiom,
    ! [Z: int] :
      ( ! [N2: nat] :
          ( Z
         != ( semiri1314217659103216013at_int @ N2 ) )
     => ~ ! [N2: nat] :
            ( Z
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% int_cases2
thf(fact_2877_Un__mono,axiom,
    ! [A: set_complex,C2: set_complex,B: set_complex,D3: set_complex] :
      ( ( ord_le211207098394363844omplex @ A @ C2 )
     => ( ( ord_le211207098394363844omplex @ B @ D3 )
       => ( ord_le211207098394363844omplex @ ( sup_sup_set_complex @ A @ B ) @ ( sup_sup_set_complex @ C2 @ D3 ) ) ) ) ).

% Un_mono
thf(fact_2878_Un__mono,axiom,
    ! [A: set_int,C2: set_int,B: set_int,D3: set_int] :
      ( ( ord_less_eq_set_int @ A @ C2 )
     => ( ( ord_less_eq_set_int @ B @ D3 )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A @ B ) @ ( sup_sup_set_int @ C2 @ D3 ) ) ) ) ).

% Un_mono
thf(fact_2879_Un__mono,axiom,
    ! [A: set_real,C2: set_real,B: set_real,D3: set_real] :
      ( ( ord_less_eq_set_real @ A @ C2 )
     => ( ( ord_less_eq_set_real @ B @ D3 )
       => ( ord_less_eq_set_real @ ( sup_sup_set_real @ A @ B ) @ ( sup_sup_set_real @ C2 @ D3 ) ) ) ) ).

% Un_mono
thf(fact_2880_Un__mono,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat,D3: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C2 )
     => ( ( ord_less_eq_set_nat @ B @ D3 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C2 @ D3 ) ) ) ) ).

% Un_mono
thf(fact_2881_Un__least,axiom,
    ! [A: set_complex,C2: set_complex,B: set_complex] :
      ( ( ord_le211207098394363844omplex @ A @ C2 )
     => ( ( ord_le211207098394363844omplex @ B @ C2 )
       => ( ord_le211207098394363844omplex @ ( sup_sup_set_complex @ A @ B ) @ C2 ) ) ) ).

% Un_least
thf(fact_2882_Un__least,axiom,
    ! [A: set_int,C2: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ C2 )
     => ( ( ord_less_eq_set_int @ B @ C2 )
       => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A @ B ) @ C2 ) ) ) ).

% Un_least
thf(fact_2883_Un__least,axiom,
    ! [A: set_real,C2: set_real,B: set_real] :
      ( ( ord_less_eq_set_real @ A @ C2 )
     => ( ( ord_less_eq_set_real @ B @ C2 )
       => ( ord_less_eq_set_real @ ( sup_sup_set_real @ A @ B ) @ C2 ) ) ) ).

% Un_least
thf(fact_2884_Un__least,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C2 )
     => ( ( ord_less_eq_set_nat @ B @ C2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 ) ) ) ).

% Un_least
thf(fact_2885_Un__upper1,axiom,
    ! [A: set_complex,B: set_complex] : ( ord_le211207098394363844omplex @ A @ ( sup_sup_set_complex @ A @ B ) ) ).

% Un_upper1
thf(fact_2886_Un__upper1,axiom,
    ! [A: set_int,B: set_int] : ( ord_less_eq_set_int @ A @ ( sup_sup_set_int @ A @ B ) ) ).

% Un_upper1
thf(fact_2887_Un__upper1,axiom,
    ! [A: set_real,B: set_real] : ( ord_less_eq_set_real @ A @ ( sup_sup_set_real @ A @ B ) ) ).

% Un_upper1
thf(fact_2888_Un__upper1,axiom,
    ! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).

% Un_upper1
thf(fact_2889_Un__upper2,axiom,
    ! [B: set_complex,A: set_complex] : ( ord_le211207098394363844omplex @ B @ ( sup_sup_set_complex @ A @ B ) ) ).

% Un_upper2
thf(fact_2890_Un__upper2,axiom,
    ! [B: set_int,A: set_int] : ( ord_less_eq_set_int @ B @ ( sup_sup_set_int @ A @ B ) ) ).

% Un_upper2
thf(fact_2891_Un__upper2,axiom,
    ! [B: set_real,A: set_real] : ( ord_less_eq_set_real @ B @ ( sup_sup_set_real @ A @ B ) ) ).

% Un_upper2
thf(fact_2892_Un__upper2,axiom,
    ! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).

% Un_upper2
thf(fact_2893_Un__absorb1,axiom,
    ! [A: set_complex,B: set_complex] :
      ( ( ord_le211207098394363844omplex @ A @ B )
     => ( ( sup_sup_set_complex @ A @ B )
        = B ) ) ).

% Un_absorb1
thf(fact_2894_Un__absorb1,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( sup_sup_set_int @ A @ B )
        = B ) ) ).

% Un_absorb1
thf(fact_2895_Un__absorb1,axiom,
    ! [A: set_real,B: set_real] :
      ( ( ord_less_eq_set_real @ A @ B )
     => ( ( sup_sup_set_real @ A @ B )
        = B ) ) ).

% Un_absorb1
thf(fact_2896_Un__absorb1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( sup_sup_set_nat @ A @ B )
        = B ) ) ).

% Un_absorb1
thf(fact_2897_Un__absorb2,axiom,
    ! [B: set_complex,A: set_complex] :
      ( ( ord_le211207098394363844omplex @ B @ A )
     => ( ( sup_sup_set_complex @ A @ B )
        = A ) ) ).

% Un_absorb2
thf(fact_2898_Un__absorb2,axiom,
    ! [B: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( sup_sup_set_int @ A @ B )
        = A ) ) ).

% Un_absorb2
thf(fact_2899_Un__absorb2,axiom,
    ! [B: set_real,A: set_real] :
      ( ( ord_less_eq_set_real @ B @ A )
     => ( ( sup_sup_set_real @ A @ B )
        = A ) ) ).

% Un_absorb2
thf(fact_2900_Un__absorb2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( sup_sup_set_nat @ A @ B )
        = A ) ) ).

% Un_absorb2
thf(fact_2901_subset__UnE,axiom,
    ! [C2: set_complex,A: set_complex,B: set_complex] :
      ( ( ord_le211207098394363844omplex @ C2 @ ( sup_sup_set_complex @ A @ B ) )
     => ~ ! [A7: set_complex] :
            ( ( ord_le211207098394363844omplex @ A7 @ A )
           => ! [B8: set_complex] :
                ( ( ord_le211207098394363844omplex @ B8 @ B )
               => ( C2
                 != ( sup_sup_set_complex @ A7 @ B8 ) ) ) ) ) ).

% subset_UnE
thf(fact_2902_subset__UnE,axiom,
    ! [C2: set_int,A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ C2 @ ( sup_sup_set_int @ A @ B ) )
     => ~ ! [A7: set_int] :
            ( ( ord_less_eq_set_int @ A7 @ A )
           => ! [B8: set_int] :
                ( ( ord_less_eq_set_int @ B8 @ B )
               => ( C2
                 != ( sup_sup_set_int @ A7 @ B8 ) ) ) ) ) ).

% subset_UnE
thf(fact_2903_subset__UnE,axiom,
    ! [C2: set_real,A: set_real,B: set_real] :
      ( ( ord_less_eq_set_real @ C2 @ ( sup_sup_set_real @ A @ B ) )
     => ~ ! [A7: set_real] :
            ( ( ord_less_eq_set_real @ A7 @ A )
           => ! [B8: set_real] :
                ( ( ord_less_eq_set_real @ B8 @ B )
               => ( C2
                 != ( sup_sup_set_real @ A7 @ B8 ) ) ) ) ) ).

% subset_UnE
thf(fact_2904_subset__UnE,axiom,
    ! [C2: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) )
     => ~ ! [A7: set_nat] :
            ( ( ord_less_eq_set_nat @ A7 @ A )
           => ! [B8: set_nat] :
                ( ( ord_less_eq_set_nat @ B8 @ B )
               => ( C2
                 != ( sup_sup_set_nat @ A7 @ B8 ) ) ) ) ) ).

% subset_UnE
thf(fact_2905_subset__Un__eq,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [A3: set_complex,B3: set_complex] :
          ( ( sup_sup_set_complex @ A3 @ B3 )
          = B3 ) ) ) ).

% subset_Un_eq
thf(fact_2906_subset__Un__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A3: set_int,B3: set_int] :
          ( ( sup_sup_set_int @ A3 @ B3 )
          = B3 ) ) ) ).

% subset_Un_eq
thf(fact_2907_subset__Un__eq,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A3: set_real,B3: set_real] :
          ( ( sup_sup_set_real @ A3 @ B3 )
          = B3 ) ) ) ).

% subset_Un_eq
thf(fact_2908_subset__Un__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( sup_sup_set_nat @ A3 @ B3 )
          = B3 ) ) ) ).

% subset_Un_eq
thf(fact_2909_word__coorder_Oextremum,axiom,
    ! [A2: word_N3645301735248828278l_num1] : ( ord_le3335648743751981014l_num1 @ zero_z3563351764282998399l_num1 @ A2 ) ).

% word_coorder.extremum
thf(fact_2910_word__coorder_Oextremum__uniqueI,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ A2 @ zero_z3563351764282998399l_num1 )
     => ( A2 = zero_z3563351764282998399l_num1 ) ) ).

% word_coorder.extremum_uniqueI
thf(fact_2911_word__zero__le,axiom,
    ! [Y: word_N3645301735248828278l_num1] : ( ord_le3335648743751981014l_num1 @ zero_z3563351764282998399l_num1 @ Y ) ).

% word_zero_le
thf(fact_2912_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_set_nat @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_2913_subset__psubset__trans,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_set_nat @ B @ C2 )
       => ( ord_less_set_nat @ A @ C2 ) ) ) ).

% subset_psubset_trans
thf(fact_2914_subset__not__subset__eq,axiom,
    ( ord_less_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ A3 @ B3 )
          & ~ ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_2915_psubset__subset__trans,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ C2 )
       => ( ord_less_set_nat @ A @ C2 ) ) ) ).

% psubset_subset_trans
thf(fact_2916_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_2917_psubset__eq,axiom,
    ( ord_less_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% psubset_eq
thf(fact_2918_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_2919_iadd__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( ( plus_p3455044024723400733d_enat @ M @ N )
        = zero_z5237406670263579293d_enat )
      = ( ( M = zero_z5237406670263579293d_enat )
        & ( N = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
thf(fact_2920_i0__lb,axiom,
    ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).

% i0_lb
thf(fact_2921_ile0__eq,axiom,
    ! [N: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
      = ( N = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
thf(fact_2922_power__decreasing,axiom,
    ! [N: nat,N3: nat,A2: real] :
      ( ( ord_less_eq_nat @ N @ N3 )
     => ( ( 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 ) @ ( power_power_real @ A2 @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2923_power__decreasing,axiom,
    ! [N: nat,N3: nat,A2: code_integer] :
      ( ( ord_less_eq_nat @ N @ N3 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
       => ( ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2924_power__decreasing,axiom,
    ! [N: nat,N3: nat,A2: rat] :
      ( ( ord_less_eq_nat @ N @ N3 )
     => ( ( 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 ) @ ( power_power_rat @ A2 @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2925_power__decreasing,axiom,
    ! [N: nat,N3: nat,A2: nat] :
      ( ( ord_less_eq_nat @ N @ N3 )
     => ( ( 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 ) @ ( power_power_nat @ A2 @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2926_power__decreasing,axiom,
    ! [N: nat,N3: nat,A2: int] :
      ( ( ord_less_eq_nat @ N @ N3 )
     => ( ( 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 ) @ ( power_power_int @ A2 @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2927_power__le__imp__le__exp,axiom,
    ! [A2: code_integer,M: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ M ) @ ( power_8256067586552552935nteger @ A2 @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_2928_power__le__imp__le__exp,axiom,
    ! [A2: real,M: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_eq_real @ ( power_power_real @ A2 @ M ) @ ( power_power_real @ A2 @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_2929_power__le__imp__le__exp,axiom,
    ! [A2: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ M ) @ ( power_power_rat @ A2 @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_2930_power__le__imp__le__exp,axiom,
    ! [A2: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_2931_power__le__imp__le__exp,axiom,
    ! [A2: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ( ord_less_eq_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_2932_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_2933_le__log__of__power,axiom,
    ! [B2: real,N: nat,M: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ B2 @ N ) @ M )
     => ( ( ord_less_real @ one_one_real @ B2 )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ M ) ) ) ) ).

% le_log_of_power
thf(fact_2934_nat__less__real__le,axiom,
    ( ord_less_nat
    = ( ^ [N4: nat,M3: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N4 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M3 ) ) ) ) ).

% nat_less_real_le
thf(fact_2935_nat__le__real__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [N4: nat,M3: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M3 ) @ one_one_real ) ) ) ) ).

% nat_le_real_less
thf(fact_2936_plus__one__helper2,axiom,
    ! [X: word_N3645301735248828278l_num1,N: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ X @ N )
     => ( ( ( plus_p361126936061061375l_num1 @ N @ one_on7727431528512463931l_num1 )
         != zero_z3563351764282998399l_num1 )
       => ( ord_le750835935415966154l_num1 @ X @ ( plus_p361126936061061375l_num1 @ N @ one_on7727431528512463931l_num1 ) ) ) ) ).

% plus_one_helper2
thf(fact_2937_Collect__imp__eq,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ( collec213857154873943460nt_int
        @ ^ [X2: product_prod_int_int] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) )
      = ( sup_su6024340866399070445nt_int @ ( uminus6221592323253981072nt_int @ ( collec213857154873943460nt_int @ P ) ) @ ( collec213857154873943460nt_int @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_2938_Collect__imp__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X2: nat] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) )
      = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) @ ( collect_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_2939_Collect__imp__eq,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( collect_complex
        @ ^ [X2: complex] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) )
      = ( sup_sup_set_complex @ ( uminus8566677241136511917omplex @ ( collect_complex @ P ) ) @ ( collect_complex @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_2940_Collect__imp__eq,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( collect_int
        @ ^ [X2: int] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) )
      = ( sup_sup_set_int @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) @ ( collect_int @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_2941_Collect__imp__eq,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ( collect_real
        @ ^ [X2: real] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) )
      = ( sup_sup_set_real @ ( uminus612125837232591019t_real @ ( collect_real @ P ) ) @ ( collect_real @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_2942_le__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% le_log2_of_power
thf(fact_2943_word__1__0,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1,X: nat] :
      ( ( ord_le3335648743751981014l_num1 @ ( plus_p361126936061061375l_num1 @ A2 @ one_on7727431528512463931l_num1 ) @ B2 )
     => ( ( ord_le750835935415966154l_num1 @ A2 @ ( semiri8819519690708144855l_num1 @ X ) )
       => ( ord_le750835935415966154l_num1 @ A2 @ B2 ) ) ) ).

% word_1_0
thf(fact_2944_dbl__def,axiom,
    ( neg_numeral_dbl_real
    = ( ^ [X2: real] : ( plus_plus_real @ X2 @ X2 ) ) ) ).

% dbl_def
thf(fact_2945_dbl__def,axiom,
    ( neg_numeral_dbl_rat
    = ( ^ [X2: rat] : ( plus_plus_rat @ X2 @ X2 ) ) ) ).

% dbl_def
thf(fact_2946_dbl__def,axiom,
    ( neg_numeral_dbl_int
    = ( ^ [X2: int] : ( plus_plus_int @ X2 @ X2 ) ) ) ).

% dbl_def
thf(fact_2947_realpow__square__minus__le,axiom,
    ! [U2: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% realpow_square_minus_le
thf(fact_2948_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_2949_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_2950_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_2951_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_2952_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_2953_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_2954_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_2955_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_2956_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_2957_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_2958_zero__neq__neg__one,axiom,
    ( zero_zero_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% zero_neq_neg_one
thf(fact_2959_zero__neq__neg__one,axiom,
    ( zero_zero_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% zero_neq_neg_one
thf(fact_2960_zero__neq__neg__one,axiom,
    ( zero_zero_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% zero_neq_neg_one
thf(fact_2961_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_2962_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_2963_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_2964_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_2965_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_2966_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_2967_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_2968_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_2969_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_2970_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_2971_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_2972_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ N )
     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% numeral_neq_neg_one
thf(fact_2973_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ N )
     != ( uminus_uminus_real @ one_one_real ) ) ).

% numeral_neq_neg_one
thf(fact_2974_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ N )
     != ( uminus_uminus_rat @ one_one_rat ) ) ).

% numeral_neq_neg_one
thf(fact_2975_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ N )
     != ( uminus_uminus_int @ one_one_int ) ) ).

% numeral_neq_neg_one
thf(fact_2976_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_2977_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_2978_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_2979_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_2980_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_2981_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_2982_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_2983_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_2984_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_2985_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_2986_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_2987_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_2988_pos__add__strict,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_real @ B2 @ C )
       => ( ord_less_real @ B2 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).

% pos_add_strict
thf(fact_2989_pos__add__strict,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).

% pos_add_strict
thf(fact_2990_pos__add__strict,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% pos_add_strict
thf(fact_2991_pos__add__strict,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ B2 @ C )
       => ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).

% pos_add_strict
thf(fact_2992_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numera7442385471795722001l_num1 @ ( bit0 @ N ) )
      = ( plus_p361126936061061375l_num1 @ ( numera7442385471795722001l_num1 @ N ) @ ( numera7442385471795722001l_num1 @ N ) ) ) ).

% numeral_Bit0
thf(fact_2993_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bit0 @ N ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).

% numeral_Bit0
thf(fact_2994_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).

% numeral_Bit0
thf(fact_2995_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).

% numeral_Bit0
thf(fact_2996_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).

% numeral_Bit0
thf(fact_2997_less__add__one,axiom,
    ! [A2: real] : ( ord_less_real @ A2 @ ( plus_plus_real @ A2 @ one_one_real ) ) ).

% less_add_one
thf(fact_2998_less__add__one,axiom,
    ! [A2: rat] : ( ord_less_rat @ A2 @ ( plus_plus_rat @ A2 @ one_one_rat ) ) ).

% less_add_one
thf(fact_2999_less__add__one,axiom,
    ! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).

% less_add_one
thf(fact_3000_less__add__one,axiom,
    ! [A2: int] : ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ one_one_int ) ) ).

% less_add_one
thf(fact_3001_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_3002_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_3003_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_3004_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_3005_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ ( numera7442385471795722001l_num1 @ X ) )
      = ( plus_p361126936061061375l_num1 @ ( numera7442385471795722001l_num1 @ X ) @ one_on7727431528512463931l_num1 ) ) ).

% one_plus_numeral_commute
thf(fact_3006_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_3007_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_3008_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_3009_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_3010_uint__0__iff,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( ( semiri7338730514057886004m1_int @ X )
        = zero_zero_int )
      = ( X = zero_z3563351764282998399l_num1 ) ) ).

% uint_0_iff
thf(fact_3011_uint__0__eq,axiom,
    ( ( semiri7338730514057886004m1_int @ zero_z3563351764282998399l_num1 )
    = zero_zero_int ) ).

% uint_0_eq
thf(fact_3012_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).

% not_numeral_le_zero
thf(fact_3013_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).

% not_numeral_le_zero
thf(fact_3014_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).

% not_numeral_le_zero
thf(fact_3015_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).

% not_numeral_le_zero
thf(fact_3016_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).

% zero_le_numeral
thf(fact_3017_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).

% zero_le_numeral
thf(fact_3018_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).

% zero_le_numeral
thf(fact_3019_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).

% zero_le_numeral
thf(fact_3020_not__one__le__zero,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% not_one_le_zero
thf(fact_3021_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_3022_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_3023_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_3024_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_3025_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_3026_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_3027_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_3028_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_3029_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_3030_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_3031_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_3032_uint__1__eq,axiom,
    ( ( semiri7338730514057886004m1_int @ one_on7727431528512463931l_num1 )
    = one_one_int ) ).

% uint_1_eq
thf(fact_3033_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).

% one_le_numeral
thf(fact_3034_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).

% one_le_numeral
thf(fact_3035_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).

% one_le_numeral
thf(fact_3036_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).

% one_le_numeral
thf(fact_3037_word__less__def,axiom,
    ( ord_le750835935415966154l_num1
    = ( ^ [A4: word_N3645301735248828278l_num1,B4: word_N3645301735248828278l_num1] : ( ord_less_int @ ( semiri7338730514057886004m1_int @ A4 ) @ ( semiri7338730514057886004m1_int @ B4 ) ) ) ) ).

% word_less_def
thf(fact_3038_zero__le__power,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ N ) ) ) ).

% zero_le_power
thf(fact_3039_zero__le__power,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ).

% zero_le_power
thf(fact_3040_zero__le__power,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ N ) ) ) ).

% zero_le_power
thf(fact_3041_zero__le__power,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A2 @ N ) ) ) ).

% zero_le_power
thf(fact_3042_zero__le__power,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ N ) ) ) ).

% zero_le_power
thf(fact_3043_power__mono,axiom,
    ! [A2: real,B2: real,N: nat] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
       => ( ord_less_eq_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ).

% power_mono
thf(fact_3044_power__mono,axiom,
    ! [A2: code_integer,B2: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ ( power_8256067586552552935nteger @ B2 @ N ) ) ) ) ).

% power_mono
thf(fact_3045_power__mono,axiom,
    ! [A2: rat,B2: rat,N: nat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
       => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ).

% power_mono
thf(fact_3046_power__mono,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
       => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ).

% power_mono
thf(fact_3047_power__mono,axiom,
    ! [A2: int,B2: int,N: nat] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
       => ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).

% power_mono
thf(fact_3048_norm__pre__pure__rule2,axiom,
    ! [B2: $o,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( B2
       => ( hoare_1429296392585015714_VEBTi @ one_one_assn @ F @ Q ) )
     => ( hoare_1429296392585015714_VEBTi @ ( pure_assn @ B2 ) @ F @ Q ) ) ).

% norm_pre_pure_rule2
thf(fact_3049_norm__pre__pure__rule2,axiom,
    ! [B2: $o,F: heap_Time_Heap_o,Q: $o > assn] :
      ( ( B2
       => ( hoare_hoare_triple_o @ one_one_assn @ F @ Q ) )
     => ( hoare_hoare_triple_o @ ( pure_assn @ B2 ) @ F @ Q ) ) ).

% norm_pre_pure_rule2
thf(fact_3050_one__le__power,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A2 )
     => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A2 @ N ) ) ) ).

% one_le_power
thf(fact_3051_one__le__power,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 )
     => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ).

% one_le_power
thf(fact_3052_one__le__power,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A2 )
     => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A2 @ N ) ) ) ).

% one_le_power
thf(fact_3053_one__le__power,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A2 )
     => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A2 @ N ) ) ) ).

% one_le_power
thf(fact_3054_one__le__power,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A2 )
     => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A2 @ N ) ) ) ).

% one_le_power
thf(fact_3055_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).

% of_nat_0_le_iff
thf(fact_3056_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( semiri4939895301339042750nteger @ N ) ) ).

% of_nat_0_le_iff
thf(fact_3057_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).

% of_nat_0_le_iff
thf(fact_3058_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).

% of_nat_0_le_iff
thf(fact_3059_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).

% of_nat_0_le_iff
thf(fact_3060_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K2 )
          & ( ( plus_plus_nat @ I @ K2 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_3061_log2__of__power__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log2_of_power_le
thf(fact_3062_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K2 )
               => ~ ( P @ I2 ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_3063_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_3064_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_3065_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_3066_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_3067_subset__singleton__iff,axiom,
    ! [X5: set_real,A2: real] :
      ( ( ord_less_eq_set_real @ X5 @ ( insert_real @ A2 @ bot_bot_set_real ) )
      = ( ( X5 = bot_bot_set_real )
        | ( X5
          = ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ).

% subset_singleton_iff
thf(fact_3068_subset__singleton__iff,axiom,
    ! [X5: set_o,A2: $o] :
      ( ( ord_less_eq_set_o @ X5 @ ( insert_o @ A2 @ bot_bot_set_o ) )
      = ( ( X5 = bot_bot_set_o )
        | ( X5
          = ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).

% subset_singleton_iff
thf(fact_3069_subset__singleton__iff,axiom,
    ! [X5: set_int,A2: int] :
      ( ( ord_less_eq_set_int @ X5 @ ( insert_int @ A2 @ bot_bot_set_int ) )
      = ( ( X5 = bot_bot_set_int )
        | ( X5
          = ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ).

% subset_singleton_iff
thf(fact_3070_subset__singleton__iff,axiom,
    ! [X5: set_nat,A2: nat] :
      ( ( ord_less_eq_set_nat @ X5 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
      = ( ( X5 = bot_bot_set_nat )
        | ( X5
          = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_3071_odd__nonzero,axiom,
    ! [Z: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
     != zero_zero_int ) ).

% odd_nonzero
thf(fact_3072_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_3073_int__gr__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_int @ K @ I )
     => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
       => ( ! [I3: int] :
              ( ( ord_less_int @ K @ I3 )
             => ( ( P @ I3 )
               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_gr_induct
thf(fact_3074_not__int__zless__negative,axiom,
    ! [N: nat,M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% not_int_zless_negative
thf(fact_3075_max__word__not__0,axiom,
    ( ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 )
   != zero_z3563351764282998399l_num1 ) ).

% max_word_not_0
thf(fact_3076_word__not__simps_I3_J,axiom,
    ! [Y: word_N3645301735248828278l_num1] :
      ~ ( ord_le750835935415966154l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ Y ) ).

% word_not_simps(3)
thf(fact_3077_word__order_Oextremum__strict,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ~ ( ord_le750835935415966154l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ A2 ) ).

% word_order.extremum_strict
thf(fact_3078_word__order_Onot__eq__extremum,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( A2
       != ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
      = ( ord_le750835935415966154l_num1 @ A2 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ) ).

% word_order.not_eq_extremum
thf(fact_3079_max__word__not__less,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ~ ( ord_le750835935415966154l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ X ) ).

% max_word_not_less
thf(fact_3080_nonneg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ~ ! [N2: nat] :
            ( K
           != ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% nonneg_int_cases
thf(fact_3081_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ? [N2: nat] :
          ( K
          = ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% zero_le_imp_eq_int
thf(fact_3082_word__induct2,axiom,
    ! [P: word_N3645301735248828278l_num1 > $o,N: word_N3645301735248828278l_num1] :
      ( ( P @ zero_z3563351764282998399l_num1 )
     => ( ! [N2: word_N3645301735248828278l_num1] :
            ( ( ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ N2 )
             != zero_z3563351764282998399l_num1 )
           => ( ( P @ N2 )
             => ( P @ ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ N2 ) ) ) )
       => ( P @ N ) ) ) ).

% word_induct2
thf(fact_3083_word__induct,axiom,
    ! [P: word_N3645301735248828278l_num1 > $o,M: word_N3645301735248828278l_num1] :
      ( ( P @ zero_z3563351764282998399l_num1 )
     => ( ! [N2: word_N3645301735248828278l_num1] :
            ( ( P @ N2 )
           => ( P @ ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ N2 ) ) )
       => ( P @ M ) ) ) ).

% word_induct
thf(fact_3084_lt1__neq0,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ one_on7727431528512463931l_num1 @ X )
      = ( X != zero_z3563351764282998399l_num1 ) ) ).

% lt1_neq0
thf(fact_3085_unsigned__eq__0__iff,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( ( semiri46416754965307273481_real @ W )
        = zero_zero_real )
      = ( W = zero_z3563351764282998399l_num1 ) ) ).

% unsigned_eq_0_iff
thf(fact_3086_unsigned__eq__0__iff,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( ( semiri6706090924480440544m1_rat @ W )
        = zero_zero_rat )
      = ( W = zero_z3563351764282998399l_num1 ) ) ).

% unsigned_eq_0_iff
thf(fact_3087_unsigned__eq__0__iff,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( ( semiri7341220984566936280m1_nat @ W )
        = zero_zero_nat )
      = ( W = zero_z3563351764282998399l_num1 ) ) ).

% unsigned_eq_0_iff
thf(fact_3088_unsigned__eq__0__iff,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( ( semiri7338730514057886004m1_int @ W )
        = zero_zero_int )
      = ( W = zero_z3563351764282998399l_num1 ) ) ).

% unsigned_eq_0_iff
thf(fact_3089_word__less__iff__unsigned,axiom,
    ( ord_le750835935415966154l_num1
    = ( ^ [A4: word_N3645301735248828278l_num1,B4: word_N3645301735248828278l_num1] : ( ord_less_int @ ( semiri7338730514057886004m1_int @ A4 ) @ ( semiri7338730514057886004m1_int @ B4 ) ) ) ) ).

% word_less_iff_unsigned
thf(fact_3090_log__of__power__le,axiom,
    ! [M: nat,B2: real,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B2 @ N ) )
     => ( ( 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 @ N ) ) ) ) ) ).

% log_of_power_le
thf(fact_3091_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_3092_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_3093_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_3094_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_3095_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_3096_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_3097_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_3098_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_3099_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_3100_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_3101_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_3102_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_3103_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numera7442385471795722001l_num1 @ ( bit0 @ N ) )
      = ( plus_p361126936061061375l_num1 @ ( numera7442385471795722001l_num1 @ N ) @ ( numera7442385471795722001l_num1 @ N ) ) ) ).

% numeral_code(2)
thf(fact_3104_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bit0 @ N ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).

% numeral_code(2)
thf(fact_3105_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).

% numeral_code(2)
thf(fact_3106_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).

% numeral_code(2)
thf(fact_3107_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).

% numeral_code(2)
thf(fact_3108_less__log__of__power,axiom,
    ! [B2: real,N: nat,M: real] :
      ( ( ord_less_real @ ( power_power_real @ B2 @ N ) @ M )
     => ( ( ord_less_real @ one_one_real @ B2 )
       => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ M ) ) ) ) ).

% less_log_of_power
thf(fact_3109_log__of__power__eq,axiom,
    ! [M: nat,B2: real,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( power_power_real @ B2 @ N ) )
     => ( ( ord_less_real @ one_one_real @ B2 )
       => ( ( semiri5074537144036343181t_real @ N )
          = ( log @ B2 @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).

% log_of_power_eq
thf(fact_3110_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).

% neg_numeral_less_zero
thf(fact_3111_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).

% neg_numeral_less_zero
thf(fact_3112_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).

% neg_numeral_less_zero
thf(fact_3113_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_3114_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_3115_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_3116_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 )
       => ? [N2: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ N2 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_3117_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 )
       => ? [N2: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B2 @ N2 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_3118_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_3119_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_3120_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_3121_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_3122_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_3123_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_3124_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_3125_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_3126_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_3127_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_3128_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_3129_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_3130_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_3131_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_3132_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_3133_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_3134_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_3135_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_3136_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_3137_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_3138_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_3139_uminus__numeral__One,axiom,
    ( ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ one ) )
    = ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ).

% uminus_numeral_One
thf(fact_3140_uminus__numeral__One,axiom,
    ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% uminus_numeral_One
thf(fact_3141_uminus__numeral__One,axiom,
    ( ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ one ) )
    = ( uminus_uminus_uint32 @ one_one_uint32 ) ) ).

% uminus_numeral_One
thf(fact_3142_uminus__numeral__One,axiom,
    ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% uminus_numeral_One
thf(fact_3143_uminus__numeral__One,axiom,
    ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% uminus_numeral_One
thf(fact_3144_uminus__numeral__One,axiom,
    ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% uminus_numeral_One
thf(fact_3145_zero__less__two,axiom,
    ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).

% zero_less_two
thf(fact_3146_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_3147_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_3148_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_3149_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_3150_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_3151_power__minus__Bit0,axiom,
    ! [X: uint32,K: num] :
      ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_uint32 @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_3152_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_3153_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_3154_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_3155_power__less__imp__less__base,axiom,
    ! [A2: code_integer,N: nat,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ ( power_8256067586552552935nteger @ B2 @ N ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ) ).

% power_less_imp_less_base
thf(fact_3156_power__less__imp__less__base,axiom,
    ! [A2: real,N: nat,B2: real] :
      ( ( ord_less_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ B2 @ N ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_real @ A2 @ B2 ) ) ) ).

% power_less_imp_less_base
thf(fact_3157_power__less__imp__less__base,axiom,
    ! [A2: rat,N: nat,B2: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ B2 @ N ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ A2 @ B2 ) ) ) ).

% power_less_imp_less_base
thf(fact_3158_power__less__imp__less__base,axiom,
    ! [A2: nat,N: nat,B2: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B2 @ N ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ A2 @ B2 ) ) ) ).

% power_less_imp_less_base
thf(fact_3159_power__less__imp__less__base,axiom,
    ! [A2: int,N: nat,B2: int] :
      ( ( ord_less_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B2 @ N ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ A2 @ B2 ) ) ) ).

% power_less_imp_less_base
thf(fact_3160_power__le__one,axiom,
    ! [A2: real,N: 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 @ N ) @ one_one_real ) ) ) ).

% power_le_one
thf(fact_3161_power__le__one,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ one_one_Code_integer ) ) ) ).

% power_le_one
thf(fact_3162_power__le__one,axiom,
    ! [A2: rat,N: 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 @ N ) @ one_one_rat ) ) ) ).

% power_le_one
thf(fact_3163_power__le__one,axiom,
    ! [A2: nat,N: 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 @ N ) @ one_one_nat ) ) ) ).

% power_le_one
thf(fact_3164_power__le__one,axiom,
    ! [A2: int,N: 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 @ N ) @ one_one_int ) ) ) ).

% power_le_one
thf(fact_3165_int__cases4,axiom,
    ! [M: int] :
      ( ! [N2: nat] :
          ( M
         != ( semiri1314217659103216013at_int @ N2 ) )
     => ~ ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( M
             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% int_cases4
thf(fact_3166_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_3167_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_3168_word__induct__less,axiom,
    ! [P: word_N3645301735248828278l_num1 > $o,M: word_N3645301735248828278l_num1] :
      ( ( P @ zero_z3563351764282998399l_num1 )
     => ( ! [N2: word_N3645301735248828278l_num1] :
            ( ( ord_le750835935415966154l_num1 @ N2 @ M )
           => ( ( P @ N2 )
             => ( P @ ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ N2 ) ) ) )
       => ( P @ M ) ) ) ).

% word_induct_less
thf(fact_3169_word__overflow,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ X @ ( plus_p361126936061061375l_num1 @ X @ one_on7727431528512463931l_num1 ) )
      | ( ( plus_p361126936061061375l_num1 @ X @ one_on7727431528512463931l_num1 )
        = zero_z3563351764282998399l_num1 ) ) ).

% word_overflow
thf(fact_3170_word__gr0__conv__Suc,axiom,
    ! [M: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ M )
     => ? [N2: word_N3645301735248828278l_num1] :
          ( M
          = ( plus_p361126936061061375l_num1 @ N2 @ one_on7727431528512463931l_num1 ) ) ) ).

% word_gr0_conv_Suc
thf(fact_3171_less__is__non__zero__p1,axiom,
    ! [A2: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ A2 @ K )
     => ( ( plus_p361126936061061375l_num1 @ A2 @ one_on7727431528512463931l_num1 )
       != zero_z3563351764282998399l_num1 ) ) ).

% less_is_non_zero_p1
thf(fact_3172_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_3173_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_3174_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_3175_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_3176_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_3177_power__eq__iff__eq__base,axiom,
    ! [N: nat,A2: real,B2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
         => ( ( ( power_power_real @ A2 @ N )
              = ( power_power_real @ B2 @ N ) )
            = ( A2 = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_3178_power__eq__iff__eq__base,axiom,
    ! [N: nat,A2: code_integer,B2: code_integer] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
         => ( ( ( power_8256067586552552935nteger @ A2 @ N )
              = ( power_8256067586552552935nteger @ B2 @ N ) )
            = ( A2 = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_3179_power__eq__iff__eq__base,axiom,
    ! [N: nat,A2: rat,B2: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
         => ( ( ( power_power_rat @ A2 @ N )
              = ( power_power_rat @ B2 @ N ) )
            = ( A2 = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_3180_power__eq__iff__eq__base,axiom,
    ! [N: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
         => ( ( ( power_power_nat @ A2 @ N )
              = ( power_power_nat @ B2 @ N ) )
            = ( A2 = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_3181_power__eq__iff__eq__base,axiom,
    ! [N: nat,A2: int,B2: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
         => ( ( ( power_power_int @ A2 @ N )
              = ( power_power_int @ B2 @ N ) )
            = ( A2 = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_3182_power__eq__imp__eq__base,axiom,
    ! [A2: real,N: nat,B2: real] :
      ( ( ( power_power_real @ A2 @ N )
        = ( power_power_real @ B2 @ N ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A2 = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_3183_power__eq__imp__eq__base,axiom,
    ! [A2: code_integer,N: nat,B2: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A2 @ N )
        = ( power_8256067586552552935nteger @ B2 @ N ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A2 = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_3184_power__eq__imp__eq__base,axiom,
    ! [A2: rat,N: nat,B2: rat] :
      ( ( ( power_power_rat @ A2 @ N )
        = ( power_power_rat @ B2 @ N ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A2 = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_3185_power__eq__imp__eq__base,axiom,
    ! [A2: nat,N: nat,B2: nat] :
      ( ( ( power_power_nat @ A2 @ N )
        = ( power_power_nat @ B2 @ N ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A2 = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_3186_power__eq__imp__eq__base,axiom,
    ! [A2: int,N: nat,B2: int] :
      ( ( ( power_power_int @ A2 @ N )
        = ( power_power_int @ B2 @ N ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A2 = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_3187_log2__of__power__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( M
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( semiri5074537144036343181t_real @ N )
        = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% log2_of_power_eq
thf(fact_3188_self__le__power,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ A2 @ ( power_power_real @ A2 @ N ) ) ) ) ).

% self_le_power
thf(fact_3189_self__le__power,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_le3102999989581377725nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ) ).

% self_le_power
thf(fact_3190_self__le__power,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_rat @ A2 @ ( power_power_rat @ A2 @ N ) ) ) ) ).

% self_le_power
thf(fact_3191_self__le__power,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ) ).

% self_le_power
thf(fact_3192_self__le__power,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ) ).

% self_le_power
thf(fact_3193_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_3194_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N2: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N2 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
       => ~ ! [N2: nat] :
              ( ( K
                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
             => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% int_cases3
thf(fact_3195_log__of__power__less,axiom,
    ! [M: nat,B2: real,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B2 @ N ) )
     => ( ( ord_less_real @ one_one_real @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_real @ ( log @ B2 @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_of_power_less
thf(fact_3196_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_3197_power2__nat__le__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% power2_nat_le_eq_le
thf(fact_3198_power2__nat__le__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% power2_nat_le_imp_le
thf(fact_3199_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_3200_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_3201_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_3202_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_3203_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_3204_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_3205_zero__le__power2,axiom,
    ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_3206_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_3207_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_3208_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_3209_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_3210_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_3211_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_3212_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_3213_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_3214_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_3215_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_3216_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_3217_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_3218_power__strict__mono,axiom,
    ! [A2: code_integer,B2: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ ( power_8256067586552552935nteger @ B2 @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_3219_power__strict__mono,axiom,
    ! [A2: real,B2: real,N: nat] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_3220_power__strict__mono,axiom,
    ! [A2: rat,B2: rat,N: nat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_3221_power__strict__mono,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_3222_power__strict__mono,axiom,
    ! [A2: int,B2: int,N: nat] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_3223_neg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ~ ! [N2: nat] :
            ( ( K
              = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% neg_int_cases
thf(fact_3224_two__realpow__ge__one,axiom,
    ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).

% two_realpow_ge_one
thf(fact_3225_less__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% less_log2_of_power
thf(fact_3226_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_3227_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_3228_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_3229_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_3230_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_3231_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_3232_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_3233_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_3234_arcosh__1,axiom,
    ( ( arcosh_real @ one_one_real )
    = zero_zero_real ) ).

% arcosh_1
thf(fact_3235_VEBT__internal_Otwo__realpow__ge__two,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) )
     => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).

% VEBT_internal.two_realpow_ge_two
thf(fact_3236_not__exp__less__eq__0__int,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).

% not_exp_less_eq_0_int
thf(fact_3237_nat__induct2,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ( P @ one_one_nat )
       => ( ! [N2: nat] :
              ( ( P @ N2 )
             => ( P @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct2
thf(fact_3238_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
       != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_left
thf(fact_3239_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_z3563351764282998399l_num1 )
     => ( ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ M )
       != zero_z3563351764282998399l_num1 ) ) ).

% exp_add_not_zero_imp_left
thf(fact_3240_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
thf(fact_3241_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
thf(fact_3242_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
       != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_right
thf(fact_3243_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_z3563351764282998399l_num1 )
     => ( ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N )
       != zero_z3563351764282998399l_num1 ) ) ).

% exp_add_not_zero_imp_right
thf(fact_3244_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
thf(fact_3245_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
thf(fact_3246_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_3247_discrete,axiom,
    ( ord_less_int
    = ( ^ [A4: int] : ( ord_less_eq_int @ ( plus_plus_int @ A4 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_3248_field__le__epsilon,axiom,
    ! [X: real,Y: real] :
      ( ! [E: real] :
          ( ( ord_less_real @ zero_zero_real @ E )
         => ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E ) ) )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% field_le_epsilon
thf(fact_3249_field__le__epsilon,axiom,
    ! [X: rat,Y: rat] :
      ( ! [E: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E )
         => ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y @ E ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_epsilon
thf(fact_3250_artanh__0,axiom,
    ( ( artanh_real @ zero_zero_real )
    = zero_zero_real ) ).

% artanh_0
thf(fact_3251_arsinh__0,axiom,
    ( ( arsinh_real @ zero_zero_real )
    = zero_zero_real ) ).

% arsinh_0
thf(fact_3252_of__nat__code,axiom,
    ( semiri8819519690708144855l_num1
    = ( ^ [N4: nat] :
          ( semiri2846968517960172219l_num1
          @ ^ [I4: word_N3645301735248828278l_num1] : ( plus_p361126936061061375l_num1 @ I4 @ one_on7727431528512463931l_num1 )
          @ N4
          @ zero_z3563351764282998399l_num1 ) ) ) ).

% of_nat_code
thf(fact_3253_of__nat__code,axiom,
    ( semiri681578069525770553at_rat
    = ( ^ [N4: nat] :
          ( semiri7787848453975740701ux_rat
          @ ^ [I4: rat] : ( plus_plus_rat @ I4 @ one_one_rat )
          @ N4
          @ zero_zero_rat ) ) ) ).

% of_nat_code
thf(fact_3254_of__nat__code,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N4: nat] :
          ( semiri8420488043553186161ux_int
          @ ^ [I4: int] : ( plus_plus_int @ I4 @ one_one_int )
          @ N4
          @ zero_zero_int ) ) ) ).

% of_nat_code
thf(fact_3255_of__nat__code,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N4: nat] :
          ( semiri7260567687927622513x_real
          @ ^ [I4: real] : ( plus_plus_real @ I4 @ one_one_real )
          @ N4
          @ zero_zero_real ) ) ) ).

% of_nat_code
thf(fact_3256_of__nat__code,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N4: nat] :
          ( semiri8422978514062236437ux_nat
          @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ one_one_nat )
          @ N4
          @ zero_zero_nat ) ) ) ).

% of_nat_code
thf(fact_3257_of__nat__code,axiom,
    ( semiri4939895301339042750nteger
    = ( ^ [N4: nat] :
          ( semiri4055485073559036834nteger
          @ ^ [I4: code_integer] : ( plus_p5714425477246183910nteger @ I4 @ one_one_Code_integer )
          @ N4
          @ zero_z3403309356797280102nteger ) ) ) ).

% of_nat_code
thf(fact_3258_of__nat__code,axiom,
    ( semiri8010041392384452111omplex
    = ( ^ [N4: nat] :
          ( semiri2816024913162550771omplex
          @ ^ [I4: complex] : ( plus_plus_complex @ I4 @ one_one_complex )
          @ N4
          @ zero_zero_complex ) ) ) ).

% of_nat_code
thf(fact_3259_subsetI,axiom,
    ! [A: set_real,B: set_real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A )
         => ( member_real @ X3 @ B ) )
     => ( ord_less_eq_set_real @ A @ B ) ) ).

% subsetI
thf(fact_3260_subsetI,axiom,
    ! [A: set_int,B: set_int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A )
         => ( member_int @ X3 @ B ) )
     => ( ord_less_eq_set_int @ A @ B ) ) ).

% subsetI
thf(fact_3261_subsetI,axiom,
    ! [A: set_complex,B: set_complex] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A )
         => ( member_complex @ X3 @ B ) )
     => ( ord_le211207098394363844omplex @ A @ B ) ) ).

% subsetI
thf(fact_3262_subsetI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_nat @ X3 @ B ) )
     => ( ord_less_eq_set_nat @ A @ B ) ) ).

% subsetI
thf(fact_3263_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_3264_ComplI,axiom,
    ! [C: real,A: set_real] :
      ( ~ ( member_real @ C @ A )
     => ( member_real @ C @ ( uminus612125837232591019t_real @ A ) ) ) ).

% ComplI
thf(fact_3265_ComplI,axiom,
    ! [C: nat,A: set_nat] :
      ( ~ ( member_nat @ C @ A )
     => ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) ) ) ).

% ComplI
thf(fact_3266_ComplI,axiom,
    ! [C: int,A: set_int] :
      ( ~ ( member_int @ C @ A )
     => ( member_int @ C @ ( uminus1532241313380277803et_int @ A ) ) ) ).

% ComplI
thf(fact_3267_ComplI,axiom,
    ! [C: complex,A: set_complex] :
      ( ~ ( member_complex @ C @ A )
     => ( member_complex @ C @ ( uminus8566677241136511917omplex @ A ) ) ) ).

% ComplI
thf(fact_3268_Compl__iff,axiom,
    ! [C: real,A: set_real] :
      ( ( member_real @ C @ ( uminus612125837232591019t_real @ A ) )
      = ( ~ ( member_real @ C @ A ) ) ) ).

% Compl_iff
thf(fact_3269_Compl__iff,axiom,
    ! [C: nat,A: set_nat] :
      ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) )
      = ( ~ ( member_nat @ C @ A ) ) ) ).

% Compl_iff
thf(fact_3270_Compl__iff,axiom,
    ! [C: int,A: set_int] :
      ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A ) )
      = ( ~ ( member_int @ C @ A ) ) ) ).

% Compl_iff
thf(fact_3271_Compl__iff,axiom,
    ! [C: complex,A: set_complex] :
      ( ( member_complex @ C @ ( uminus8566677241136511917omplex @ A ) )
      = ( ~ ( member_complex @ C @ A ) ) ) ).

% Compl_iff
thf(fact_3272_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% nat_add_left_cancel_le
thf(fact_3273_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_3274_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_3275_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ~ ( ( ord_less_eq_nat @ M @ N )
         => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).

% add_leE
thf(fact_3276_le__add1,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).

% le_add1
thf(fact_3277_le__add2,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).

% le_add2
thf(fact_3278_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_3279_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% add_leD1
thf(fact_3280_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ K @ N ) ) ).

% add_leD2
thf(fact_3281_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I @ K ) ) ) ).

% le_trans
thf(fact_3282_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_3283_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N2: nat] :
          ( L
          = ( plus_plus_nat @ K @ N2 ) ) ) ).

% le_Suc_ex
thf(fact_3284_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( M = N ) ) ) ).

% le_antisym
thf(fact_3285_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_le_mono
thf(fact_3286_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_le_mono1
thf(fact_3287_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
      | ( ord_less_eq_nat @ N @ M ) ) ).

% nat_le_linear
thf(fact_3288_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_le_add1
thf(fact_3289_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_le_add2
thf(fact_3290_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N4: nat] :
        ? [K3: nat] :
          ( N4
          = ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).

% nat_le_iff_add
thf(fact_3291_complete__real,axiom,
    ! [S: set_real] :
      ( ? [X6: real] : ( member_real @ X6 @ S )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( member_real @ X3 @ S )
           => ( ord_less_eq_real @ X3 @ Z5 ) )
       => ? [Y4: real] :
            ( ! [X6: real] :
                ( ( member_real @ X6 @ S )
               => ( ord_less_eq_real @ X6 @ Y4 ) )
            & ! [Z5: real] :
                ( ! [X3: real] :
                    ( ( member_real @ X3 @ S )
                   => ( ord_less_eq_real @ X3 @ Z5 ) )
               => ( ord_less_eq_real @ Y4 @ Z5 ) ) ) ) ) ).

% complete_real
thf(fact_3292_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B2: nat] :
      ( ( P @ K )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B2 ) )
       => ? [X3: nat] :
            ( ( P @ X3 )
            & ! [Y5: nat] :
                ( ( P @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_3293_ComplD,axiom,
    ! [C: real,A: set_real] :
      ( ( member_real @ C @ ( uminus612125837232591019t_real @ A ) )
     => ~ ( member_real @ C @ A ) ) ).

% ComplD
thf(fact_3294_ComplD,axiom,
    ! [C: nat,A: set_nat] :
      ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) )
     => ~ ( member_nat @ C @ A ) ) ).

% ComplD
thf(fact_3295_ComplD,axiom,
    ! [C: int,A: set_int] :
      ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A ) )
     => ~ ( member_int @ C @ A ) ) ).

% ComplD
thf(fact_3296_ComplD,axiom,
    ! [C: complex,A: set_complex] :
      ( ( member_complex @ C @ ( uminus8566677241136511917omplex @ A ) )
     => ~ ( member_complex @ C @ A ) ) ).

% ComplD
thf(fact_3297_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_3298_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_3299_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_3300_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_3301_subsetD,axiom,
    ! [A: set_real,B: set_real,C: real] :
      ( ( ord_less_eq_set_real @ A @ B )
     => ( ( member_real @ C @ A )
       => ( member_real @ C @ B ) ) ) ).

% subsetD
thf(fact_3302_subsetD,axiom,
    ! [A: set_int,B: set_int,C: int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( member_int @ C @ A )
       => ( member_int @ C @ B ) ) ) ).

% subsetD
thf(fact_3303_subsetD,axiom,
    ! [A: set_complex,B: set_complex,C: complex] :
      ( ( ord_le211207098394363844omplex @ A @ B )
     => ( ( member_complex @ C @ A )
       => ( member_complex @ C @ B ) ) ) ).

% subsetD
thf(fact_3304_subsetD,axiom,
    ! [A: set_nat,B: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( member_nat @ C @ A )
       => ( member_nat @ C @ B ) ) ) ).

% subsetD
thf(fact_3305_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_3306_subset__eq,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A3: set_real,B3: set_real] :
        ! [X2: real] :
          ( ( member_real @ X2 @ A3 )
         => ( member_real @ X2 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_3307_subset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A3: set_int,B3: set_int] :
        ! [X2: int] :
          ( ( member_int @ X2 @ A3 )
         => ( member_int @ X2 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_3308_subset__eq,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [A3: set_complex,B3: set_complex] :
        ! [X2: complex] :
          ( ( member_complex @ X2 @ A3 )
         => ( member_complex @ X2 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_3309_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ A3 )
         => ( member_nat @ X2 @ B3 ) ) ) ) ).

% subset_eq
thf(fact_3310_equalityD1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
     => ( ord_less_eq_set_nat @ A @ B ) ) ).

% equalityD1
thf(fact_3311_Set_OequalityD2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
     => ( ord_less_eq_set_nat @ B @ A ) ) ).

% Set.equalityD2
thf(fact_3312_subset__iff,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A3: set_real,B3: set_real] :
        ! [T2: real] :
          ( ( member_real @ T2 @ A3 )
         => ( member_real @ T2 @ B3 ) ) ) ) ).

% subset_iff
thf(fact_3313_subset__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A3: set_int,B3: set_int] :
        ! [T2: int] :
          ( ( member_int @ T2 @ A3 )
         => ( member_int @ T2 @ B3 ) ) ) ) ).

% subset_iff
thf(fact_3314_subset__iff,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [A3: set_complex,B3: set_complex] :
        ! [T2: complex] :
          ( ( member_complex @ T2 @ A3 )
         => ( member_complex @ T2 @ B3 ) ) ) ) ).

% subset_iff
thf(fact_3315_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
        ! [T2: nat] :
          ( ( member_nat @ T2 @ A3 )
         => ( member_nat @ T2 @ B3 ) ) ) ) ).

% subset_iff
thf(fact_3316_subset__refl,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% subset_refl
thf(fact_3317_Collect__mono,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ! [X3: complex] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).

% Collect_mono
thf(fact_3318_Collect__mono,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ! [X3: product_prod_int_int] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) ) ) ).

% Collect_mono
thf(fact_3319_Collect__mono,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).

% Collect_mono
thf(fact_3320_Collect__mono,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_3321_subset__trans,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ C2 )
       => ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).

% subset_trans
thf(fact_3322_set__eq__subset,axiom,
    ( ( ^ [Y3: set_nat,Z2: set_nat] : Y3 = Z2 )
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ A3 @ B3 )
          & ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).

% set_eq_subset
thf(fact_3323_uminus__set__def,axiom,
    ( uminus612125837232591019t_real
    = ( ^ [A3: set_real] :
          ( collect_real
          @ ( uminus_uminus_real_o
            @ ^ [X2: real] : ( member_real @ X2 @ A3 ) ) ) ) ) ).

% uminus_set_def
thf(fact_3324_uminus__set__def,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A3: set_nat] :
          ( collect_nat
          @ ( uminus_uminus_nat_o
            @ ^ [X2: nat] : ( member_nat @ X2 @ A3 ) ) ) ) ) ).

% uminus_set_def
thf(fact_3325_uminus__set__def,axiom,
    ( uminus8566677241136511917omplex
    = ( ^ [A3: set_complex] :
          ( collect_complex
          @ ( uminus1680532995456772888plex_o
            @ ^ [X2: complex] : ( member_complex @ X2 @ A3 ) ) ) ) ) ).

% uminus_set_def
thf(fact_3326_uminus__set__def,axiom,
    ( uminus6221592323253981072nt_int
    = ( ^ [A3: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ( uminus7117520113953359693_int_o
            @ ^ [X2: product_prod_int_int] : ( member5262025264175285858nt_int @ X2 @ A3 ) ) ) ) ) ).

% uminus_set_def
thf(fact_3327_uminus__set__def,axiom,
    ( uminus1532241313380277803et_int
    = ( ^ [A3: set_int] :
          ( collect_int
          @ ( uminus_uminus_int_o
            @ ^ [X2: int] : ( member_int @ X2 @ A3 ) ) ) ) ) ).

% uminus_set_def
thf(fact_3328_less__eq__set__def,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A3: set_real,B3: set_real] :
          ( ord_less_eq_real_o
          @ ^ [X2: real] : ( member_real @ X2 @ A3 )
          @ ^ [X2: real] : ( member_real @ X2 @ B3 ) ) ) ) ).

% less_eq_set_def
thf(fact_3329_less__eq__set__def,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A3: set_int,B3: set_int] :
          ( ord_less_eq_int_o
          @ ^ [X2: int] : ( member_int @ X2 @ A3 )
          @ ^ [X2: int] : ( member_int @ X2 @ B3 ) ) ) ) ).

% less_eq_set_def
thf(fact_3330_less__eq__set__def,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [A3: set_complex,B3: set_complex] :
          ( ord_le4573692005234683329plex_o
          @ ^ [X2: complex] : ( member_complex @ X2 @ A3 )
          @ ^ [X2: complex] : ( member_complex @ X2 @ B3 ) ) ) ) ).

% less_eq_set_def
thf(fact_3331_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X2: nat] : ( member_nat @ X2 @ A3 )
          @ ^ [X2: nat] : ( member_nat @ X2 @ B3 ) ) ) ) ).

% less_eq_set_def
thf(fact_3332_Collect__mono__iff,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
      = ( ! [X2: complex] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_3333_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 ) )
      = ( ! [X2: product_prod_int_int] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_3334_Collect__mono__iff,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
      = ( ! [X2: int] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_3335_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X2: nat] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_3336_verit__la__generic,axiom,
    ! [A2: int,X: int] :
      ( ( ord_less_eq_int @ A2 @ X )
      | ( A2 = X )
      | ( ord_less_eq_int @ X @ A2 ) ) ).

% verit_la_generic
thf(fact_3337_pred__subset__eq,axiom,
    ! [R: set_real,S: set_real] :
      ( ( ord_less_eq_real_o
        @ ^ [X2: real] : ( member_real @ X2 @ R )
        @ ^ [X2: real] : ( member_real @ X2 @ S ) )
      = ( ord_less_eq_set_real @ R @ S ) ) ).

% pred_subset_eq
thf(fact_3338_pred__subset__eq,axiom,
    ! [R: set_int,S: set_int] :
      ( ( ord_less_eq_int_o
        @ ^ [X2: int] : ( member_int @ X2 @ R )
        @ ^ [X2: int] : ( member_int @ X2 @ S ) )
      = ( ord_less_eq_set_int @ R @ S ) ) ).

% pred_subset_eq
thf(fact_3339_pred__subset__eq,axiom,
    ! [R: set_complex,S: set_complex] :
      ( ( ord_le4573692005234683329plex_o
        @ ^ [X2: complex] : ( member_complex @ X2 @ R )
        @ ^ [X2: complex] : ( member_complex @ X2 @ S ) )
      = ( ord_le211207098394363844omplex @ R @ S ) ) ).

% pred_subset_eq
thf(fact_3340_pred__subset__eq,axiom,
    ! [R: set_nat,S: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X2: nat] : ( member_nat @ X2 @ R )
        @ ^ [X2: nat] : ( member_nat @ X2 @ S ) )
      = ( ord_less_eq_set_nat @ R @ S ) ) ).

% pred_subset_eq
thf(fact_3341_Collect__neg__eq,axiom,
    ! [P: nat > $o] :
      ( ( collect_nat
        @ ^ [X2: nat] :
            ~ ( P @ X2 ) )
      = ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_3342_Collect__neg__eq,axiom,
    ! [P: complex > $o] :
      ( ( collect_complex
        @ ^ [X2: complex] :
            ~ ( P @ X2 ) )
      = ( uminus8566677241136511917omplex @ ( collect_complex @ P ) ) ) ).

% Collect_neg_eq
thf(fact_3343_Collect__neg__eq,axiom,
    ! [P: product_prod_int_int > $o] :
      ( ( collec213857154873943460nt_int
        @ ^ [X2: product_prod_int_int] :
            ~ ( P @ X2 ) )
      = ( uminus6221592323253981072nt_int @ ( collec213857154873943460nt_int @ P ) ) ) ).

% Collect_neg_eq
thf(fact_3344_Collect__neg__eq,axiom,
    ! [P: int > $o] :
      ( ( collect_int
        @ ^ [X2: int] :
            ~ ( P @ X2 ) )
      = ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ).

% Collect_neg_eq
thf(fact_3345_Compl__eq,axiom,
    ( uminus612125837232591019t_real
    = ( ^ [A3: set_real] :
          ( collect_real
          @ ^ [X2: real] :
              ~ ( member_real @ X2 @ A3 ) ) ) ) ).

% Compl_eq
thf(fact_3346_Compl__eq,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A3: set_nat] :
          ( collect_nat
          @ ^ [X2: nat] :
              ~ ( member_nat @ X2 @ A3 ) ) ) ) ).

% Compl_eq
thf(fact_3347_Compl__eq,axiom,
    ( uminus8566677241136511917omplex
    = ( ^ [A3: set_complex] :
          ( collect_complex
          @ ^ [X2: complex] :
              ~ ( member_complex @ X2 @ A3 ) ) ) ) ).

% Compl_eq
thf(fact_3348_Compl__eq,axiom,
    ( uminus6221592323253981072nt_int
    = ( ^ [A3: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ^ [X2: product_prod_int_int] :
              ~ ( member5262025264175285858nt_int @ X2 @ A3 ) ) ) ) ).

% Compl_eq
thf(fact_3349_Compl__eq,axiom,
    ( uminus1532241313380277803et_int
    = ( ^ [A3: set_int] :
          ( collect_int
          @ ^ [X2: int] :
              ~ ( member_int @ X2 @ A3 ) ) ) ) ).

% Compl_eq
thf(fact_3350_Collect__subset,axiom,
    ! [A: set_real,P: real > $o] :
      ( ord_less_eq_set_real
      @ ( collect_real
        @ ^ [X2: real] :
            ( ( member_real @ X2 @ A )
            & ( P @ X2 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_3351_Collect__subset,axiom,
    ! [A: set_complex,P: complex > $o] :
      ( ord_le211207098394363844omplex
      @ ( collect_complex
        @ ^ [X2: complex] :
            ( ( member_complex @ X2 @ A )
            & ( P @ X2 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_3352_Collect__subset,axiom,
    ! [A: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
      ( ord_le2843351958646193337nt_int
      @ ( collec213857154873943460nt_int
        @ ^ [X2: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X2 @ A )
            & ( P @ X2 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_3353_Collect__subset,axiom,
    ! [A: set_int,P: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X2: int] :
            ( ( member_int @ X2 @ A )
            & ( P @ X2 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_3354_Collect__subset,axiom,
    ! [A: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X2: nat] :
            ( ( member_nat @ X2 @ A )
            & ( P @ X2 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_3355_ucast__1,axiom,
    ( ( semiri1312839663145358974l_num1 @ one_on7727431528512463931l_num1 )
    = one_on7727431528512463931l_num1 ) ).

% ucast_1
thf(fact_3356_ucast__0__I,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( X = zero_z3563351764282998399l_num1 )
     => ( ( semiri1312839663145358974l_num1 @ X )
        = zero_z3563351764282998399l_num1 ) ) ).

% ucast_0_I
thf(fact_3357_ucast__0,axiom,
    ( ( semiri1312839663145358974l_num1 @ zero_z3563351764282998399l_num1 )
    = zero_z3563351764282998399l_num1 ) ).

% ucast_0
thf(fact_3358_word__unat__Rep__inject1,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( ( semiri7341220984566936280m1_nat @ X )
        = ( semiri7341220984566936280m1_nat @ one_on7727431528512463931l_num1 ) )
      = ( X = one_on7727431528512463931l_num1 ) ) ).

% word_unat_Rep_inject1
thf(fact_3359_un__ui__le,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( ord_less_eq_nat @ ( semiri7341220984566936280m1_nat @ A2 ) @ ( semiri7341220984566936280m1_nat @ B2 ) )
      = ( ord_less_eq_int @ ( semiri7338730514057886004m1_int @ A2 ) @ ( semiri7338730514057886004m1_int @ B2 ) ) ) ).

% un_ui_le
thf(fact_3360_unat__eq__zero,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( ( semiri7341220984566936280m1_nat @ X )
        = zero_zero_nat )
      = ( X = zero_z3563351764282998399l_num1 ) ) ).

% unat_eq_zero
thf(fact_3361_unat__0,axiom,
    ( ( semiri7341220984566936280m1_nat @ zero_z3563351764282998399l_num1 )
    = zero_zero_nat ) ).

% unat_0
thf(fact_3362_unat__1,axiom,
    ( ( semiri7341220984566936280m1_nat @ one_on7727431528512463931l_num1 )
    = one_one_nat ) ).

% unat_1
thf(fact_3363_uint__nat,axiom,
    ( semiri7338730514057886004m1_int
    = ( ^ [W2: word_N3645301735248828278l_num1] : ( semiri1314217659103216013at_int @ ( semiri7341220984566936280m1_nat @ W2 ) ) ) ) ).

% uint_nat
thf(fact_3364_unat__1__0,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ one_on7727431528512463931l_num1 @ X )
      = ( ord_less_nat @ zero_zero_nat @ ( semiri7341220984566936280m1_nat @ X ) ) ) ).

% unat_1_0
thf(fact_3365_unat__max__word__pos,axiom,
    ord_less_nat @ zero_zero_nat @ ( semiri7341220984566936280m1_nat @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ).

% unat_max_word_pos
thf(fact_3366_unat__gt__0,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( ord_less_nat @ zero_zero_nat @ ( semiri7341220984566936280m1_nat @ X ) )
      = ( X != zero_z3563351764282998399l_num1 ) ) ).

% unat_gt_0
thf(fact_3367_word__overflow__unat,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( ( semiri7341220984566936280m1_nat @ ( plus_p361126936061061375l_num1 @ X @ one_on7727431528512463931l_num1 ) )
        = ( plus_plus_nat @ ( semiri7341220984566936280m1_nat @ X ) @ one_one_nat ) )
      | ( ( plus_p361126936061061375l_num1 @ X @ one_on7727431528512463931l_num1 )
        = zero_z3563351764282998399l_num1 ) ) ).

% word_overflow_unat
thf(fact_3368_linordered__field__no__ub,axiom,
    ! [X6: real] :
    ? [X_1: real] : ( ord_less_real @ X6 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_3369_linordered__field__no__ub,axiom,
    ! [X6: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X6 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_3370_linordered__field__no__lb,axiom,
    ! [X6: real] :
    ? [Y4: real] : ( ord_less_real @ Y4 @ X6 ) ).

% linordered_field_no_lb
thf(fact_3371_linordered__field__no__lb,axiom,
    ! [X6: rat] :
    ? [Y4: rat] : ( ord_less_rat @ Y4 @ X6 ) ).

% linordered_field_no_lb
thf(fact_3372_lt__plus__1__le__word,axiom,
    ! [N: nat,MaxBound: word_N3645301735248828278l_num1,X: word_N3645301735248828278l_num1] :
      ( ( ord_less_nat @ N @ ( semiri7341220984566936280m1_nat @ MaxBound ) )
     => ( ( ord_le750835935415966154l_num1 @ X @ ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ ( semiri8819519690708144855l_num1 @ N ) ) )
        = ( ord_le3335648743751981014l_num1 @ X @ ( semiri8819519690708144855l_num1 @ N ) ) ) ) ).

% lt_plus_1_le_word
thf(fact_3373_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,N: 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 @ N ) @ one_one_int ) )
          = ( ( 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 ) ) ) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
thf(fact_3374_ceiling__log__nat__eq__if,axiom,
    ! [B2: nat,N: nat,K: 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 ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
         => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).

% ceiling_log_nat_eq_if
thf(fact_3375_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,N: 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 @ N ) )
          = ( ( 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 ) ) ) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
thf(fact_3376_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_3377_floor__zero,axiom,
    ( ( archim6058952711729229775r_real @ zero_zero_real )
    = zero_zero_int ) ).

% floor_zero
thf(fact_3378_floor__zero,axiom,
    ( ( archim3151403230148437115or_rat @ zero_zero_rat )
    = zero_zero_int ) ).

% floor_zero
thf(fact_3379_floor__numeral,axiom,
    ! [V: num] :
      ( ( archim6058952711729229775r_real @ ( numeral_numeral_real @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% floor_numeral
thf(fact_3380_floor__numeral,axiom,
    ! [V: num] :
      ( ( archim3151403230148437115or_rat @ ( numeral_numeral_rat @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% floor_numeral
thf(fact_3381_ceiling__zero,axiom,
    ( ( archim2889992004027027881ng_rat @ zero_zero_rat )
    = zero_zero_int ) ).

% ceiling_zero
thf(fact_3382_ceiling__zero,axiom,
    ( ( archim7802044766580827645g_real @ zero_zero_real )
    = zero_zero_int ) ).

% ceiling_zero
thf(fact_3383_ceiling__numeral,axiom,
    ! [V: num] :
      ( ( archim7802044766580827645g_real @ ( numeral_numeral_real @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% ceiling_numeral
thf(fact_3384_ceiling__numeral,axiom,
    ! [V: num] :
      ( ( archim2889992004027027881ng_rat @ ( numeral_numeral_rat @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% ceiling_numeral
thf(fact_3385_floor__one,axiom,
    ( ( archim6058952711729229775r_real @ one_one_real )
    = one_one_int ) ).

% floor_one
thf(fact_3386_floor__one,axiom,
    ( ( archim3151403230148437115or_rat @ one_one_rat )
    = one_one_int ) ).

% floor_one
thf(fact_3387_ceiling__one,axiom,
    ( ( archim2889992004027027881ng_rat @ one_one_rat )
    = one_one_int ) ).

% ceiling_one
thf(fact_3388_ceiling__one,axiom,
    ( ( archim7802044766580827645g_real @ one_one_real )
    = one_one_int ) ).

% ceiling_one
thf(fact_3389_floor__of__nat,axiom,
    ! [N: nat] :
      ( ( archim6058952711729229775r_real @ ( semiri5074537144036343181t_real @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% floor_of_nat
thf(fact_3390_floor__of__nat,axiom,
    ! [N: nat] :
      ( ( archim3151403230148437115or_rat @ ( semiri681578069525770553at_rat @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% floor_of_nat
thf(fact_3391_ceiling__of__nat,axiom,
    ! [N: nat] :
      ( ( archim7802044766580827645g_real @ ( semiri5074537144036343181t_real @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% ceiling_of_nat
thf(fact_3392_zero__le__floor,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% zero_le_floor
thf(fact_3393_zero__le__floor,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ X ) ) ).

% zero_le_floor
thf(fact_3394_numeral__le__floor,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( numeral_numeral_real @ V ) @ X ) ) ).

% numeral_le_floor
thf(fact_3395_numeral__le__floor,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).

% numeral_le_floor
thf(fact_3396_floor__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% floor_less_zero
thf(fact_3397_floor__less__zero,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
      = ( ord_less_rat @ X @ zero_zero_rat ) ) ).

% floor_less_zero
thf(fact_3398_floor__less__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_real @ X @ ( numeral_numeral_real @ V ) ) ) ).

% floor_less_numeral
thf(fact_3399_floor__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).

% floor_less_numeral
thf(fact_3400_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_3401_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_3402_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_3403_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_3404_zero__less__floor,axiom,
    ! [X: real] :
      ( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ one_one_real @ X ) ) ).

% zero_less_floor
thf(fact_3405_zero__less__floor,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).

% zero_less_floor
thf(fact_3406_floor__le__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ zero_zero_int )
      = ( ord_less_real @ X @ one_one_real ) ) ).

% floor_le_zero
thf(fact_3407_floor__le__zero,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
      = ( ord_less_rat @ X @ one_one_rat ) ) ).

% floor_le_zero
thf(fact_3408_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_3409_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_3410_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_3411_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_3412_one__le__floor,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ one_one_real @ X ) ) ).

% one_le_floor
thf(fact_3413_one__le__floor,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).

% one_le_floor
thf(fact_3414_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_3415_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_3416_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_3417_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_3418_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_3419_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_3420_floor__less__one,axiom,
    ! [X: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
      = ( ord_less_real @ X @ one_one_real ) ) ).

% floor_less_one
thf(fact_3421_floor__less__one,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
      = ( ord_less_rat @ X @ one_one_rat ) ) ).

% floor_less_one
thf(fact_3422_floor__neg__numeral,axiom,
    ! [V: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).

% floor_neg_numeral
thf(fact_3423_floor__neg__numeral,axiom,
    ! [V: num] :
      ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).

% floor_neg_numeral
thf(fact_3424_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_3425_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_3426_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_3427_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_3428_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_3429_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_3430_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_3431_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_3432_floor__numeral__power,axiom,
    ! [X: num,N: nat] :
      ( ( archim6058952711729229775r_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
      = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).

% floor_numeral_power
thf(fact_3433_floor__numeral__power,axiom,
    ! [X: num,N: nat] :
      ( ( archim3151403230148437115or_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
      = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).

% floor_numeral_power
thf(fact_3434_ceiling__numeral__power,axiom,
    ! [X: num,N: nat] :
      ( ( archim7802044766580827645g_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
      = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).

% ceiling_numeral_power
thf(fact_3435_ceiling__numeral__power,axiom,
    ! [X: num,N: nat] :
      ( ( archim2889992004027027881ng_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
      = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).

% ceiling_numeral_power
thf(fact_3436_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_3437_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_3438_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_3439_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_3440_numeral__less__floor,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).

% numeral_less_floor
thf(fact_3441_numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).

% numeral_less_floor
thf(fact_3442_floor__le__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_real @ X @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).

% floor_le_numeral
thf(fact_3443_floor__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_rat @ X @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).

% floor_le_numeral
thf(fact_3444_one__less__floor,axiom,
    ! [X: real] :
      ( ( ord_less_int @ one_one_int @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ).

% one_less_floor
thf(fact_3445_one__less__floor,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) ) ).

% one_less_floor
thf(fact_3446_floor__le__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
      = ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% floor_le_one
thf(fact_3447_floor__le__one,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
      = ( ord_less_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% floor_le_one
thf(fact_3448_neg__numeral__le__floor,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).

% neg_numeral_le_floor
thf(fact_3449_neg__numeral__le__floor,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).

% neg_numeral_le_floor
thf(fact_3450_floor__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).

% floor_less_neg_numeral
thf(fact_3451_floor__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).

% floor_less_neg_numeral
thf(fact_3452_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_3453_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_3454_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_3455_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_3456_neg__numeral__less__floor,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).

% neg_numeral_less_floor
thf(fact_3457_neg__numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).

% neg_numeral_less_floor
thf(fact_3458_floor__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_real @ X @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).

% floor_le_neg_numeral
thf(fact_3459_floor__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_rat @ X @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).

% floor_le_neg_numeral
thf(fact_3460_floor__le__ceiling,axiom,
    ! [X: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).

% floor_le_ceiling
thf(fact_3461_floor__le__ceiling,axiom,
    ! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim2889992004027027881ng_rat @ X ) ) ).

% floor_le_ceiling
thf(fact_3462_ceiling__minus,axiom,
    ! [X: real] :
      ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ X ) )
      = ( uminus_uminus_int @ ( archim6058952711729229775r_real @ X ) ) ) ).

% ceiling_minus
thf(fact_3463_ceiling__minus,axiom,
    ! [X: rat] :
      ( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ X ) )
      = ( uminus_uminus_int @ ( archim3151403230148437115or_rat @ X ) ) ) ).

% ceiling_minus
thf(fact_3464_floor__minus,axiom,
    ! [X: real] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ X ) )
      = ( uminus_uminus_int @ ( archim7802044766580827645g_real @ X ) ) ) ).

% floor_minus
thf(fact_3465_floor__minus,axiom,
    ! [X: rat] :
      ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ X ) )
      = ( uminus_uminus_int @ ( archim2889992004027027881ng_rat @ X ) ) ) ).

% floor_minus
thf(fact_3466_ceiling__def,axiom,
    ( archim7802044766580827645g_real
    = ( ^ [X2: real] : ( uminus_uminus_int @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ X2 ) ) ) ) ) ).

% ceiling_def
thf(fact_3467_ceiling__def,axiom,
    ( archim2889992004027027881ng_rat
    = ( ^ [X2: rat] : ( uminus_uminus_int @ ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ X2 ) ) ) ) ) ).

% ceiling_def
thf(fact_3468_floor__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) ).

% floor_mono
thf(fact_3469_floor__mono,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) ).

% floor_mono
thf(fact_3470_floor__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) )
     => ( ord_less_real @ X @ Y ) ) ).

% floor_less_cancel
thf(fact_3471_floor__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) )
     => ( ord_less_rat @ X @ Y ) ) ).

% floor_less_cancel
thf(fact_3472_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_3473_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_3474_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_3475_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_3476_le__floor__add,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) @ ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) ) ) ).

% le_floor_add
thf(fact_3477_le__floor__add,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) ) ) ).

% le_floor_add
thf(fact_3478_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_3479_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_3480_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_3481_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_3482_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_3483_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_3484_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_3485_set__notEmptyE,axiom,
    ! [S: set_complex] :
      ( ( S != bot_bot_set_complex )
     => ~ ! [X3: complex] :
            ~ ( member_complex @ X3 @ S ) ) ).

% set_notEmptyE
thf(fact_3486_set__notEmptyE,axiom,
    ! [S: set_real] :
      ( ( S != bot_bot_set_real )
     => ~ ! [X3: real] :
            ~ ( member_real @ X3 @ S ) ) ).

% set_notEmptyE
thf(fact_3487_set__notEmptyE,axiom,
    ! [S: set_o] :
      ( ( S != bot_bot_set_o )
     => ~ ! [X3: $o] :
            ~ ( member_o @ X3 @ S ) ) ).

% set_notEmptyE
thf(fact_3488_set__notEmptyE,axiom,
    ! [S: set_nat] :
      ( ( S != bot_bot_set_nat )
     => ~ ! [X3: nat] :
            ~ ( member_nat @ X3 @ S ) ) ).

% set_notEmptyE
thf(fact_3489_set__notEmptyE,axiom,
    ! [S: set_int] :
      ( ( S != bot_bot_set_int )
     => ~ ! [X3: int] :
            ~ ( member_int @ X3 @ S ) ) ).

% set_notEmptyE
thf(fact_3490_one__add__floor,axiom,
    ! [X: real] :
      ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).

% one_add_floor
thf(fact_3491_one__add__floor,axiom,
    ! [X: rat] :
      ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ one_one_rat ) ) ) ).

% one_add_floor
thf(fact_3492_exists__leI,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [N5: nat] :
            ( ( ord_less_nat @ N5 @ N )
           => ~ ( P @ N5 ) )
       => ( P @ N ) )
     => ? [N6: nat] :
          ( ( ord_less_eq_nat @ N6 @ N )
          & ( P @ N6 ) ) ) ).

% exists_leI
thf(fact_3493_floor__log__nat__eq__if,axiom,
    ! [B2: nat,N: nat,K: 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 ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
         => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% floor_log_nat_eq_if
thf(fact_3494_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_3495_floor__log2__div2,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).

% floor_log2_div2
thf(fact_3496_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8115118780965096967l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ K ) ) )
      = ( uminus8244633308260627903l_num1 @ ( neg_nu93272222329896523l_num1 @ ( numera7442385471795722001l_num1 @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_3497_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_3498_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu4269007558841261821uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ K ) ) )
      = ( uminus_uminus_uint32 @ ( neg_nu965353292909893953uint32 @ ( numera9087168376688890119uint32 @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_3499_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
      = ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_3500_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
      = ( uminus_uminus_rat @ ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_3501_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_3502_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu93272222329896523l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ K ) ) )
      = ( uminus8244633308260627903l_num1 @ ( neg_nu8115118780965096967l_num1 @ ( numera7442385471795722001l_num1 @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_3503_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_3504_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu965353292909893953uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ K ) ) )
      = ( uminus_uminus_uint32 @ ( neg_nu4269007558841261821uint32 @ ( numera9087168376688890119uint32 @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_3505_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
      = ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_3506_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
      = ( uminus_uminus_rat @ ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_3507_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_3508_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_3509_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_3510_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_3511_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_3512_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_3513_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_3514_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_3515_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_3516_word__2p__lem,axiom,
    ! [N: nat,W: word_N3645301735248828278l_num1] :
      ( ( ord_less_nat @ N @ ( size_s8261804613246490634l_num1 @ W ) )
     => ( ( ord_le750835935415966154l_num1 @ W @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N ) )
        = ( ord_less_int @ ( semiri7338730514057886004m1_int @ W ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% word_2p_lem
thf(fact_3517_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A2 ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_3518_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A2 ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_3519_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A2 ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_3520_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A2 ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_3521_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_3522_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_3523_Diff__cancel,axiom,
    ! [A: set_real] :
      ( ( minus_minus_set_real @ A @ A )
      = bot_bot_set_real ) ).

% Diff_cancel
thf(fact_3524_Diff__cancel,axiom,
    ! [A: set_o] :
      ( ( minus_minus_set_o @ A @ A )
      = bot_bot_set_o ) ).

% Diff_cancel
thf(fact_3525_Diff__cancel,axiom,
    ! [A: set_int] :
      ( ( minus_minus_set_int @ A @ A )
      = bot_bot_set_int ) ).

% Diff_cancel
thf(fact_3526_Diff__cancel,axiom,
    ! [A: set_nat] :
      ( ( minus_minus_set_nat @ A @ A )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_3527_empty__Diff,axiom,
    ! [A: set_real] :
      ( ( minus_minus_set_real @ bot_bot_set_real @ A )
      = bot_bot_set_real ) ).

% empty_Diff
thf(fact_3528_empty__Diff,axiom,
    ! [A: set_o] :
      ( ( minus_minus_set_o @ bot_bot_set_o @ A )
      = bot_bot_set_o ) ).

% empty_Diff
thf(fact_3529_empty__Diff,axiom,
    ! [A: set_int] :
      ( ( minus_minus_set_int @ bot_bot_set_int @ A )
      = bot_bot_set_int ) ).

% empty_Diff
thf(fact_3530_empty__Diff,axiom,
    ! [A: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_3531_Diff__empty,axiom,
    ! [A: set_real] :
      ( ( minus_minus_set_real @ A @ bot_bot_set_real )
      = A ) ).

% Diff_empty
thf(fact_3532_Diff__empty,axiom,
    ! [A: set_o] :
      ( ( minus_minus_set_o @ A @ bot_bot_set_o )
      = A ) ).

% Diff_empty
thf(fact_3533_Diff__empty,axiom,
    ! [A: set_int] :
      ( ( minus_minus_set_int @ A @ bot_bot_set_int )
      = A ) ).

% Diff_empty
thf(fact_3534_Diff__empty,axiom,
    ! [A: set_nat] :
      ( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
      = A ) ).

% Diff_empty
thf(fact_3535_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).

% diff_diff_left
thf(fact_3536_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_3537_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_3538_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_3539_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_3540_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_3541_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_3542_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_3543_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_3544_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_3545_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_3546_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_3547_of__int__eq__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ( ring_1_of_int_real @ W )
        = ( ring_1_of_int_real @ Z ) )
      = ( W = Z ) ) ).

% of_int_eq_iff
thf(fact_3548_of__int__eq__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ( ring_1_of_int_rat @ W )
        = ( ring_1_of_int_rat @ Z ) )
      = ( W = Z ) ) ).

% of_int_eq_iff
thf(fact_3549_of__int__eq__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ( ring_17405671764205052669omplex @ W )
        = ( ring_17405671764205052669omplex @ Z ) )
      = ( W = Z ) ) ).

% of_int_eq_iff
thf(fact_3550_Un__Diff__cancel2,axiom,
    ! [B: set_complex,A: set_complex] :
      ( ( sup_sup_set_complex @ ( minus_811609699411566653omplex @ B @ A ) @ A )
      = ( sup_sup_set_complex @ B @ A ) ) ).

% Un_Diff_cancel2
thf(fact_3551_Un__Diff__cancel2,axiom,
    ! [B: set_int,A: set_int] :
      ( ( sup_sup_set_int @ ( minus_minus_set_int @ B @ A ) @ A )
      = ( sup_sup_set_int @ B @ A ) ) ).

% Un_Diff_cancel2
thf(fact_3552_Un__Diff__cancel2,axiom,
    ! [B: set_real,A: set_real] :
      ( ( sup_sup_set_real @ ( minus_minus_set_real @ B @ A ) @ A )
      = ( sup_sup_set_real @ B @ A ) ) ).

% Un_Diff_cancel2
thf(fact_3553_Un__Diff__cancel2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B @ A ) @ A )
      = ( sup_sup_set_nat @ B @ A ) ) ).

% Un_Diff_cancel2
thf(fact_3554_Un__Diff__cancel,axiom,
    ! [A: set_complex,B: set_complex] :
      ( ( sup_sup_set_complex @ A @ ( minus_811609699411566653omplex @ B @ A ) )
      = ( sup_sup_set_complex @ A @ B ) ) ).

% Un_Diff_cancel
thf(fact_3555_Un__Diff__cancel,axiom,
    ! [A: set_int,B: set_int] :
      ( ( sup_sup_set_int @ A @ ( minus_minus_set_int @ B @ A ) )
      = ( sup_sup_set_int @ A @ B ) ) ).

% Un_Diff_cancel
thf(fact_3556_Un__Diff__cancel,axiom,
    ! [A: set_real,B: set_real] :
      ( ( sup_sup_set_real @ A @ ( minus_minus_set_real @ B @ A ) )
      = ( sup_sup_set_real @ A @ B ) ) ).

% Un_Diff_cancel
thf(fact_3557_Un__Diff__cancel,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
      = ( sup_sup_set_nat @ A @ B ) ) ).

% Un_Diff_cancel
thf(fact_3558_idiff__0__right,axiom,
    ! [N: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
      = N ) ).

% idiff_0_right
thf(fact_3559_idiff__0,axiom,
    ! [N: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
      = zero_z5237406670263579293d_enat ) ).

% idiff_0
thf(fact_3560_diff__self,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( minus_4019991460397169231l_num1 @ A2 @ A2 )
      = zero_z3563351764282998399l_num1 ) ).

% diff_self
thf(fact_3561_diff__self,axiom,
    ! [A2: real] :
      ( ( minus_minus_real @ A2 @ A2 )
      = zero_zero_real ) ).

% diff_self
thf(fact_3562_diff__self,axiom,
    ! [A2: rat] :
      ( ( minus_minus_rat @ A2 @ A2 )
      = zero_zero_rat ) ).

% diff_self
thf(fact_3563_diff__self,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ A2 @ A2 )
      = zero_zero_int ) ).

% diff_self
thf(fact_3564_diff__0__right,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( minus_4019991460397169231l_num1 @ A2 @ zero_z3563351764282998399l_num1 )
      = A2 ) ).

% diff_0_right
thf(fact_3565_diff__0__right,axiom,
    ! [A2: real] :
      ( ( minus_minus_real @ A2 @ zero_zero_real )
      = A2 ) ).

% diff_0_right
thf(fact_3566_diff__0__right,axiom,
    ! [A2: rat] :
      ( ( minus_minus_rat @ A2 @ zero_zero_rat )
      = A2 ) ).

% diff_0_right
thf(fact_3567_diff__0__right,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% diff_0_right
thf(fact_3568_zero__diff,axiom,
    ! [A2: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_3569_diff__zero,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( minus_4019991460397169231l_num1 @ A2 @ zero_z3563351764282998399l_num1 )
      = A2 ) ).

% diff_zero
thf(fact_3570_diff__zero,axiom,
    ! [A2: real] :
      ( ( minus_minus_real @ A2 @ zero_zero_real )
      = A2 ) ).

% diff_zero
thf(fact_3571_diff__zero,axiom,
    ! [A2: rat] :
      ( ( minus_minus_rat @ A2 @ zero_zero_rat )
      = A2 ) ).

% diff_zero
thf(fact_3572_diff__zero,axiom,
    ! [A2: nat] :
      ( ( minus_minus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% diff_zero
thf(fact_3573_diff__zero,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% diff_zero
thf(fact_3574_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( minus_4019991460397169231l_num1 @ A2 @ A2 )
      = zero_z3563351764282998399l_num1 ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_3575_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_3576_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_3577_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_3578_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_3579_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_3580_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_3581_divide__cancel__left,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( ( divide_divide_real @ C @ A2 )
        = ( divide_divide_real @ C @ B2 ) )
      = ( ( C = zero_zero_real )
        | ( A2 = B2 ) ) ) ).

% divide_cancel_left
thf(fact_3582_divide__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ( divide_divide_rat @ C @ A2 )
        = ( divide_divide_rat @ C @ B2 ) )
      = ( ( C = zero_zero_rat )
        | ( A2 = B2 ) ) ) ).

% divide_cancel_left
thf(fact_3583_divide__cancel__right,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( ( divide_divide_real @ A2 @ C )
        = ( divide_divide_real @ B2 @ C ) )
      = ( ( C = zero_zero_real )
        | ( A2 = B2 ) ) ) ).

% divide_cancel_right
thf(fact_3584_divide__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ( divide_divide_rat @ A2 @ C )
        = ( divide_divide_rat @ B2 @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A2 = B2 ) ) ) ).

% divide_cancel_right
thf(fact_3585_bits__div__0,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( divide1791077408188789448l_num1 @ zero_z3563351764282998399l_num1 @ A2 )
      = zero_z3563351764282998399l_num1 ) ).

% bits_div_0
thf(fact_3586_bits__div__0,axiom,
    ! [A2: uint32] :
      ( ( divide_divide_uint32 @ zero_zero_uint32 @ A2 )
      = zero_zero_uint32 ) ).

% bits_div_0
thf(fact_3587_bits__div__0,axiom,
    ! [A2: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% bits_div_0
thf(fact_3588_bits__div__0,axiom,
    ! [A2: int] :
      ( ( divide_divide_int @ zero_zero_int @ A2 )
      = zero_zero_int ) ).

% bits_div_0
thf(fact_3589_bits__div__by__0,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( divide1791077408188789448l_num1 @ A2 @ zero_z3563351764282998399l_num1 )
      = zero_z3563351764282998399l_num1 ) ).

% bits_div_by_0
thf(fact_3590_bits__div__by__0,axiom,
    ! [A2: uint32] :
      ( ( divide_divide_uint32 @ A2 @ zero_zero_uint32 )
      = zero_zero_uint32 ) ).

% bits_div_by_0
thf(fact_3591_bits__div__by__0,axiom,
    ! [A2: nat] :
      ( ( divide_divide_nat @ A2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% bits_div_by_0
thf(fact_3592_bits__div__by__0,axiom,
    ! [A2: int] :
      ( ( divide_divide_int @ A2 @ zero_zero_int )
      = zero_zero_int ) ).

% bits_div_by_0
thf(fact_3593_division__ring__divide__zero,axiom,
    ! [A2: real] :
      ( ( divide_divide_real @ A2 @ zero_zero_real )
      = zero_zero_real ) ).

% division_ring_divide_zero
thf(fact_3594_division__ring__divide__zero,axiom,
    ! [A2: rat] :
      ( ( divide_divide_rat @ A2 @ zero_zero_rat )
      = zero_zero_rat ) ).

% division_ring_divide_zero
thf(fact_3595_div__0,axiom,
    ! [A2: real] :
      ( ( divide_divide_real @ zero_zero_real @ A2 )
      = zero_zero_real ) ).

% div_0
thf(fact_3596_div__0,axiom,
    ! [A2: rat] :
      ( ( divide_divide_rat @ zero_zero_rat @ A2 )
      = zero_zero_rat ) ).

% div_0
thf(fact_3597_div__0,axiom,
    ! [A2: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% div_0
thf(fact_3598_div__0,axiom,
    ! [A2: int] :
      ( ( divide_divide_int @ zero_zero_int @ A2 )
      = zero_zero_int ) ).

% div_0
thf(fact_3599_div__by__0,axiom,
    ! [A2: real] :
      ( ( divide_divide_real @ A2 @ zero_zero_real )
      = zero_zero_real ) ).

% div_by_0
thf(fact_3600_div__by__0,axiom,
    ! [A2: rat] :
      ( ( divide_divide_rat @ A2 @ zero_zero_rat )
      = zero_zero_rat ) ).

% div_by_0
thf(fact_3601_div__by__0,axiom,
    ! [A2: nat] :
      ( ( divide_divide_nat @ A2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% div_by_0
thf(fact_3602_div__by__0,axiom,
    ! [A2: int] :
      ( ( divide_divide_int @ A2 @ zero_zero_int )
      = zero_zero_int ) ).

% div_by_0
thf(fact_3603_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_3604_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_3605_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_3606_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_3607_add__diff__cancel__right,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ C ) )
      = ( minus_minus_real @ A2 @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_3608_add__diff__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B2 @ C ) )
      = ( minus_minus_rat @ A2 @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_3609_add__diff__cancel__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
      = ( minus_minus_nat @ A2 @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_3610_add__diff__cancel__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
      = ( minus_minus_int @ A2 @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_3611_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_3612_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_3613_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_3614_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_3615_add__diff__cancel__left,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B2 ) )
      = ( minus_minus_real @ A2 @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_3616_add__diff__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B2 ) )
      = ( minus_minus_rat @ A2 @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_3617_add__diff__cancel__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
      = ( minus_minus_nat @ A2 @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_3618_add__diff__cancel__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
      = ( minus_minus_int @ A2 @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_3619_diff__add__cancel,axiom,
    ! [A2: real,B2: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% diff_add_cancel
thf(fact_3620_diff__add__cancel,axiom,
    ! [A2: rat,B2: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% diff_add_cancel
thf(fact_3621_diff__add__cancel,axiom,
    ! [A2: int,B2: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% diff_add_cancel
thf(fact_3622_add__diff__cancel,axiom,
    ! [A2: real,B2: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel
thf(fact_3623_add__diff__cancel,axiom,
    ! [A2: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel
thf(fact_3624_add__diff__cancel,axiom,
    ! [A2: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel
thf(fact_3625_bits__div__by__1,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( divide1791077408188789448l_num1 @ A2 @ one_on7727431528512463931l_num1 )
      = A2 ) ).

% bits_div_by_1
thf(fact_3626_bits__div__by__1,axiom,
    ! [A2: uint32] :
      ( ( divide_divide_uint32 @ A2 @ one_one_uint32 )
      = A2 ) ).

% bits_div_by_1
thf(fact_3627_bits__div__by__1,axiom,
    ! [A2: nat] :
      ( ( divide_divide_nat @ A2 @ one_one_nat )
      = A2 ) ).

% bits_div_by_1
thf(fact_3628_bits__div__by__1,axiom,
    ! [A2: int] :
      ( ( divide_divide_int @ A2 @ one_one_int )
      = A2 ) ).

% bits_div_by_1
thf(fact_3629_div__by__1,axiom,
    ! [A2: real] :
      ( ( divide_divide_real @ A2 @ one_one_real )
      = A2 ) ).

% div_by_1
thf(fact_3630_div__by__1,axiom,
    ! [A2: rat] :
      ( ( divide_divide_rat @ A2 @ one_one_rat )
      = A2 ) ).

% div_by_1
thf(fact_3631_div__by__1,axiom,
    ! [A2: nat] :
      ( ( divide_divide_nat @ A2 @ one_one_nat )
      = A2 ) ).

% div_by_1
thf(fact_3632_div__by__1,axiom,
    ! [A2: int] :
      ( ( divide_divide_int @ A2 @ one_one_int )
      = A2 ) ).

% div_by_1
thf(fact_3633_dvd__0__right,axiom,
    ! [A2: uint32] : ( dvd_dvd_uint32 @ A2 @ zero_zero_uint32 ) ).

% dvd_0_right
thf(fact_3634_dvd__0__right,axiom,
    ! [A2: code_integer] : ( dvd_dvd_Code_integer @ A2 @ zero_z3403309356797280102nteger ) ).

% dvd_0_right
thf(fact_3635_dvd__0__right,axiom,
    ! [A2: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ A2 @ zero_z3563351764282998399l_num1 ) ).

% dvd_0_right
thf(fact_3636_dvd__0__right,axiom,
    ! [A2: real] : ( dvd_dvd_real @ A2 @ zero_zero_real ) ).

% dvd_0_right
thf(fact_3637_dvd__0__right,axiom,
    ! [A2: rat] : ( dvd_dvd_rat @ A2 @ zero_zero_rat ) ).

% dvd_0_right
thf(fact_3638_dvd__0__right,axiom,
    ! [A2: nat] : ( dvd_dvd_nat @ A2 @ zero_zero_nat ) ).

% dvd_0_right
thf(fact_3639_dvd__0__right,axiom,
    ! [A2: int] : ( dvd_dvd_int @ A2 @ zero_zero_int ) ).

% dvd_0_right
thf(fact_3640_dvd__0__left__iff,axiom,
    ! [A2: uint32] :
      ( ( dvd_dvd_uint32 @ zero_zero_uint32 @ A2 )
      = ( A2 = zero_zero_uint32 ) ) ).

% dvd_0_left_iff
thf(fact_3641_dvd__0__left__iff,axiom,
    ! [A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A2 )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left_iff
thf(fact_3642_dvd__0__left__iff,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ zero_z3563351764282998399l_num1 @ A2 )
      = ( A2 = zero_z3563351764282998399l_num1 ) ) ).

% dvd_0_left_iff
thf(fact_3643_dvd__0__left__iff,axiom,
    ! [A2: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A2 )
      = ( A2 = zero_zero_real ) ) ).

% dvd_0_left_iff
thf(fact_3644_dvd__0__left__iff,axiom,
    ! [A2: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A2 )
      = ( A2 = zero_zero_rat ) ) ).

% dvd_0_left_iff
thf(fact_3645_dvd__0__left__iff,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
      = ( A2 = zero_zero_nat ) ) ).

% dvd_0_left_iff
thf(fact_3646_dvd__0__left__iff,axiom,
    ! [A2: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A2 )
      = ( A2 = zero_zero_int ) ) ).

% dvd_0_left_iff
thf(fact_3647_minus__diff__eq,axiom,
    ! [A2: complex,B2: complex] :
      ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A2 @ B2 ) )
      = ( minus_minus_complex @ B2 @ A2 ) ) ).

% minus_diff_eq
thf(fact_3648_minus__diff__eq,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( uminus_uminus_uint32 @ ( minus_minus_uint32 @ A2 @ B2 ) )
      = ( minus_minus_uint32 @ B2 @ A2 ) ) ).

% minus_diff_eq
thf(fact_3649_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_3650_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_3651_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_3652_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
      = ( ord_less_nat @ M @ N ) ) ).

% zero_less_diff
thf(fact_3653_dvd__add__triv__right__iff,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( dvd_dvd_uint32 @ A2 @ ( plus_plus_uint32 @ B2 @ A2 ) )
      = ( dvd_dvd_uint32 @ A2 @ B2 ) ) ).

% dvd_add_triv_right_iff
thf(fact_3654_dvd__add__triv__right__iff,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A2 @ ( plus_p361126936061061375l_num1 @ B2 @ A2 ) )
      = ( dvd_dv6812691276156420380l_num1 @ A2 @ B2 ) ) ).

% dvd_add_triv_right_iff
thf(fact_3655_dvd__add__triv__right__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ ( plus_p5714425477246183910nteger @ B2 @ A2 ) )
      = ( dvd_dvd_Code_integer @ A2 @ B2 ) ) ).

% dvd_add_triv_right_iff
thf(fact_3656_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_3657_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_3658_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_3659_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_3660_dvd__add__triv__left__iff,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( dvd_dvd_uint32 @ A2 @ ( plus_plus_uint32 @ A2 @ B2 ) )
      = ( dvd_dvd_uint32 @ A2 @ B2 ) ) ).

% dvd_add_triv_left_iff
thf(fact_3661_dvd__add__triv__left__iff,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A2 @ ( plus_p361126936061061375l_num1 @ A2 @ B2 ) )
      = ( dvd_dv6812691276156420380l_num1 @ A2 @ B2 ) ) ).

% dvd_add_triv_left_iff
thf(fact_3662_dvd__add__triv__left__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) )
      = ( dvd_dvd_Code_integer @ A2 @ B2 ) ) ).

% dvd_add_triv_left_iff
thf(fact_3663_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_3664_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_3665_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_3666_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_3667_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_3668_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% diff_is_0_eq
thf(fact_3669_div__dvd__div,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ B2 )
     => ( ( dvd_dvd_Code_integer @ A2 @ C )
       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B2 @ A2 ) @ ( divide6298287555418463151nteger @ C @ A2 ) )
          = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ) ).

% div_dvd_div
thf(fact_3670_div__dvd__div,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A2 @ B2 )
     => ( ( dvd_dvd_nat @ A2 @ C )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B2 @ A2 ) @ ( divide_divide_nat @ C @ A2 ) )
          = ( dvd_dvd_nat @ B2 @ C ) ) ) ) ).

% div_dvd_div
thf(fact_3671_div__dvd__div,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A2 @ B2 )
     => ( ( dvd_dvd_int @ A2 @ C )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ B2 @ A2 ) @ ( divide_divide_int @ C @ A2 ) )
          = ( dvd_dvd_int @ B2 @ C ) ) ) ) ).

% div_dvd_div
thf(fact_3672_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_3673_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_3674_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_3675_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_3676_minus__dvd__iff,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( uminus8244633308260627903l_num1 @ X ) @ Y )
      = ( dvd_dv6812691276156420380l_num1 @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_3677_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_3678_minus__dvd__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X ) @ Y )
      = ( dvd_dvd_complex @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_3679_minus__dvd__iff,axiom,
    ! [X: uint32,Y: uint32] :
      ( ( dvd_dvd_uint32 @ ( uminus_uminus_uint32 @ X ) @ Y )
      = ( dvd_dvd_uint32 @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_3680_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_3681_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_3682_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_3683_dvd__minus__iff,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ X @ ( uminus8244633308260627903l_num1 @ Y ) )
      = ( dvd_dv6812691276156420380l_num1 @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_3684_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_3685_dvd__minus__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( dvd_dvd_complex @ X @ ( uminus1482373934393186551omplex @ Y ) )
      = ( dvd_dvd_complex @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_3686_dvd__minus__iff,axiom,
    ! [X: uint32,Y: uint32] :
      ( ( dvd_dvd_uint32 @ X @ ( uminus_uminus_uint32 @ Y ) )
      = ( dvd_dvd_uint32 @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_3687_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_3688_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_3689_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_3690_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_3691_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_3692_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_3693_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_3694_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_3695_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_3696_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_3697_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_3698_of__int__diff,axiom,
    ! [W: int,Z: int] :
      ( ( ring_17408606157368542149l_num1 @ ( minus_minus_int @ W @ Z ) )
      = ( minus_4019991460397169231l_num1 @ ( ring_17408606157368542149l_num1 @ W ) @ ( ring_17408606157368542149l_num1 @ Z ) ) ) ).

% of_int_diff
thf(fact_3699_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_3700_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_3701_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_3702_word__of__int__numeral,axiom,
    ! [Bin: num] :
      ( ( ring_17408606157368542149l_num1 @ ( numeral_numeral_int @ Bin ) )
      = ( numera7442385471795722001l_num1 @ Bin ) ) ).

% word_of_int_numeral
thf(fact_3703_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_3704_powr__0,axiom,
    ! [Z: real] :
      ( ( powr_real @ zero_zero_real @ Z )
      = zero_zero_real ) ).

% powr_0
thf(fact_3705_powr__eq__0__iff,axiom,
    ! [W: real,Z: real] :
      ( ( ( powr_real @ W @ Z )
        = zero_zero_real )
      = ( W = zero_zero_real ) ) ).

% powr_eq_0_iff
thf(fact_3706_word__of__int__0,axiom,
    ( ( ring_17408606157368542149l_num1 @ zero_zero_int )
    = zero_z3563351764282998399l_num1 ) ).

% word_of_int_0
thf(fact_3707_powr__one__eq__one,axiom,
    ! [A2: real] :
      ( ( powr_real @ one_one_real @ A2 )
      = one_one_real ) ).

% powr_one_eq_one
thf(fact_3708_word__of__int__1,axiom,
    ( ( ring_17408606157368542149l_num1 @ one_one_int )
    = one_on7727431528512463931l_num1 ) ).

% word_of_int_1
thf(fact_3709_of__int__floor__cancel,axiom,
    ! [X: real] :
      ( ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) )
        = X )
      = ( ? [N4: int] :
            ( X
            = ( ring_1_of_int_real @ N4 ) ) ) ) ).

% of_int_floor_cancel
thf(fact_3710_of__int__floor__cancel,axiom,
    ! [X: rat] :
      ( ( ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) )
        = X )
      = ( ? [N4: int] :
            ( X
            = ( ring_1_of_int_rat @ N4 ) ) ) ) ).

% of_int_floor_cancel
thf(fact_3711_floor__of__int,axiom,
    ! [Z: int] :
      ( ( archim6058952711729229775r_real @ ( ring_1_of_int_real @ Z ) )
      = Z ) ).

% floor_of_int
thf(fact_3712_floor__of__int,axiom,
    ! [Z: int] :
      ( ( archim3151403230148437115or_rat @ ( ring_1_of_int_rat @ Z ) )
      = Z ) ).

% floor_of_int
thf(fact_3713_of__int__ceiling__cancel,axiom,
    ! [X: rat] :
      ( ( ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) )
        = X )
      = ( ? [N4: int] :
            ( X
            = ( ring_1_of_int_rat @ N4 ) ) ) ) ).

% of_int_ceiling_cancel
thf(fact_3714_of__int__ceiling__cancel,axiom,
    ! [X: real] :
      ( ( ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) )
        = X )
      = ( ? [N4: int] :
            ( X
            = ( ring_1_of_int_real @ N4 ) ) ) ) ).

% of_int_ceiling_cancel
thf(fact_3715_ceiling__of__int,axiom,
    ! [Z: int] :
      ( ( archim2889992004027027881ng_rat @ ( ring_1_of_int_rat @ Z ) )
      = Z ) ).

% ceiling_of_int
thf(fact_3716_ceiling__of__int,axiom,
    ! [Z: int] :
      ( ( archim7802044766580827645g_real @ ( ring_1_of_int_real @ Z ) )
      = Z ) ).

% ceiling_of_int
thf(fact_3717_Compl__Diff__eq,axiom,
    ! [A: set_complex,B: set_complex] :
      ( ( uminus8566677241136511917omplex @ ( minus_811609699411566653omplex @ A @ B ) )
      = ( sup_sup_set_complex @ ( uminus8566677241136511917omplex @ A ) @ B ) ) ).

% Compl_Diff_eq
thf(fact_3718_Compl__Diff__eq,axiom,
    ! [A: set_int,B: set_int] :
      ( ( uminus1532241313380277803et_int @ ( minus_minus_set_int @ A @ B ) )
      = ( sup_sup_set_int @ ( uminus1532241313380277803et_int @ A ) @ B ) ) ).

% Compl_Diff_eq
thf(fact_3719_Compl__Diff__eq,axiom,
    ! [A: set_real,B: set_real] :
      ( ( uminus612125837232591019t_real @ ( minus_minus_set_real @ A @ B ) )
      = ( sup_sup_set_real @ ( uminus612125837232591019t_real @ A ) @ B ) ) ).

% Compl_Diff_eq
thf(fact_3720_Compl__Diff__eq,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( minus_minus_set_nat @ A @ B ) )
      = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ B ) ) ).

% Compl_Diff_eq
thf(fact_3721_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu93272222329896523l_num1 @ one_on7727431528512463931l_num1 )
    = one_on7727431528512463931l_num1 ) ).

% dbl_dec_simps(3)
thf(fact_3722_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ one_one_real )
    = one_one_real ) ).

% dbl_dec_simps(3)
thf(fact_3723_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
    = one_one_rat ) ).

% dbl_dec_simps(3)
thf(fact_3724_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ one_one_int )
    = one_one_int ) ).

% dbl_dec_simps(3)
thf(fact_3725_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_3726_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_3727_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_3728_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_3729_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_3730_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_3731_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_3732_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_3733_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_3734_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_3735_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_3736_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_3737_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_3738_diff__numeral__special_I9_J,axiom,
    ( ( minus_4019991460397169231l_num1 @ one_on7727431528512463931l_num1 @ one_on7727431528512463931l_num1 )
    = zero_z3563351764282998399l_num1 ) ).

% diff_numeral_special(9)
thf(fact_3739_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_real @ one_one_real @ one_one_real )
    = zero_zero_real ) ).

% diff_numeral_special(9)
thf(fact_3740_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
    = zero_zero_rat ) ).

% diff_numeral_special(9)
thf(fact_3741_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_3742_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_3743_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_3744_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_3745_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_3746_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_3747_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_3748_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_3749_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_3750_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_3751_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_3752_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_3753_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_3754_divide__self,axiom,
    ! [A2: real] :
      ( ( A2 != zero_zero_real )
     => ( ( divide_divide_real @ A2 @ A2 )
        = one_one_real ) ) ).

% divide_self
thf(fact_3755_divide__self,axiom,
    ! [A2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( divide_divide_rat @ A2 @ A2 )
        = one_one_rat ) ) ).

% divide_self
thf(fact_3756_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_3757_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_3758_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_3759_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_3760_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_3761_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_3762_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_3763_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_3764_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_3765_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_3766_div__self,axiom,
    ! [A2: real] :
      ( ( A2 != zero_zero_real )
     => ( ( divide_divide_real @ A2 @ A2 )
        = one_one_real ) ) ).

% div_self
thf(fact_3767_div__self,axiom,
    ! [A2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( divide_divide_rat @ A2 @ A2 )
        = one_one_rat ) ) ).

% div_self
thf(fact_3768_div__self,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( divide_divide_nat @ A2 @ A2 )
        = one_one_nat ) ) ).

% div_self
thf(fact_3769_div__self,axiom,
    ! [A2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( divide_divide_int @ A2 @ A2 )
        = one_one_int ) ) ).

% div_self
thf(fact_3770_diff__0,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( minus_4019991460397169231l_num1 @ zero_z3563351764282998399l_num1 @ A2 )
      = ( uminus8244633308260627903l_num1 @ A2 ) ) ).

% diff_0
thf(fact_3771_diff__0,axiom,
    ! [A2: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ A2 )
      = ( uminus1482373934393186551omplex @ A2 ) ) ).

% diff_0
thf(fact_3772_diff__0,axiom,
    ! [A2: uint32] :
      ( ( minus_minus_uint32 @ zero_zero_uint32 @ A2 )
      = ( uminus_uminus_uint32 @ A2 ) ) ).

% diff_0
thf(fact_3773_diff__0,axiom,
    ! [A2: real] :
      ( ( minus_minus_real @ zero_zero_real @ A2 )
      = ( uminus_uminus_real @ A2 ) ) ).

% diff_0
thf(fact_3774_diff__0,axiom,
    ! [A2: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ A2 )
      = ( uminus_uminus_rat @ A2 ) ) ).

% diff_0
thf(fact_3775_diff__0,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ zero_zero_int @ A2 )
      = ( uminus_uminus_int @ A2 ) ) ).

% diff_0
thf(fact_3776_verit__minus__simplify_I3_J,axiom,
    ! [B2: word_N3645301735248828278l_num1] :
      ( ( minus_4019991460397169231l_num1 @ zero_z3563351764282998399l_num1 @ B2 )
      = ( uminus8244633308260627903l_num1 @ B2 ) ) ).

% verit_minus_simplify(3)
thf(fact_3777_verit__minus__simplify_I3_J,axiom,
    ! [B2: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ B2 )
      = ( uminus1482373934393186551omplex @ B2 ) ) ).

% verit_minus_simplify(3)
thf(fact_3778_verit__minus__simplify_I3_J,axiom,
    ! [B2: uint32] :
      ( ( minus_minus_uint32 @ zero_zero_uint32 @ B2 )
      = ( uminus_uminus_uint32 @ B2 ) ) ).

% verit_minus_simplify(3)
thf(fact_3779_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_3780_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_3781_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_3782_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_3783_uminus__add__conv__diff,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ A2 ) @ B2 )
      = ( minus_minus_uint32 @ B2 @ A2 ) ) ).

% uminus_add_conv_diff
thf(fact_3784_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_3785_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_3786_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_3787_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_3788_diff__minus__eq__add,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( minus_minus_uint32 @ A2 @ ( uminus_uminus_uint32 @ B2 ) )
      = ( plus_plus_uint32 @ A2 @ B2 ) ) ).

% diff_minus_eq_add
thf(fact_3789_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_3790_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_3791_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_3792_divide__minus1,axiom,
    ! [X: complex] :
      ( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ X ) ) ).

% divide_minus1
thf(fact_3793_divide__minus1,axiom,
    ! [X: real] :
      ( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ X ) ) ).

% divide_minus1
thf(fact_3794_divide__minus1,axiom,
    ! [X: rat] :
      ( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ X ) ) ).

% divide_minus1
thf(fact_3795_div__add,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A2 )
     => ( ( dvd_dvd_Code_integer @ C @ B2 )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ C )
          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A2 @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ) ) ).

% div_add
thf(fact_3796_div__add,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ C @ A2 )
     => ( ( dvd_dvd_nat @ C @ B2 )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
          = ( plus_plus_nat @ ( divide_divide_nat @ A2 @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ) ).

% div_add
thf(fact_3797_div__add,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( dvd_dvd_int @ C @ A2 )
     => ( ( dvd_dvd_int @ C @ B2 )
       => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
          = ( plus_plus_int @ ( divide_divide_int @ A2 @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ) ).

% div_add
thf(fact_3798_unit__div__1__div__1,axiom,
    ! [A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
     => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A2 ) )
        = A2 ) ) ).

% unit_div_1_div_1
thf(fact_3799_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_3800_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_3801_unit__div__1__unit,axiom,
    ! [A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
     => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A2 ) @ one_one_Code_integer ) ) ).

% unit_div_1_unit
thf(fact_3802_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_3803_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_3804_unit__div,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ one_one_Code_integer ) ) ) ).

% unit_div
thf(fact_3805_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_3806_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_3807_div__diff,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A2 )
     => ( ( dvd_dvd_Code_integer @ C @ B2 )
       => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ C )
          = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A2 @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ) ) ).

% div_diff
thf(fact_3808_div__diff,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( dvd_dvd_int @ C @ A2 )
     => ( ( dvd_dvd_int @ C @ B2 )
       => ( ( divide_divide_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
          = ( minus_minus_int @ ( divide_divide_int @ A2 @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ) ).

% div_diff
thf(fact_3809_of__int__0,axiom,
    ( ( ring_1_of_int_int @ zero_zero_int )
    = zero_zero_int ) ).

% of_int_0
thf(fact_3810_of__int__0,axiom,
    ( ( ring_1_of_int_real @ zero_zero_int )
    = zero_zero_real ) ).

% of_int_0
thf(fact_3811_of__int__0,axiom,
    ( ( ring_1_of_int_rat @ zero_zero_int )
    = zero_zero_rat ) ).

% of_int_0
thf(fact_3812_of__int__0,axiom,
    ( ( ring_17405671764205052669omplex @ zero_zero_int )
    = zero_zero_complex ) ).

% of_int_0
thf(fact_3813_of__int__0,axiom,
    ( ( ring_17408606157368542149l_num1 @ zero_zero_int )
    = zero_z3563351764282998399l_num1 ) ).

% of_int_0
thf(fact_3814_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_3815_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_3816_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_3817_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_3818_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_3819_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_3820_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_3821_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_3822_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ( ring_17405671764205052669omplex @ Z )
        = ( numera6690914467698888265omplex @ N ) )
      = ( Z
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_3823_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ( ring_1_of_int_real @ Z )
        = ( numeral_numeral_real @ N ) )
      = ( Z
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_3824_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ( ring_1_of_int_rat @ Z )
        = ( numeral_numeral_rat @ N ) )
      = ( Z
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_3825_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ( ring_1_of_int_int @ Z )
        = ( numeral_numeral_int @ N ) )
      = ( Z
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_3826_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
      = ( numera6690914467698888265omplex @ K ) ) ).

% of_int_numeral
thf(fact_3827_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_17408606157368542149l_num1 @ ( numeral_numeral_int @ K ) )
      = ( numera7442385471795722001l_num1 @ K ) ) ).

% of_int_numeral
thf(fact_3828_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_real @ K ) ) ).

% of_int_numeral
thf(fact_3829_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_rat @ K ) ) ).

% of_int_numeral
thf(fact_3830_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% of_int_numeral
thf(fact_3831_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_3832_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_3833_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_3834_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_3835_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_3836_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_3837_of__int__1,axiom,
    ( ( ring_1_of_int_int @ one_one_int )
    = one_one_int ) ).

% of_int_1
thf(fact_3838_of__int__1,axiom,
    ( ( ring_1_of_int_real @ one_one_int )
    = one_one_real ) ).

% of_int_1
thf(fact_3839_of__int__1,axiom,
    ( ( ring_1_of_int_rat @ one_one_int )
    = one_one_rat ) ).

% of_int_1
thf(fact_3840_of__int__1,axiom,
    ( ( ring_17405671764205052669omplex @ one_one_int )
    = one_one_complex ) ).

% of_int_1
thf(fact_3841_of__int__1,axiom,
    ( ( ring_17408606157368542149l_num1 @ one_one_int )
    = one_on7727431528512463931l_num1 ) ).

% of_int_1
thf(fact_3842_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_int @ Z )
        = one_one_int )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_3843_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_real @ Z )
        = one_one_real )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_3844_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_rat @ Z )
        = one_one_rat )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_3845_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_17405671764205052669omplex @ Z )
        = one_one_complex )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_3846_powr__zero__eq__one,axiom,
    ! [X: real] :
      ( ( ( X = zero_zero_real )
       => ( ( powr_real @ X @ zero_zero_real )
          = zero_zero_real ) )
      & ( ( X != zero_zero_real )
       => ( ( powr_real @ X @ zero_zero_real )
          = one_one_real ) ) ) ).

% powr_zero_eq_one
thf(fact_3847_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_3848_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_3849_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_3850_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_3851_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_17408606157368542149l_num1 @ ( plus_plus_int @ W @ Z ) )
      = ( plus_p361126936061061375l_num1 @ ( ring_17408606157368542149l_num1 @ W ) @ ( ring_17408606157368542149l_num1 @ Z ) ) ) ).

% of_int_add
thf(fact_3852_of__int__minus,axiom,
    ! [Z: int] :
      ( ( ring_17408606157368542149l_num1 @ ( uminus_uminus_int @ Z ) )
      = ( uminus8244633308260627903l_num1 @ ( ring_17408606157368542149l_num1 @ Z ) ) ) ).

% of_int_minus
thf(fact_3853_of__int__minus,axiom,
    ! [Z: int] :
      ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ Z ) )
      = ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ Z ) ) ) ).

% of_int_minus
thf(fact_3854_of__int__minus,axiom,
    ! [Z: int] :
      ( ( ring_1_of_int_uint32 @ ( uminus_uminus_int @ Z ) )
      = ( uminus_uminus_uint32 @ ( ring_1_of_int_uint32 @ Z ) ) ) ).

% of_int_minus
thf(fact_3855_of__int__minus,axiom,
    ! [Z: int] :
      ( ( ring_1_of_int_real @ ( uminus_uminus_int @ Z ) )
      = ( uminus_uminus_real @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_minus
thf(fact_3856_of__int__minus,axiom,
    ! [Z: int] :
      ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ Z ) )
      = ( uminus_uminus_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_minus
thf(fact_3857_of__int__minus,axiom,
    ! [Z: int] :
      ( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z ) )
      = ( uminus_uminus_int @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_minus
thf(fact_3858_of__int__of__nat__eq,axiom,
    ! [N: nat] :
      ( ( ring_1_of_int_rat @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri681578069525770553at_rat @ N ) ) ).

% of_int_of_nat_eq
thf(fact_3859_of__int__of__nat__eq,axiom,
    ! [N: nat] :
      ( ( ring_17408606157368542149l_num1 @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri8819519690708144855l_num1 @ N ) ) ).

% of_int_of_nat_eq
thf(fact_3860_of__int__of__nat__eq,axiom,
    ! [N: nat] :
      ( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% of_int_of_nat_eq
thf(fact_3861_of__int__of__nat__eq,axiom,
    ! [N: nat] :
      ( ( ring_1_of_int_real @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri5074537144036343181t_real @ N ) ) ).

% of_int_of_nat_eq
thf(fact_3862_of__int__of__nat__eq,axiom,
    ! [N: nat] :
      ( ( ring_18347121197199848620nteger @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri4939895301339042750nteger @ N ) ) ).

% of_int_of_nat_eq
thf(fact_3863_of__int__of__nat__eq,axiom,
    ! [N: nat] :
      ( ( ring_17405671764205052669omplex @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri8010041392384452111omplex @ N ) ) ).

% of_int_of_nat_eq
thf(fact_3864_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_3865_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_3866_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_3867_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_3868_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_3869_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_3870_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_3871_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_3872_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_3873_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_3874_of__int__power,axiom,
    ! [Z: int,N: nat] :
      ( ( ring_1_of_int_rat @ ( power_power_int @ Z @ N ) )
      = ( power_power_rat @ ( ring_1_of_int_rat @ Z ) @ N ) ) ).

% of_int_power
thf(fact_3875_of__int__power,axiom,
    ! [Z: int,N: nat] :
      ( ( ring_17408606157368542149l_num1 @ ( power_power_int @ Z @ N ) )
      = ( power_2184487114949457152l_num1 @ ( ring_17408606157368542149l_num1 @ Z ) @ N ) ) ).

% of_int_power
thf(fact_3876_of__int__power,axiom,
    ! [Z: int,N: nat] :
      ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N ) )
      = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N ) ) ).

% of_int_power
thf(fact_3877_of__int__power,axiom,
    ! [Z: int,N: nat] :
      ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N ) )
      = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N ) ) ).

% of_int_power
thf(fact_3878_of__int__power,axiom,
    ! [Z: int,N: nat] :
      ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z @ N ) )
      = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z ) @ N ) ) ).

% of_int_power
thf(fact_3879_of__int__power,axiom,
    ! [Z: int,N: nat] :
      ( ( ring_18347121197199848620nteger @ ( power_power_int @ Z @ N ) )
      = ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ Z ) @ N ) ) ).

% of_int_power
thf(fact_3880_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_3881_Word_Oof__int__uint,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( ring_1_of_int_real @ ( semiri7338730514057886004m1_int @ W ) )
      = ( semiri46416754965307273481_real @ W ) ) ).

% Word.of_int_uint
thf(fact_3882_Word_Oof__int__uint,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( ring_1_of_int_rat @ ( semiri7338730514057886004m1_int @ W ) )
      = ( semiri6706090924480440544m1_rat @ W ) ) ).

% Word.of_int_uint
thf(fact_3883_Word_Oof__int__uint,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( ring_17405671764205052669omplex @ ( semiri7338730514057886004m1_int @ W ) )
      = ( semiri7067251934024306614omplex @ W ) ) ).

% Word.of_int_uint
thf(fact_3884_Word_Oof__int__uint,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( ring_17408606157368542149l_num1 @ ( semiri7338730514057886004m1_int @ W ) )
      = ( semiri1312839663145358974l_num1 @ W ) ) ).

% Word.of_int_uint
thf(fact_3885_Word_Oof__int__uint,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( ring_1_of_int_int @ ( semiri7338730514057886004m1_int @ W ) )
      = ( semiri7338730514057886004m1_int @ W ) ) ).

% Word.of_int_uint
thf(fact_3886_floor__diff__of__int,axiom,
    ! [X: real,Z: int] :
      ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
      = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ Z ) ) ).

% floor_diff_of_int
thf(fact_3887_floor__diff__of__int,axiom,
    ! [X: rat,Z: int] :
      ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) )
      = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ Z ) ) ).

% floor_diff_of_int
thf(fact_3888_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_3889_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_3890_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_3891_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_3892_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_3893_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_3894_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_3895_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_3896_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_3897_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_3898_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_3899_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_3900_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_3901_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_3902_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_3903_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_3904_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_3905_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_3906_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_3907_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_3908_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_3909_diff__numeral__special_I12_J,axiom,
    ( ( minus_4019991460397169231l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
    = zero_z3563351764282998399l_num1 ) ).

% diff_numeral_special(12)
thf(fact_3910_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_3911_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ ( uminus_uminus_uint32 @ one_one_uint32 ) )
    = zero_zero_uint32 ) ).

% diff_numeral_special(12)
thf(fact_3912_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_3913_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_3914_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_3915_floor__diff__numeral,axiom,
    ! [X: real,V: num] :
      ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ ( numeral_numeral_real @ V ) ) )
      = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% floor_diff_numeral
thf(fact_3916_floor__diff__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
      = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% floor_diff_numeral
thf(fact_3917_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_3918_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_3919_floor__diff__one,axiom,
    ! [X: real] :
      ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X @ one_one_real ) )
      = ( minus_minus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ).

% floor_diff_one
thf(fact_3920_floor__diff__one,axiom,
    ! [X: rat] :
      ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
      = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ).

% floor_diff_one
thf(fact_3921_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_4019991460397169231l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ M ) ) @ ( numera7442385471795722001l_num1 @ N ) )
      = ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_3922_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_3923_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ M ) ) @ ( numera9087168376688890119uint32 @ N ) )
      = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_3924_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_3925_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_3926_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_3927_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_4019991460397169231l_num1 @ ( numera7442385471795722001l_num1 @ M ) @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ N ) ) )
      = ( numera7442385471795722001l_num1 @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_3928_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_3929_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_uint32 @ ( numera9087168376688890119uint32 @ M ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ N ) ) )
      = ( numera9087168376688890119uint32 @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_3930_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_3931_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_3932_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_3933_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_3934_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_3935_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_3936_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_3937_floor__uminus__of__int,axiom,
    ! [Z: int] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( ring_1_of_int_real @ Z ) ) )
      = ( uminus_uminus_int @ Z ) ) ).

% floor_uminus_of_int
thf(fact_3938_floor__uminus__of__int,axiom,
    ! [Z: int] :
      ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ Z ) ) )
      = ( uminus_uminus_int @ Z ) ) ).

% floor_uminus_of_int
thf(fact_3939_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_3940_powr__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ one_one_real )
        = X ) ) ).

% powr_one
thf(fact_3941_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_3942_word__of__int__neg__numeral,axiom,
    ! [Bin: num] :
      ( ( ring_17408606157368542149l_num1 @ ( uminus_uminus_int @ ( numeral_numeral_int @ Bin ) ) )
      = ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ Bin ) ) ) ).

% word_of_int_neg_numeral
thf(fact_3943_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_3944_numeral__powr__numeral__real,axiom,
    ! [M: num,N: num] :
      ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).

% numeral_powr_numeral_real
thf(fact_3945_word__le__sub1__numberof,axiom,
    ! [W: num] :
      ( ( ( numera7442385471795722001l_num1 @ W )
       != zero_z3563351764282998399l_num1 )
     => ( ( ord_le3335648743751981014l_num1 @ one_on7727431528512463931l_num1 @ ( numera7442385471795722001l_num1 @ W ) )
        = ( ord_le3335648743751981014l_num1 @ zero_z3563351764282998399l_num1 @ ( minus_4019991460397169231l_num1 @ ( numera7442385471795722001l_num1 @ W ) @ one_on7727431528512463931l_num1 ) ) ) ) ).

% word_le_sub1_numberof
thf(fact_3946_word__of__int__neg__1,axiom,
    ( ( ring_17408606157368542149l_num1 @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ).

% word_of_int_neg_1
thf(fact_3947_word__less__sub1__numberof,axiom,
    ! [W: num] :
      ( ( ( numera7442385471795722001l_num1 @ W )
       != zero_z3563351764282998399l_num1 )
     => ( ( ord_le750835935415966154l_num1 @ one_on7727431528512463931l_num1 @ ( numera7442385471795722001l_num1 @ W ) )
        = ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ ( minus_4019991460397169231l_num1 @ ( numera7442385471795722001l_num1 @ W ) @ one_on7727431528512463931l_num1 ) ) ) ) ).

% word_less_sub1_numberof
thf(fact_3948_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu93272222329896523l_num1 @ zero_z3563351764282998399l_num1 )
    = ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ).

% dbl_dec_simps(2)
thf(fact_3949_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_dec_simps(2)
thf(fact_3950_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu965353292909893953uint32 @ zero_zero_uint32 )
    = ( uminus_uminus_uint32 @ one_one_uint32 ) ) ).

% dbl_dec_simps(2)
thf(fact_3951_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_dec_simps(2)
thf(fact_3952_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_dec_simps(2)
thf(fact_3953_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_dec_simps(2)
thf(fact_3954_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_3955_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_3956_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_3957_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_3958_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_3959_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_3960_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_3961_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_3962_even__add,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_uint32 @ A2 @ B2 ) )
      = ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A2 )
        = ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_add
thf(fact_3963_even__add,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_add
thf(fact_3964_even__add,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( plus_p361126936061061375l_num1 @ A2 @ B2 ) )
      = ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A2 )
        = ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_add
thf(fact_3965_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_3966_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_3967_odd__add,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_uint32 @ A2 @ B2 ) ) )
      = ( ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A2 ) )
       != ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).

% odd_add
thf(fact_3968_odd__add,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) ) )
      = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 ) )
       != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).

% odd_add
thf(fact_3969_odd__add,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( plus_p361126936061061375l_num1 @ A2 @ B2 ) ) )
      = ( ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A2 ) )
       != ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).

% odd_add
thf(fact_3970_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_3971_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_3972_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_3973_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_3974_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_3975_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_3976_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_3977_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_3978_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_3979_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_3980_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_3981_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_3982_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_3983_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_3984_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_3985_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_3986_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_3987_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_3988_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_3989_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_3990_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_3991_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_3992_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_3993_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_3994_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_3995_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_3996_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_3997_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_3998_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_3999_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_4000_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_4001_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_4002_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_4003_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_4004_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_4005_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_4006_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_4007_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_4008_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_4009_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_18347121197199848620nteger @ Y )
        = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_4010_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_real @ Y )
        = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_4011_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_rat @ Y )
        = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_4012_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_int @ Y )
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_4013_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
        = ( ring_17405671764205052669omplex @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_4014_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N )
        = ( ring_18347121197199848620nteger @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_4015_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
        = ( ring_1_of_int_real @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_4016_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
        = ( ring_1_of_int_rat @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_4017_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = ( ring_1_of_int_int @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_4018_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_4019_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_4020_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_4021_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_4022_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_4023_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_4024_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_4025_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_4026_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_4027_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_4028_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_4029_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_4030_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_4031_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_4032_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_4033_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_4034_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_4035_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_4036_bits__1__div__2,axiom,
    ( ( divide1791077408188789448l_num1 @ one_on7727431528512463931l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
    = zero_z3563351764282998399l_num1 ) ).

% bits_1_div_2
thf(fact_4037_bits__1__div__2,axiom,
    ( ( divide_divide_uint32 @ one_one_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
    = zero_zero_uint32 ) ).

% bits_1_div_2
thf(fact_4038_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_4039_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_4040_diff__numeral__special_I10_J,axiom,
    ( ( minus_4019991460397169231l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ one_on7727431528512463931l_num1 )
    = ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_4041_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_4042_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ one_one_uint32 )
    = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_4043_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_4044_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_4045_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_4046_diff__numeral__special_I11_J,axiom,
    ( ( minus_4019991460397169231l_num1 @ one_on7727431528512463931l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
    = ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_4047_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_4048_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_uint32 @ one_one_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) )
    = ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_4049_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_4050_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_4051_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_4052_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_4053_even__plus__one__iff,axiom,
    ! [A2: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_uint32 @ A2 @ one_one_uint32 ) )
      = ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A2 ) ) ) ).

% even_plus_one_iff
thf(fact_4054_even__plus__one__iff,axiom,
    ! [A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A2 @ one_one_Code_integer ) )
      = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 ) ) ) ).

% even_plus_one_iff
thf(fact_4055_even__plus__one__iff,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( plus_p361126936061061375l_num1 @ A2 @ one_on7727431528512463931l_num1 ) )
      = ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A2 ) ) ) ).

% even_plus_one_iff
thf(fact_4056_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_4057_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_4058_even__diff,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( minus_minus_uint32 @ A2 @ B2 ) )
      = ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_uint32 @ A2 @ B2 ) ) ) ).

% even_diff
thf(fact_4059_even__diff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A2 @ B2 ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) ) ) ).

% even_diff
thf(fact_4060_even__diff,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( minus_4019991460397169231l_num1 @ A2 @ B2 ) )
      = ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( plus_p361126936061061375l_num1 @ A2 @ B2 ) ) ) ).

% even_diff
thf(fact_4061_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_4062_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A2: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A2 ) @ N )
        = ( power_8256067586552552935nteger @ A2 @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_4063_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A2: complex] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A2 ) @ N )
        = ( power_power_complex @ A2 @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_4064_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A2: uint32] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ A2 ) @ N )
        = ( power_power_uint32 @ A2 @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_4065_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A2: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ A2 ) @ N )
        = ( power_power_real @ A2 @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_4066_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A2: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ A2 ) @ N )
        = ( power_power_rat @ A2 @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_4067_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A2: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ A2 ) @ N )
        = ( power_power_int @ A2 @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_4068_power__minus__odd,axiom,
    ! [N: nat,A2: code_integer] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A2 ) @ N )
        = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ) ).

% power_minus_odd
thf(fact_4069_power__minus__odd,axiom,
    ! [N: nat,A2: complex] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A2 ) @ N )
        = ( uminus1482373934393186551omplex @ ( power_power_complex @ A2 @ N ) ) ) ) ).

% power_minus_odd
thf(fact_4070_power__minus__odd,axiom,
    ! [N: nat,A2: uint32] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ A2 ) @ N )
        = ( uminus_uminus_uint32 @ ( power_power_uint32 @ A2 @ N ) ) ) ) ).

% power_minus_odd
thf(fact_4071_power__minus__odd,axiom,
    ! [N: nat,A2: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ A2 ) @ N )
        = ( uminus_uminus_real @ ( power_power_real @ A2 @ N ) ) ) ) ).

% power_minus_odd
thf(fact_4072_power__minus__odd,axiom,
    ! [N: nat,A2: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ A2 ) @ N )
        = ( uminus_uminus_rat @ ( power_power_rat @ A2 @ N ) ) ) ) ).

% power_minus_odd
thf(fact_4073_power__minus__odd,axiom,
    ! [N: nat,A2: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ A2 ) @ N )
        = ( uminus_uminus_int @ ( power_power_int @ A2 @ N ) ) ) ) ).

% power_minus_odd
thf(fact_4074_even__diff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).

% even_diff_nat
thf(fact_4075_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_4019991460397169231l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ M ) ) @ one_on7727431528512463931l_num1 )
      = ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_4076_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_4077_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ M ) ) @ one_one_uint32 )
      = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_4078_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_4079_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_4080_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_4081_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_4019991460397169231l_num1 @ one_on7727431528512463931l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ N ) ) )
      = ( numera7442385471795722001l_num1 @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_4082_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_4083_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_uint32 @ one_one_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ N ) ) )
      = ( numera9087168376688890119uint32 @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_4084_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_4085_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_4086_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_4087_powr__numeral,axiom,
    ! [X: real,N: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( numeral_numeral_real @ N ) )
        = ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ).

% powr_numeral
thf(fact_4088_even__succ__div__two,axiom,
    ! [A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A2 @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_4089_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_4090_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_4091_odd__succ__div__two,axiom,
    ! [A2: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A2 @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% odd_succ_div_two
thf(fact_4092_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_4093_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_4094_even__succ__div__2,axiom,
    ! [A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A2 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_4095_even__succ__div__2,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A2 )
     => ( ( divide1791077408188789448l_num1 @ ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ A2 ) @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
        = ( divide1791077408188789448l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_4096_even__succ__div__2,axiom,
    ! [A2: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A2 )
     => ( ( divide_divide_uint32 @ ( plus_plus_uint32 @ one_one_uint32 @ A2 ) @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
        = ( divide_divide_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_4097_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_4098_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_4099_even__power,axiom,
    ! [A2: uint32,N: nat] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( power_power_uint32 @ A2 @ N ) )
      = ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A2 )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% even_power
thf(fact_4100_even__power,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A2 @ N ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% even_power
thf(fact_4101_even__power,axiom,
    ! [A2: word_N3645301735248828278l_num1,N: nat] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( power_2184487114949457152l_num1 @ A2 @ N ) )
      = ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A2 )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% even_power
thf(fact_4102_even__power,axiom,
    ! [A2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A2 @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% even_power
thf(fact_4103_even__power,axiom,
    ! [A2: int,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A2 @ N ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% even_power
thf(fact_4104_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_4105_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_4106_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_4107_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_4108_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_4109_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_4110_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_4111_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_4112_power__less__zero__eq,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ zero_z3403309356797280102nteger )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        & ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ) ).

% power_less_zero_eq
thf(fact_4113_power__less__zero__eq,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_real @ ( power_power_real @ A2 @ N ) @ zero_zero_real )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        & ( ord_less_real @ A2 @ zero_zero_real ) ) ) ).

% power_less_zero_eq
thf(fact_4114_power__less__zero__eq,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ A2 @ N ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        & ( ord_less_rat @ A2 @ zero_zero_rat ) ) ) ).

% power_less_zero_eq
thf(fact_4115_power__less__zero__eq,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_int @ ( power_power_int @ A2 @ N ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        & ( ord_less_int @ A2 @ zero_zero_int ) ) ) ).

% power_less_zero_eq
thf(fact_4116_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_2184487114949457152l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ N )
        = one_on7727431528512463931l_num1 ) ) ).

% neg_one_even_power
thf(fact_4117_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
        = one_one_Code_integer ) ) ).

% neg_one_even_power
thf(fact_4118_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
        = one_one_complex ) ) ).

% neg_one_even_power
thf(fact_4119_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N )
        = one_one_uint32 ) ) ).

% neg_one_even_power
thf(fact_4120_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
        = one_one_real ) ) ).

% neg_one_even_power
thf(fact_4121_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
        = one_one_rat ) ) ).

% neg_one_even_power
thf(fact_4122_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
        = one_one_int ) ) ).

% neg_one_even_power
thf(fact_4123_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_2184487114949457152l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ N )
        = ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ) ).

% neg_one_odd_power
thf(fact_4124_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).

% neg_one_odd_power
thf(fact_4125_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
        = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).

% neg_one_odd_power
thf(fact_4126_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N )
        = ( uminus_uminus_uint32 @ one_one_uint32 ) ) ) ).

% neg_one_odd_power
thf(fact_4127_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
        = ( uminus_uminus_real @ one_one_real ) ) ) ).

% neg_one_odd_power
thf(fact_4128_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
        = ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% neg_one_odd_power
thf(fact_4129_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% neg_one_odd_power
thf(fact_4130_even__of__nat,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( semiri2565882477558803405uint32 @ N ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% even_of_nat
thf(fact_4131_even__of__nat,axiom,
    ! [N: nat] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( semiri8819519690708144855l_num1 @ N ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% even_of_nat
thf(fact_4132_even__of__nat,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% even_of_nat
thf(fact_4133_even__of__nat,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% even_of_nat
thf(fact_4134_even__of__nat,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% even_of_nat
thf(fact_4135_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_4136_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_4137_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_4138_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_4139_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_4140_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A2 ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_4141_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_4142_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A2 ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_4143_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) @ ( ring_18347121197199848620nteger @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A2 ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_4144_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A2 ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_4145_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A2 ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_4146_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A2 ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_4147_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_4148_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A2 ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_4149_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_4150_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A2 ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_4151_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) @ ( ring_18347121197199848620nteger @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A2 ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_4152_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A2 ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_4153_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A2 ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_4154_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A2 ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_4155_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_18347121197199848620nteger @ Y )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_4156_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_4157_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_real @ Y )
        = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_4158_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_rat @ Y )
        = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_4159_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_int @ Y )
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_4160_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N )
        = ( ring_18347121197199848620nteger @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_4161_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N )
        = ( ring_17405671764205052669omplex @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_4162_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N )
        = ( ring_1_of_int_real @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_4163_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N )
        = ( ring_1_of_int_rat @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_4164_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = ( ring_1_of_int_int @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_4165_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( minus_minus_uint32 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N ) @ one_one_uint32 ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_4166_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_4167_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( minus_4019991460397169231l_num1 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N ) @ one_on7727431528512463931l_num1 ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_4168_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_4169_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_4170_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_4171_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_4172_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_4173_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_4174_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_4175_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A2 ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_4176_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_4177_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A2 ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_4178_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A2 ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_4179_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A2 ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_4180_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A2 ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_4181_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A2 ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_4182_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_4183_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A2 ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_4184_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_4185_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A2 ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_4186_even__succ__div__exp,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
          = ( divide6298287555418463151nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_4187_even__succ__div__exp,axiom,
    ! [A2: word_N3645301735248828278l_num1,N: nat] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide1791077408188789448l_num1 @ ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ A2 ) @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N ) )
          = ( divide1791077408188789448l_num1 @ A2 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_4188_even__succ__div__exp,axiom,
    ! [A2: uint32,N: nat] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide_divide_uint32 @ ( plus_plus_uint32 @ one_one_uint32 @ A2 ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N ) )
          = ( divide_divide_uint32 @ A2 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_4189_even__succ__div__exp,axiom,
    ! [A2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
          = ( divide_divide_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_4190_even__succ__div__exp,axiom,
    ! [A2: int,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
          = ( divide_divide_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_4191_dvd__diffD,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
     => ( ( dvd_dvd_nat @ K @ N )
       => ( ( ord_less_eq_nat @ N @ M )
         => ( dvd_dvd_nat @ K @ M ) ) ) ) ).

% dvd_diffD
thf(fact_4192_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
     => ( ( dvd_dvd_nat @ K @ M )
       => ( ( ord_less_eq_nat @ N @ M )
         => ( dvd_dvd_nat @ K @ N ) ) ) ) ).

% dvd_diffD1
thf(fact_4193_less__eq__dvd__minus,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( dvd_dvd_nat @ M @ N )
        = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).

% less_eq_dvd_minus
thf(fact_4194_word__sub__wi,axiom,
    ( minus_4019991460397169231l_num1
    = ( ^ [A4: word_N3645301735248828278l_num1,B4: word_N3645301735248828278l_num1] : ( ring_17408606157368542149l_num1 @ ( minus_minus_int @ ( semiri7338730514057886004m1_int @ A4 ) @ ( semiri7338730514057886004m1_int @ B4 ) ) ) ) ) ).

% word_sub_wi
thf(fact_4195_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_4196_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_4197_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_4198_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_4199_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_4200_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_4201_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_4202_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_4203_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_4204_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_4205_floor__divide__of__int__eq,axiom,
    ! [K: int,L: int] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( ring_1_of_int_real @ K ) @ ( ring_1_of_int_real @ L ) ) )
      = ( divide_divide_int @ K @ L ) ) ).

% floor_divide_of_int_eq
thf(fact_4206_floor__divide__of__int__eq,axiom,
    ! [K: int,L: int] :
      ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ K ) @ ( ring_1_of_int_rat @ L ) ) )
      = ( divide_divide_int @ K @ L ) ) ).

% floor_divide_of_int_eq
thf(fact_4207_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_4208_size__neq__size__imp__neq,axiom,
    ! [X: uint32,Y: uint32] :
      ( ( ( size_size_uint32 @ X )
       != ( size_size_uint32 @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_4209_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_4210_div__div__div__same,axiom,
    ! [D: code_integer,B2: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ B2 )
     => ( ( dvd_dvd_Code_integer @ B2 @ A2 )
       => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A2 @ D ) @ ( divide6298287555418463151nteger @ B2 @ D ) )
          = ( divide6298287555418463151nteger @ A2 @ B2 ) ) ) ) ).

% div_div_div_same
thf(fact_4211_div__div__div__same,axiom,
    ! [D: nat,B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ D @ B2 )
     => ( ( dvd_dvd_nat @ B2 @ A2 )
       => ( ( divide_divide_nat @ ( divide_divide_nat @ A2 @ D ) @ ( divide_divide_nat @ B2 @ D ) )
          = ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).

% div_div_div_same
thf(fact_4212_div__div__div__same,axiom,
    ! [D: int,B2: int,A2: int] :
      ( ( dvd_dvd_int @ D @ B2 )
     => ( ( dvd_dvd_int @ B2 @ A2 )
       => ( ( divide_divide_int @ ( divide_divide_int @ A2 @ D ) @ ( divide_divide_int @ B2 @ D ) )
          = ( divide_divide_int @ A2 @ B2 ) ) ) ) ).

% div_div_div_same
thf(fact_4213_dvd__div__eq__cancel,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( ( ( divide6298287555418463151nteger @ A2 @ C )
        = ( divide6298287555418463151nteger @ B2 @ C ) )
     => ( ( dvd_dvd_Code_integer @ C @ A2 )
       => ( ( dvd_dvd_Code_integer @ C @ B2 )
         => ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_4214_dvd__div__eq__cancel,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( ( divide_divide_real @ A2 @ C )
        = ( divide_divide_real @ B2 @ C ) )
     => ( ( dvd_dvd_real @ C @ A2 )
       => ( ( dvd_dvd_real @ C @ B2 )
         => ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_4215_dvd__div__eq__cancel,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ( divide_divide_rat @ A2 @ C )
        = ( divide_divide_rat @ B2 @ C ) )
     => ( ( dvd_dvd_rat @ C @ A2 )
       => ( ( dvd_dvd_rat @ C @ B2 )
         => ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_4216_dvd__div__eq__cancel,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ( divide_divide_nat @ A2 @ C )
        = ( divide_divide_nat @ B2 @ C ) )
     => ( ( dvd_dvd_nat @ C @ A2 )
       => ( ( dvd_dvd_nat @ C @ B2 )
         => ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_4217_dvd__div__eq__cancel,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ( divide_divide_int @ A2 @ C )
        = ( divide_divide_int @ B2 @ C ) )
     => ( ( dvd_dvd_int @ C @ A2 )
       => ( ( dvd_dvd_int @ C @ B2 )
         => ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_4218_dvd__div__eq__iff,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A2 )
     => ( ( dvd_dvd_Code_integer @ C @ B2 )
       => ( ( ( divide6298287555418463151nteger @ A2 @ C )
            = ( divide6298287555418463151nteger @ B2 @ C ) )
          = ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_4219_dvd__div__eq__iff,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( dvd_dvd_real @ C @ A2 )
     => ( ( dvd_dvd_real @ C @ B2 )
       => ( ( ( divide_divide_real @ A2 @ C )
            = ( divide_divide_real @ B2 @ C ) )
          = ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_4220_dvd__div__eq__iff,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( dvd_dvd_rat @ C @ A2 )
     => ( ( dvd_dvd_rat @ C @ B2 )
       => ( ( ( divide_divide_rat @ A2 @ C )
            = ( divide_divide_rat @ B2 @ C ) )
          = ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_4221_dvd__div__eq__iff,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ C @ A2 )
     => ( ( dvd_dvd_nat @ C @ B2 )
       => ( ( ( divide_divide_nat @ A2 @ C )
            = ( divide_divide_nat @ B2 @ C ) )
          = ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_4222_dvd__div__eq__iff,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( dvd_dvd_int @ C @ A2 )
     => ( ( dvd_dvd_int @ C @ B2 )
       => ( ( ( divide_divide_int @ A2 @ C )
            = ( divide_divide_int @ B2 @ C ) )
          = ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_4223_dvd__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A2 @ B2 )
     => ( ( dvd_dvd_nat @ B2 @ C )
       => ( dvd_dvd_nat @ A2 @ C ) ) ) ).

% dvd_trans
thf(fact_4224_dvd__trans,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A2 @ B2 )
     => ( ( dvd_dvd_int @ B2 @ C )
       => ( dvd_dvd_int @ A2 @ C ) ) ) ).

% dvd_trans
thf(fact_4225_dvd__trans,axiom,
    ! [A2: uint32,B2: uint32,C: uint32] :
      ( ( dvd_dvd_uint32 @ A2 @ B2 )
     => ( ( dvd_dvd_uint32 @ B2 @ C )
       => ( dvd_dvd_uint32 @ A2 @ C ) ) ) ).

% dvd_trans
thf(fact_4226_dvd__trans,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A2 @ B2 )
     => ( ( dvd_dv6812691276156420380l_num1 @ B2 @ C )
       => ( dvd_dv6812691276156420380l_num1 @ A2 @ C ) ) ) ).

% dvd_trans
thf(fact_4227_dvd__trans,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ B2 )
     => ( ( dvd_dvd_Code_integer @ B2 @ C )
       => ( dvd_dvd_Code_integer @ A2 @ C ) ) ) ).

% dvd_trans
thf(fact_4228_dvd__refl,axiom,
    ! [A2: nat] : ( dvd_dvd_nat @ A2 @ A2 ) ).

% dvd_refl
thf(fact_4229_dvd__refl,axiom,
    ! [A2: int] : ( dvd_dvd_int @ A2 @ A2 ) ).

% dvd_refl
thf(fact_4230_dvd__refl,axiom,
    ! [A2: uint32] : ( dvd_dvd_uint32 @ A2 @ A2 ) ).

% dvd_refl
thf(fact_4231_dvd__refl,axiom,
    ! [A2: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ A2 @ A2 ) ).

% dvd_refl
thf(fact_4232_dvd__refl,axiom,
    ! [A2: code_integer] : ( dvd_dvd_Code_integer @ A2 @ A2 ) ).

% dvd_refl
thf(fact_4233_dvd__diff,axiom,
    ! [X: uint32,Y: uint32,Z: uint32] :
      ( ( dvd_dvd_uint32 @ X @ Y )
     => ( ( dvd_dvd_uint32 @ X @ Z )
       => ( dvd_dvd_uint32 @ X @ ( minus_minus_uint32 @ Y @ Z ) ) ) ) ).

% dvd_diff
thf(fact_4234_dvd__diff,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1,Z: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ X @ Y )
     => ( ( dvd_dv6812691276156420380l_num1 @ X @ Z )
       => ( dvd_dv6812691276156420380l_num1 @ X @ ( minus_4019991460397169231l_num1 @ Y @ Z ) ) ) ) ).

% dvd_diff
thf(fact_4235_dvd__diff,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( dvd_dvd_Code_integer @ X @ Y )
     => ( ( dvd_dvd_Code_integer @ X @ Z )
       => ( dvd_dvd_Code_integer @ X @ ( minus_8373710615458151222nteger @ Y @ Z ) ) ) ) ).

% dvd_diff
thf(fact_4236_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_4237_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_4238_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_4239_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ M )
     => ( ( dvd_dvd_nat @ K @ N )
       => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% dvd_diff_nat
thf(fact_4240_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ M @ N )
     => ( ( dvd_dvd_nat @ N @ M )
       => ( M = N ) ) ) ).

% dvd_antisym
thf(fact_4241_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A2 @ C ) @ B2 )
      = ( minus_minus_real @ ( minus_minus_real @ A2 @ B2 ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_4242_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A2 @ C ) @ B2 )
      = ( minus_minus_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_4243_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B2 )
      = ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_4244_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B2 )
      = ( minus_minus_int @ ( minus_minus_int @ A2 @ B2 ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_4245_diff__eq__diff__eq,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A2 @ B2 )
        = ( minus_minus_real @ C @ D ) )
     => ( ( A2 = B2 )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_4246_diff__eq__diff__eq,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A2 @ B2 )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( A2 = B2 )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_4247_diff__eq__diff__eq,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A2 @ B2 )
        = ( minus_minus_int @ C @ D ) )
     => ( ( A2 = B2 )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_4248_dvd__minus__self,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
      = ( ( ord_less_nat @ N @ M )
        | ( dvd_dvd_nat @ M @ N ) ) ) ).

% dvd_minus_self
thf(fact_4249_div__power,axiom,
    ! [B2: code_integer,A2: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ B2 @ A2 )
     => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ N )
        = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ ( power_8256067586552552935nteger @ B2 @ N ) ) ) ) ).

% div_power
thf(fact_4250_div__power,axiom,
    ! [B2: nat,A2: nat,N: nat] :
      ( ( dvd_dvd_nat @ B2 @ A2 )
     => ( ( power_power_nat @ ( divide_divide_nat @ A2 @ B2 ) @ N )
        = ( divide_divide_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ).

% div_power
thf(fact_4251_div__power,axiom,
    ! [B2: int,A2: int,N: nat] :
      ( ( dvd_dvd_int @ B2 @ A2 )
     => ( ( power_power_int @ ( divide_divide_int @ A2 @ B2 ) @ N )
        = ( divide_divide_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).

% div_power
thf(fact_4252_dvd__div__unit__iff,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A2 @ ( divide6298287555418463151nteger @ C @ B2 ) )
        = ( dvd_dvd_Code_integer @ A2 @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_4253_dvd__div__unit__iff,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ A2 @ ( divide_divide_nat @ C @ B2 ) )
        = ( dvd_dvd_nat @ A2 @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_4254_dvd__div__unit__iff,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ A2 @ ( divide_divide_int @ C @ B2 ) )
        = ( dvd_dvd_int @ A2 @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_4255_div__unit__dvd__iff,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ C )
        = ( dvd_dvd_Code_integer @ A2 @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_4256_div__unit__dvd__iff,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C )
        = ( dvd_dvd_nat @ A2 @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_4257_div__unit__dvd__iff,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ ( divide_divide_int @ A2 @ B2 ) @ C )
        = ( dvd_dvd_int @ A2 @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_4258_unit__div__cancel,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ B2 @ A2 )
          = ( divide6298287555418463151nteger @ C @ A2 ) )
        = ( B2 = C ) ) ) ).

% unit_div_cancel
thf(fact_4259_unit__div__cancel,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( ( ( divide_divide_nat @ B2 @ A2 )
          = ( divide_divide_nat @ C @ A2 ) )
        = ( B2 = C ) ) ) ).

% unit_div_cancel
thf(fact_4260_unit__div__cancel,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( ( ( divide_divide_int @ B2 @ A2 )
          = ( divide_divide_int @ C @ A2 ) )
        = ( B2 = C ) ) ) ).

% unit_div_cancel
thf(fact_4261_dvd__div__eq__0__iff,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ A2 )
     => ( ( ( divide6298287555418463151nteger @ A2 @ B2 )
          = zero_z3403309356797280102nteger )
        = ( A2 = zero_z3403309356797280102nteger ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_4262_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_4263_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_4264_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_4265_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_4266_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_4267_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_4268_unit__div__eq__0__iff,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ A2 @ B2 )
          = zero_z3403309356797280102nteger )
        = ( A2 = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
thf(fact_4269_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_4270_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_4271_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).

% of_nat_diff
thf(fact_4272_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% of_nat_diff
thf(fact_4273_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% of_nat_diff
thf(fact_4274_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).

% of_nat_diff
thf(fact_4275_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri4939895301339042750nteger @ ( minus_minus_nat @ M @ N ) )
        = ( minus_8373710615458151222nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ) ).

% of_nat_diff
thf(fact_4276_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri8010041392384452111omplex @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).

% of_nat_diff
thf(fact_4277_uint__sub__lem,axiom,
    ! [Y: word_N3645301735248828278l_num1,X: word_N3645301735248828278l_num1] :
      ( ( ord_less_eq_int @ ( semiri7338730514057886004m1_int @ Y ) @ ( semiri7338730514057886004m1_int @ X ) )
      = ( ( semiri7338730514057886004m1_int @ ( minus_4019991460397169231l_num1 @ X @ Y ) )
        = ( minus_minus_int @ ( semiri7338730514057886004m1_int @ X ) @ ( semiri7338730514057886004m1_int @ Y ) ) ) ) ).

% uint_sub_lem
thf(fact_4278_uint__sub__ge,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] : ( ord_less_eq_int @ ( minus_minus_int @ ( semiri7338730514057886004m1_int @ X ) @ ( semiri7338730514057886004m1_int @ Y ) ) @ ( semiri7338730514057886004m1_int @ ( minus_4019991460397169231l_num1 @ X @ Y ) ) ) ).

% uint_sub_ge
thf(fact_4279_uint__minus__simple__iff,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ ( minus_4019991460397169231l_num1 @ X @ Y ) @ X )
      = ( ( semiri7338730514057886004m1_int @ ( minus_4019991460397169231l_num1 @ X @ Y ) )
        = ( minus_minus_int @ ( semiri7338730514057886004m1_int @ X ) @ ( semiri7338730514057886004m1_int @ Y ) ) ) ) ).

% uint_minus_simple_iff
thf(fact_4280_uint__minus__simple__alt,axiom,
    ( ord_le3335648743751981014l_num1
    = ( ^ [Y2: word_N3645301735248828278l_num1,X2: word_N3645301735248828278l_num1] :
          ( ( semiri7338730514057886004m1_int @ ( minus_4019991460397169231l_num1 @ X2 @ Y2 ) )
          = ( minus_minus_int @ ( semiri7338730514057886004m1_int @ X2 ) @ ( semiri7338730514057886004m1_int @ Y2 ) ) ) ) ) ).

% uint_minus_simple_alt
thf(fact_4281_ex__le__of__int,axiom,
    ! [X: real] :
    ? [Z3: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ).

% ex_le_of_int
thf(fact_4282_ex__le__of__int,axiom,
    ! [X: rat] :
    ? [Z3: int] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z3 ) ) ).

% ex_le_of_int
thf(fact_4283_ex__of__int__less,axiom,
    ! [X: real] :
    ? [Z3: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z3 ) @ X ) ).

% ex_of_int_less
thf(fact_4284_ex__of__int__less,axiom,
    ! [X: rat] :
    ? [Z3: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z3 ) @ X ) ).

% ex_of_int_less
thf(fact_4285_ex__less__of__int,axiom,
    ! [X: real] :
    ? [Z3: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z3 ) ) ).

% ex_less_of_int
thf(fact_4286_ex__less__of__int,axiom,
    ! [X: rat] :
    ? [Z3: int] : ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z3 ) ) ).

% ex_less_of_int
thf(fact_4287_dvd__field__iff,axiom,
    ( dvd_dvd_real
    = ( ^ [A4: real,B4: real] :
          ( ( A4 = zero_zero_real )
         => ( B4 = zero_zero_real ) ) ) ) ).

% dvd_field_iff
thf(fact_4288_dvd__field__iff,axiom,
    ( dvd_dvd_rat
    = ( ^ [A4: rat,B4: rat] :
          ( ( A4 = zero_zero_rat )
         => ( B4 = zero_zero_rat ) ) ) ) ).

% dvd_field_iff
thf(fact_4289_dvd__0__left,axiom,
    ! [A2: uint32] :
      ( ( dvd_dvd_uint32 @ zero_zero_uint32 @ A2 )
     => ( A2 = zero_zero_uint32 ) ) ).

% dvd_0_left
thf(fact_4290_dvd__0__left,axiom,
    ! [A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A2 )
     => ( A2 = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left
thf(fact_4291_dvd__0__left,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ zero_z3563351764282998399l_num1 @ A2 )
     => ( A2 = zero_z3563351764282998399l_num1 ) ) ).

% dvd_0_left
thf(fact_4292_dvd__0__left,axiom,
    ! [A2: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A2 )
     => ( A2 = zero_zero_real ) ) ).

% dvd_0_left
thf(fact_4293_dvd__0__left,axiom,
    ! [A2: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A2 )
     => ( A2 = zero_zero_rat ) ) ).

% dvd_0_left
thf(fact_4294_dvd__0__left,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
     => ( A2 = zero_zero_nat ) ) ).

% dvd_0_left
thf(fact_4295_dvd__0__left,axiom,
    ! [A2: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A2 )
     => ( A2 = zero_zero_int ) ) ).

% dvd_0_left
thf(fact_4296_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y3: word_N3645301735248828278l_num1,Z2: word_N3645301735248828278l_num1] : Y3 = Z2 )
    = ( ^ [A4: word_N3645301735248828278l_num1,B4: word_N3645301735248828278l_num1] :
          ( ( minus_4019991460397169231l_num1 @ A4 @ B4 )
          = zero_z3563351764282998399l_num1 ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_4297_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y3: real,Z2: real] : Y3 = Z2 )
    = ( ^ [A4: real,B4: real] :
          ( ( minus_minus_real @ A4 @ B4 )
          = zero_zero_real ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_4298_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y3: rat,Z2: rat] : Y3 = Z2 )
    = ( ^ [A4: rat,B4: rat] :
          ( ( minus_minus_rat @ A4 @ B4 )
          = zero_zero_rat ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_4299_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y3: int,Z2: int] : Y3 = Z2 )
    = ( ^ [A4: int,B4: int] :
          ( ( minus_minus_int @ A4 @ B4 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_4300_diff__eq__diff__less__eq,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A2 @ B2 )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_eq_real @ A2 @ B2 )
        = ( ord_less_eq_real @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_4301_diff__eq__diff__less__eq,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A2 @ B2 )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_eq_rat @ A2 @ B2 )
        = ( ord_less_eq_rat @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_4302_diff__eq__diff__less__eq,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A2 @ B2 )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_eq_int @ A2 @ B2 )
        = ( ord_less_eq_int @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_4303_diff__right__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ord_less_eq_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B2 @ C ) ) ) ).

% diff_right_mono
thf(fact_4304_diff__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B2 @ C ) ) ) ).

% diff_right_mono
thf(fact_4305_diff__right__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).

% diff_right_mono
thf(fact_4306_diff__left__mono,axiom,
    ! [B2: real,A2: real,C: real] :
      ( ( ord_less_eq_real @ B2 @ A2 )
     => ( ord_less_eq_real @ ( minus_minus_real @ C @ A2 ) @ ( minus_minus_real @ C @ B2 ) ) ) ).

% diff_left_mono
thf(fact_4307_diff__left__mono,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A2 ) @ ( minus_minus_rat @ C @ B2 ) ) ) ).

% diff_left_mono
thf(fact_4308_diff__left__mono,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ord_less_eq_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B2 ) ) ) ).

% diff_left_mono
thf(fact_4309_diff__mono,axiom,
    ! [A2: real,B2: real,D: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ D @ C )
       => ( ord_less_eq_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B2 @ D ) ) ) ) ).

% diff_mono
thf(fact_4310_diff__mono,axiom,
    ! [A2: rat,B2: rat,D: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ D @ C )
       => ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B2 @ D ) ) ) ) ).

% diff_mono
thf(fact_4311_diff__mono,axiom,
    ! [A2: int,B2: int,D: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ D @ C )
       => ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).

% diff_mono
thf(fact_4312_dvd__add__right__iff,axiom,
    ! [A2: uint32,B2: uint32,C: uint32] :
      ( ( dvd_dvd_uint32 @ A2 @ B2 )
     => ( ( dvd_dvd_uint32 @ A2 @ ( plus_plus_uint32 @ B2 @ C ) )
        = ( dvd_dvd_uint32 @ A2 @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_4313_dvd__add__right__iff,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A2 @ B2 )
     => ( ( dvd_dv6812691276156420380l_num1 @ A2 @ ( plus_p361126936061061375l_num1 @ B2 @ C ) )
        = ( dvd_dv6812691276156420380l_num1 @ A2 @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_4314_dvd__add__right__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ B2 )
     => ( ( dvd_dvd_Code_integer @ A2 @ ( plus_p5714425477246183910nteger @ B2 @ C ) )
        = ( dvd_dvd_Code_integer @ A2 @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_4315_dvd__add__right__iff,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( dvd_dvd_real @ A2 @ B2 )
     => ( ( dvd_dvd_real @ A2 @ ( plus_plus_real @ B2 @ C ) )
        = ( dvd_dvd_real @ A2 @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_4316_dvd__add__right__iff,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( dvd_dvd_rat @ A2 @ B2 )
     => ( ( dvd_dvd_rat @ A2 @ ( plus_plus_rat @ B2 @ C ) )
        = ( dvd_dvd_rat @ A2 @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_4317_dvd__add__right__iff,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A2 @ B2 )
     => ( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) )
        = ( dvd_dvd_nat @ A2 @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_4318_dvd__add__right__iff,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A2 @ B2 )
     => ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B2 @ C ) )
        = ( dvd_dvd_int @ A2 @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_4319_dvd__add__left__iff,axiom,
    ! [A2: uint32,C: uint32,B2: uint32] :
      ( ( dvd_dvd_uint32 @ A2 @ C )
     => ( ( dvd_dvd_uint32 @ A2 @ ( plus_plus_uint32 @ B2 @ C ) )
        = ( dvd_dvd_uint32 @ A2 @ B2 ) ) ) ).

% dvd_add_left_iff
thf(fact_4320_dvd__add__left__iff,axiom,
    ! [A2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A2 @ C )
     => ( ( dvd_dv6812691276156420380l_num1 @ A2 @ ( plus_p361126936061061375l_num1 @ B2 @ C ) )
        = ( dvd_dv6812691276156420380l_num1 @ A2 @ B2 ) ) ) ).

% dvd_add_left_iff
thf(fact_4321_dvd__add__left__iff,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ C )
     => ( ( dvd_dvd_Code_integer @ A2 @ ( plus_p5714425477246183910nteger @ B2 @ C ) )
        = ( dvd_dvd_Code_integer @ A2 @ B2 ) ) ) ).

% dvd_add_left_iff
thf(fact_4322_dvd__add__left__iff,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( dvd_dvd_real @ A2 @ C )
     => ( ( dvd_dvd_real @ A2 @ ( plus_plus_real @ B2 @ C ) )
        = ( dvd_dvd_real @ A2 @ B2 ) ) ) ).

% dvd_add_left_iff
thf(fact_4323_dvd__add__left__iff,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( dvd_dvd_rat @ A2 @ C )
     => ( ( dvd_dvd_rat @ A2 @ ( plus_plus_rat @ B2 @ C ) )
        = ( dvd_dvd_rat @ A2 @ B2 ) ) ) ).

% dvd_add_left_iff
thf(fact_4324_dvd__add__left__iff,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ C )
     => ( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) )
        = ( dvd_dvd_nat @ A2 @ B2 ) ) ) ).

% dvd_add_left_iff
thf(fact_4325_dvd__add__left__iff,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ C )
     => ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B2 @ C ) )
        = ( dvd_dvd_int @ A2 @ B2 ) ) ) ).

% dvd_add_left_iff
thf(fact_4326_dvd__add,axiom,
    ! [A2: uint32,B2: uint32,C: uint32] :
      ( ( dvd_dvd_uint32 @ A2 @ B2 )
     => ( ( dvd_dvd_uint32 @ A2 @ C )
       => ( dvd_dvd_uint32 @ A2 @ ( plus_plus_uint32 @ B2 @ C ) ) ) ) ).

% dvd_add
thf(fact_4327_dvd__add,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A2 @ B2 )
     => ( ( dvd_dv6812691276156420380l_num1 @ A2 @ C )
       => ( dvd_dv6812691276156420380l_num1 @ A2 @ ( plus_p361126936061061375l_num1 @ B2 @ C ) ) ) ) ).

% dvd_add
thf(fact_4328_dvd__add,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ B2 )
     => ( ( dvd_dvd_Code_integer @ A2 @ C )
       => ( dvd_dvd_Code_integer @ A2 @ ( plus_p5714425477246183910nteger @ B2 @ C ) ) ) ) ).

% dvd_add
thf(fact_4329_dvd__add,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( dvd_dvd_real @ A2 @ B2 )
     => ( ( dvd_dvd_real @ A2 @ C )
       => ( dvd_dvd_real @ A2 @ ( plus_plus_real @ B2 @ C ) ) ) ) ).

% dvd_add
thf(fact_4330_dvd__add,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( dvd_dvd_rat @ A2 @ B2 )
     => ( ( dvd_dvd_rat @ A2 @ C )
       => ( dvd_dvd_rat @ A2 @ ( plus_plus_rat @ B2 @ C ) ) ) ) ).

% dvd_add
thf(fact_4331_dvd__add,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A2 @ B2 )
     => ( ( dvd_dvd_nat @ A2 @ C )
       => ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ) ).

% dvd_add
thf(fact_4332_dvd__add,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A2 @ B2 )
     => ( ( dvd_dvd_int @ A2 @ C )
       => ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ) ).

% dvd_add
thf(fact_4333_one__dvd,axiom,
    ! [A2: uint32] : ( dvd_dvd_uint32 @ one_one_uint32 @ A2 ) ).

% one_dvd
thf(fact_4334_one__dvd,axiom,
    ! [A2: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A2 ) ).

% one_dvd
thf(fact_4335_one__dvd,axiom,
    ! [A2: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ one_on7727431528512463931l_num1 @ A2 ) ).

% one_dvd
thf(fact_4336_one__dvd,axiom,
    ! [A2: real] : ( dvd_dvd_real @ one_one_real @ A2 ) ).

% one_dvd
thf(fact_4337_one__dvd,axiom,
    ! [A2: rat] : ( dvd_dvd_rat @ one_one_rat @ A2 ) ).

% one_dvd
thf(fact_4338_one__dvd,axiom,
    ! [A2: nat] : ( dvd_dvd_nat @ one_one_nat @ A2 ) ).

% one_dvd
thf(fact_4339_one__dvd,axiom,
    ! [A2: int] : ( dvd_dvd_int @ one_one_int @ A2 ) ).

% one_dvd
thf(fact_4340_unit__imp__dvd,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( dvd_dvd_Code_integer @ B2 @ A2 ) ) ).

% unit_imp_dvd
thf(fact_4341_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_4342_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_4343_dvd__unit__imp__unit,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ B2 )
     => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer ) ) ) ).

% dvd_unit_imp_unit
thf(fact_4344_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_4345_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_4346_diff__strict__right__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ord_less_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B2 @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_4347_diff__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ord_less_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B2 @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_4348_diff__strict__right__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_4349_diff__strict__left__mono,axiom,
    ! [B2: real,A2: real,C: real] :
      ( ( ord_less_real @ B2 @ A2 )
     => ( ord_less_real @ ( minus_minus_real @ C @ A2 ) @ ( minus_minus_real @ C @ B2 ) ) ) ).

% diff_strict_left_mono
thf(fact_4350_diff__strict__left__mono,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ( ord_less_rat @ ( minus_minus_rat @ C @ A2 ) @ ( minus_minus_rat @ C @ B2 ) ) ) ).

% diff_strict_left_mono
thf(fact_4351_diff__strict__left__mono,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ord_less_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B2 ) ) ) ).

% diff_strict_left_mono
thf(fact_4352_diff__eq__diff__less,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A2 @ B2 )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_real @ A2 @ B2 )
        = ( ord_less_real @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_4353_diff__eq__diff__less,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A2 @ B2 )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_rat @ A2 @ B2 )
        = ( ord_less_rat @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_4354_diff__eq__diff__less,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A2 @ B2 )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_int @ A2 @ B2 )
        = ( ord_less_int @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_4355_diff__strict__mono,axiom,
    ! [A2: real,B2: real,D: real,C: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_real @ D @ C )
       => ( ord_less_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B2 @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_4356_diff__strict__mono,axiom,
    ! [A2: rat,B2: rat,D: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ D @ C )
       => ( ord_less_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B2 @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_4357_diff__strict__mono,axiom,
    ! [A2: int,B2: int,D: int,C: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ D @ C )
       => ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_4358_diff__diff__eq,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A2 @ B2 ) @ C )
      = ( minus_minus_real @ A2 @ ( plus_plus_real @ B2 @ C ) ) ) ).

% diff_diff_eq
thf(fact_4359_diff__diff__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C )
      = ( minus_minus_rat @ A2 @ ( plus_plus_rat @ B2 @ C ) ) ) ).

% diff_diff_eq
thf(fact_4360_diff__diff__eq,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C )
      = ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% diff_diff_eq
thf(fact_4361_diff__diff__eq,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
      = ( minus_minus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).

% diff_diff_eq
thf(fact_4362_add__diff__add,axiom,
    ! [A2: real,C: real,B2: real,D: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ D ) )
      = ( plus_plus_real @ ( minus_minus_real @ A2 @ B2 ) @ ( minus_minus_real @ C @ D ) ) ) ).

% add_diff_add
thf(fact_4363_add__diff__add,axiom,
    ! [A2: rat,C: rat,B2: rat,D: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B2 @ D ) )
      = ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B2 ) @ ( minus_minus_rat @ C @ D ) ) ) ).

% add_diff_add
thf(fact_4364_add__diff__add,axiom,
    ! [A2: int,C: int,B2: int,D: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) )
      = ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ ( minus_minus_int @ C @ D ) ) ) ).

% add_diff_add
thf(fact_4365_add__implies__diff,axiom,
    ! [C: real,B2: real,A2: real] :
      ( ( ( plus_plus_real @ C @ B2 )
        = A2 )
     => ( C
        = ( minus_minus_real @ A2 @ B2 ) ) ) ).

% add_implies_diff
thf(fact_4366_add__implies__diff,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( ( ( plus_plus_rat @ C @ B2 )
        = A2 )
     => ( C
        = ( minus_minus_rat @ A2 @ B2 ) ) ) ).

% add_implies_diff
thf(fact_4367_add__implies__diff,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( ( ( plus_plus_nat @ C @ B2 )
        = A2 )
     => ( C
        = ( minus_minus_nat @ A2 @ B2 ) ) ) ).

% add_implies_diff
thf(fact_4368_add__implies__diff,axiom,
    ! [C: int,B2: int,A2: int] :
      ( ( ( plus_plus_int @ C @ B2 )
        = A2 )
     => ( C
        = ( minus_minus_int @ A2 @ B2 ) ) ) ).

% add_implies_diff
thf(fact_4369_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( minus_minus_real @ A2 @ ( plus_plus_real @ B2 @ C ) )
      = ( minus_minus_real @ ( minus_minus_real @ A2 @ C ) @ B2 ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_4370_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( minus_minus_rat @ A2 @ ( plus_plus_rat @ B2 @ C ) )
      = ( minus_minus_rat @ ( minus_minus_rat @ A2 @ C ) @ B2 ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_4371_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( minus_minus_int @ A2 @ ( plus_plus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B2 ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_4372_diff__add__eq,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A2 @ B2 ) @ C )
      = ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ B2 ) ) ).

% diff_add_eq
thf(fact_4373_diff__add__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C )
      = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C ) @ B2 ) ) ).

% diff_add_eq
thf(fact_4374_diff__add__eq,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
      = ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ).

% diff_add_eq
thf(fact_4375_diff__diff__eq2,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( minus_minus_real @ A2 @ ( minus_minus_real @ B2 @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ B2 ) ) ).

% diff_diff_eq2
thf(fact_4376_diff__diff__eq2,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( minus_minus_rat @ A2 @ ( minus_minus_rat @ B2 @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C ) @ B2 ) ) ).

% diff_diff_eq2
thf(fact_4377_diff__diff__eq2,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( minus_minus_int @ A2 @ ( minus_minus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ).

% diff_diff_eq2
thf(fact_4378_add__diff__eq,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( plus_plus_real @ A2 @ ( minus_minus_real @ B2 @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A2 @ B2 ) @ C ) ) ).

% add_diff_eq
thf(fact_4379_add__diff__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( plus_plus_rat @ A2 @ ( minus_minus_rat @ B2 @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C ) ) ).

% add_diff_eq
thf(fact_4380_add__diff__eq,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( plus_plus_int @ A2 @ ( minus_minus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ C ) ) ).

% add_diff_eq
thf(fact_4381_eq__diff__eq,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( A2
        = ( minus_minus_real @ C @ B2 ) )
      = ( ( plus_plus_real @ A2 @ B2 )
        = C ) ) ).

% eq_diff_eq
thf(fact_4382_eq__diff__eq,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( A2
        = ( minus_minus_rat @ C @ B2 ) )
      = ( ( plus_plus_rat @ A2 @ B2 )
        = C ) ) ).

% eq_diff_eq
thf(fact_4383_eq__diff__eq,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( A2
        = ( minus_minus_int @ C @ B2 ) )
      = ( ( plus_plus_int @ A2 @ B2 )
        = C ) ) ).

% eq_diff_eq
thf(fact_4384_diff__eq__eq,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ( minus_minus_real @ A2 @ B2 )
        = C )
      = ( A2
        = ( plus_plus_real @ C @ B2 ) ) ) ).

% diff_eq_eq
thf(fact_4385_diff__eq__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ( minus_minus_rat @ A2 @ B2 )
        = C )
      = ( A2
        = ( plus_plus_rat @ C @ B2 ) ) ) ).

% diff_eq_eq
thf(fact_4386_diff__eq__eq,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ( minus_minus_int @ A2 @ B2 )
        = C )
      = ( A2
        = ( plus_plus_int @ C @ B2 ) ) ) ).

% diff_eq_eq
thf(fact_4387_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_4388_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_4389_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_4390_dvd__power__same,axiom,
    ! [X: uint32,Y: uint32,N: nat] :
      ( ( dvd_dvd_uint32 @ X @ Y )
     => ( dvd_dvd_uint32 @ ( power_power_uint32 @ X @ N ) @ ( power_power_uint32 @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_4391_dvd__power__same,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1,N: nat] :
      ( ( dvd_dv6812691276156420380l_num1 @ X @ Y )
     => ( dvd_dv6812691276156420380l_num1 @ ( power_2184487114949457152l_num1 @ X @ N ) @ ( power_2184487114949457152l_num1 @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_4392_dvd__power__same,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( dvd_dvd_nat @ X @ Y )
     => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_4393_dvd__power__same,axiom,
    ! [X: real,Y: real,N: nat] :
      ( ( dvd_dvd_real @ X @ Y )
     => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_4394_dvd__power__same,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_4395_dvd__power__same,axiom,
    ! [X: complex,Y: complex,N: nat] :
      ( ( dvd_dvd_complex @ X @ Y )
     => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_4396_dvd__power__same,axiom,
    ! [X: code_integer,Y: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ X @ Y )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_4397_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_4398_minus__diff__minus,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( minus_minus_uint32 @ ( uminus_uminus_uint32 @ A2 ) @ ( uminus_uminus_uint32 @ B2 ) )
      = ( uminus_uminus_uint32 @ ( minus_minus_uint32 @ A2 @ B2 ) ) ) ).

% minus_diff_minus
thf(fact_4399_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_4400_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_4401_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_4402_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_4403_minus__diff__commute,axiom,
    ! [B2: uint32,A2: uint32] :
      ( ( minus_minus_uint32 @ ( uminus_uminus_uint32 @ B2 ) @ A2 )
      = ( minus_minus_uint32 @ ( uminus_uminus_uint32 @ A2 ) @ B2 ) ) ).

% minus_diff_commute
thf(fact_4404_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_4405_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_4406_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_4407_power__divide,axiom,
    ! [A2: complex,B2: complex,N: nat] :
      ( ( power_power_complex @ ( divide1717551699836669952omplex @ A2 @ B2 ) @ N )
      = ( divide1717551699836669952omplex @ ( power_power_complex @ A2 @ N ) @ ( power_power_complex @ B2 @ N ) ) ) ).

% power_divide
thf(fact_4408_power__divide,axiom,
    ! [A2: real,B2: real,N: nat] :
      ( ( power_power_real @ ( divide_divide_real @ A2 @ B2 ) @ N )
      = ( divide_divide_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ B2 @ N ) ) ) ).

% power_divide
thf(fact_4409_power__divide,axiom,
    ! [A2: rat,B2: rat,N: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ A2 @ B2 ) @ N )
      = ( divide_divide_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ).

% power_divide
thf(fact_4410_powr__minus__divide,axiom,
    ! [X: real,A2: real] :
      ( ( powr_real @ X @ ( uminus_uminus_real @ A2 ) )
      = ( divide_divide_real @ one_one_real @ ( powr_real @ X @ A2 ) ) ) ).

% powr_minus_divide
thf(fact_4411_even__of__int__iff,axiom,
    ! [K: int] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( ring_1_of_int_uint32 @ K ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).

% even_of_int_iff
thf(fact_4412_even__of__int__iff,axiom,
    ! [K: int] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ K ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).

% even_of_int_iff
thf(fact_4413_even__of__int__iff,axiom,
    ! [K: int] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( ring_17408606157368542149l_num1 @ K ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).

% even_of_int_iff
thf(fact_4414_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_4415_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N @ M )
          = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
thf(fact_4416_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_4417_Diff__mono,axiom,
    ! [A: set_nat,C2: set_nat,D3: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C2 )
     => ( ( ord_less_eq_set_nat @ D3 @ B )
       => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ C2 @ D3 ) ) ) ) ).

% Diff_mono
thf(fact_4418_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_4419_double__diff,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ C2 )
       => ( ( minus_minus_set_nat @ B @ ( minus_minus_set_nat @ C2 @ A ) )
          = A ) ) ) ).

% double_diff
thf(fact_4420_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ J @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).

% less_imp_diff_less
thf(fact_4421_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ord_less_nat @ M @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_4422_diff__diff__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ M @ ( minus_minus_nat @ M @ N ) ) )
      = ( ( ord_less_nat @ I @ M )
        & ( ord_less_nat @ I @ N ) ) ) ).

% diff_diff_less
thf(fact_4423_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_4424_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_4425_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_4426_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_4427_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_4428_minus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( minus_minus_int @ K @ zero_zero_int )
      = K ) ).

% minus_int_code(1)
thf(fact_4429_word__of__int__uint,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( ring_17408606157368542149l_num1 @ ( semiri7338730514057886004m1_int @ W ) )
      = W ) ).

% word_of_int_uint
thf(fact_4430_More__Word_Oof__int__uint,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( ring_17408606157368542149l_num1 @ ( semiri7338730514057886004m1_int @ X ) )
      = X ) ).

% More_Word.of_int_uint
thf(fact_4431_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% Nat.diff_cancel
thf(fact_4432_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_cancel2
thf(fact_4433_diff__add__inverse,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
      = M ) ).

% diff_add_inverse
thf(fact_4434_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
      = M ) ).

% diff_add_inverse2
thf(fact_4435_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N @ K ) )
          = ( M = N ) ) ) ) ).

% eq_diff_iff
thf(fact_4436_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% le_diff_iff
thf(fact_4437_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( minus_minus_nat @ M @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_4438_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_4439_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_4440_le__diff__iff_H,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ C )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
          = ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).

% le_diff_iff'
thf(fact_4441_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_4442_Un__Diff,axiom,
    ! [A: set_complex,B: set_complex,C2: set_complex] :
      ( ( minus_811609699411566653omplex @ ( sup_sup_set_complex @ A @ B ) @ C2 )
      = ( sup_sup_set_complex @ ( minus_811609699411566653omplex @ A @ C2 ) @ ( minus_811609699411566653omplex @ B @ C2 ) ) ) ).

% Un_Diff
thf(fact_4443_Un__Diff,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( ( minus_minus_set_int @ ( sup_sup_set_int @ A @ B ) @ C2 )
      = ( sup_sup_set_int @ ( minus_minus_set_int @ A @ C2 ) @ ( minus_minus_set_int @ B @ C2 ) ) ) ).

% Un_Diff
thf(fact_4444_Un__Diff,axiom,
    ! [A: set_real,B: set_real,C2: set_real] :
      ( ( minus_minus_set_real @ ( sup_sup_set_real @ A @ B ) @ C2 )
      = ( sup_sup_set_real @ ( minus_minus_set_real @ A @ C2 ) @ ( minus_minus_set_real @ B @ C2 ) ) ) ).

% Un_Diff
thf(fact_4445_Un__Diff,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
      = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ C2 ) @ ( minus_minus_set_nat @ B @ C2 ) ) ) ).

% Un_Diff
thf(fact_4446_set__diff__diff__left,axiom,
    ! [A: set_complex,B: set_complex,C2: set_complex] :
      ( ( minus_811609699411566653omplex @ ( minus_811609699411566653omplex @ A @ B ) @ C2 )
      = ( minus_811609699411566653omplex @ A @ ( sup_sup_set_complex @ B @ C2 ) ) ) ).

% set_diff_diff_left
thf(fact_4447_set__diff__diff__left,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( ( minus_minus_set_int @ ( minus_minus_set_int @ A @ B ) @ C2 )
      = ( minus_minus_set_int @ A @ ( sup_sup_set_int @ B @ C2 ) ) ) ).

% set_diff_diff_left
thf(fact_4448_set__diff__diff__left,axiom,
    ! [A: set_real,B: set_real,C2: set_real] :
      ( ( minus_minus_set_real @ ( minus_minus_set_real @ A @ B ) @ C2 )
      = ( minus_minus_set_real @ A @ ( sup_sup_set_real @ B @ C2 ) ) ) ).

% set_diff_diff_left
thf(fact_4449_set__diff__diff__left,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B ) @ C2 )
      = ( minus_minus_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).

% set_diff_diff_left
thf(fact_4450_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M4: nat,N2: nat] :
          ( Z
         != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% int_diff_cases
thf(fact_4451_psubset__imp__ex__mem,axiom,
    ! [A: set_real,B: set_real] :
      ( ( ord_less_set_real @ A @ B )
     => ? [B6: real] : ( member_real @ B6 @ ( minus_minus_set_real @ B @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_4452_psubset__imp__ex__mem,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_set_int @ A @ B )
     => ? [B6: int] : ( member_int @ B6 @ ( minus_minus_set_int @ B @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_4453_psubset__imp__ex__mem,axiom,
    ! [A: set_complex,B: set_complex] :
      ( ( ord_less_set_complex @ A @ B )
     => ? [B6: complex] : ( member_complex @ B6 @ ( minus_811609699411566653omplex @ B @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_4454_psubset__imp__ex__mem,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ? [B6: nat] : ( member_nat @ B6 @ ( minus_minus_set_nat @ B @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_4455_dbl__dec__def,axiom,
    ( neg_nu93272222329896523l_num1
    = ( ^ [X2: word_N3645301735248828278l_num1] : ( minus_4019991460397169231l_num1 @ ( plus_p361126936061061375l_num1 @ X2 @ X2 ) @ one_on7727431528512463931l_num1 ) ) ) ).

% dbl_dec_def
thf(fact_4456_dbl__dec__def,axiom,
    ( neg_nu6075765906172075777c_real
    = ( ^ [X2: real] : ( minus_minus_real @ ( plus_plus_real @ X2 @ X2 ) @ one_one_real ) ) ) ).

% dbl_dec_def
thf(fact_4457_dbl__dec__def,axiom,
    ( neg_nu3179335615603231917ec_rat
    = ( ^ [X2: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).

% dbl_dec_def
thf(fact_4458_dbl__dec__def,axiom,
    ( neg_nu3811975205180677377ec_int
    = ( ^ [X2: int] : ( minus_minus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).

% dbl_dec_def
thf(fact_4459_bit__eq__rec,axiom,
    ( ( ^ [Y3: code_integer,Z2: code_integer] : Y3 = Z2 )
    = ( ^ [A4: code_integer,B4: code_integer] :
          ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 )
            = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B4 ) )
          & ( ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = ( divide6298287555418463151nteger @ B4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_4460_bit__eq__rec,axiom,
    ( ( ^ [Y3: word_N3645301735248828278l_num1,Z2: word_N3645301735248828278l_num1] : Y3 = Z2 )
    = ( ^ [A4: word_N3645301735248828278l_num1,B4: word_N3645301735248828278l_num1] :
          ( ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A4 )
            = ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ B4 ) )
          & ( ( divide1791077408188789448l_num1 @ A4 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
            = ( divide1791077408188789448l_num1 @ B4 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_4461_bit__eq__rec,axiom,
    ( ( ^ [Y3: uint32,Z2: uint32] : Y3 = Z2 )
    = ( ^ [A4: uint32,B4: uint32] :
          ( ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A4 )
            = ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B4 ) )
          & ( ( divide_divide_uint32 @ A4 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
            = ( divide_divide_uint32 @ B4 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_4462_bit__eq__rec,axiom,
    ( ( ^ [Y3: nat,Z2: nat] : Y3 = Z2 )
    = ( ^ [A4: nat,B4: nat] :
          ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 )
            = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) )
          & ( ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( divide_divide_nat @ B4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_4463_bit__eq__rec,axiom,
    ( ( ^ [Y3: int,Z2: int] : Y3 = Z2 )
    = ( ^ [A4: int,B4: int] :
          ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 )
            = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) )
          & ( ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = ( divide_divide_int @ B4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_4464_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_4465_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_4466_power__diff,axiom,
    ! [A2: complex,N: nat,M: nat] :
      ( ( A2 != zero_zero_complex )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_complex @ A2 @ ( minus_minus_nat @ M @ N ) )
          = ( divide1717551699836669952omplex @ ( power_power_complex @ A2 @ M ) @ ( power_power_complex @ A2 @ N ) ) ) ) ) ).

% power_diff
thf(fact_4467_power__diff,axiom,
    ! [A2: code_integer,N: nat,M: nat] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_8256067586552552935nteger @ A2 @ ( minus_minus_nat @ M @ N ) )
          = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A2 @ M ) @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ) ) ).

% power_diff
thf(fact_4468_power__diff,axiom,
    ! [A2: real,N: nat,M: nat] :
      ( ( A2 != zero_zero_real )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_real @ A2 @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_real @ ( power_power_real @ A2 @ M ) @ ( power_power_real @ A2 @ N ) ) ) ) ) ).

% power_diff
thf(fact_4469_power__diff,axiom,
    ! [A2: rat,N: nat,M: nat] :
      ( ( A2 != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_rat @ A2 @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_rat @ ( power_power_rat @ A2 @ M ) @ ( power_power_rat @ A2 @ N ) ) ) ) ) ).

% power_diff
thf(fact_4470_power__diff,axiom,
    ! [A2: nat,N: nat,M: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_nat @ A2 @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) ) ) ) ) ).

% power_diff
thf(fact_4471_power__diff,axiom,
    ! [A2: int,N: nat,M: nat] :
      ( ( A2 != zero_zero_int )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_int @ A2 @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) ) ) ) ) ).

% power_diff
thf(fact_4472_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_4473_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_4474_subset__divisors__dvd,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( ord_le2219237028632753026uint32
        @ ( collect_uint32
          @ ^ [C5: uint32] : ( dvd_dvd_uint32 @ C5 @ A2 ) )
        @ ( collect_uint32
          @ ^ [C5: uint32] : ( dvd_dvd_uint32 @ C5 @ B2 ) ) )
      = ( dvd_dvd_uint32 @ A2 @ B2 ) ) ).

% subset_divisors_dvd
thf(fact_4475_subset__divisors__dvd,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( ord_le5203802739334966412l_num1
        @ ( collec7814023847061821259l_num1
          @ ^ [C5: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ C5 @ A2 ) )
        @ ( collec7814023847061821259l_num1
          @ ^ [C5: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ C5 @ B2 ) ) )
      = ( dvd_dv6812691276156420380l_num1 @ A2 @ B2 ) ) ).

% subset_divisors_dvd
thf(fact_4476_subset__divisors__dvd,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le7084787975880047091nteger
        @ ( collect_Code_integer
          @ ^ [C5: code_integer] : ( dvd_dvd_Code_integer @ C5 @ A2 ) )
        @ ( collect_Code_integer
          @ ^ [C5: code_integer] : ( dvd_dvd_Code_integer @ C5 @ B2 ) ) )
      = ( dvd_dvd_Code_integer @ A2 @ B2 ) ) ).

% subset_divisors_dvd
thf(fact_4477_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_4478_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_4479_floor__divide__of__nat__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) ).

% floor_divide_of_nat_eq
thf(fact_4480_floor__divide__of__nat__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) ).

% floor_divide_of_nat_eq
thf(fact_4481_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_4482_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_4483_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_4484_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_4485_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_4486_strict__subset__divisors__dvd,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( ord_less_set_uint32
        @ ( collect_uint32
          @ ^ [C5: uint32] : ( dvd_dvd_uint32 @ C5 @ A2 ) )
        @ ( collect_uint32
          @ ^ [C5: uint32] : ( dvd_dvd_uint32 @ C5 @ B2 ) ) )
      = ( ( dvd_dvd_uint32 @ A2 @ B2 )
        & ~ ( dvd_dvd_uint32 @ B2 @ A2 ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_4487_strict__subset__divisors__dvd,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( ord_le6726900395242856064l_num1
        @ ( collec7814023847061821259l_num1
          @ ^ [C5: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ C5 @ A2 ) )
        @ ( collec7814023847061821259l_num1
          @ ^ [C5: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ C5 @ B2 ) ) )
      = ( ( dvd_dv6812691276156420380l_num1 @ A2 @ B2 )
        & ~ ( dvd_dv6812691276156420380l_num1 @ B2 @ A2 ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_4488_strict__subset__divisors__dvd,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le1307284697595431911nteger
        @ ( collect_Code_integer
          @ ^ [C5: code_integer] : ( dvd_dvd_Code_integer @ C5 @ A2 ) )
        @ ( collect_Code_integer
          @ ^ [C5: code_integer] : ( dvd_dvd_Code_integer @ C5 @ B2 ) ) )
      = ( ( dvd_dvd_Code_integer @ A2 @ B2 )
        & ~ ( dvd_dvd_Code_integer @ B2 @ A2 ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_4489_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% even_mask_div_iff'
thf(fact_4490_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% even_mask_div_iff'
thf(fact_4491_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% even_mask_div_iff'
thf(fact_4492_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_4493_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_4494_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_4495_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_4496_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_4497_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_4498_ceiling__split,axiom,
    ! [P: int > $o,T: real] :
      ( ( P @ ( archim7802044766580827645g_real @ T ) )
      = ( ! [I4: int] :
            ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I4 ) @ one_one_real ) @ T )
              & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I4 ) ) )
           => ( P @ I4 ) ) ) ) ).

% ceiling_split
thf(fact_4499_ceiling__split,axiom,
    ! [P: int > $o,T: rat] :
      ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
      = ( ! [I4: int] :
            ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I4 ) @ one_one_rat ) @ T )
              & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I4 ) ) )
           => ( P @ I4 ) ) ) ) ).

% ceiling_split
thf(fact_4500_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_4501_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_4502_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_4503_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_4504_powr__non__neg,axiom,
    ! [A2: real,X: real] :
      ~ ( ord_less_real @ ( powr_real @ A2 @ X ) @ zero_zero_real ) ).

% powr_non_neg
thf(fact_4505_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_4506_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_4507_powr__ge__pzero,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X @ Y ) ) ).

% powr_ge_pzero
thf(fact_4508_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_4509_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_4510_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_4511_not__is__unit__0,axiom,
    ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).

% not_is_unit_0
thf(fact_4512_not__is__unit__0,axiom,
    ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).

% not_is_unit_0
thf(fact_4513_not__is__unit__0,axiom,
    ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).

% not_is_unit_0
thf(fact_4514_le__iff__diff__le__0,axiom,
    ( ord_less_eq_real
    = ( ^ [A4: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).

% le_iff_diff_le_0
thf(fact_4515_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A4: rat,B4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A4 @ B4 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_4516_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A4: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_4517_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_4518_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_4519_divide__right__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).

% divide_right_mono
thf(fact_4520_divide__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ A2 @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ) ).

% divide_right_mono
thf(fact_4521_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_4522_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_4523_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_4524_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_4525_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_4526_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_4527_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_4528_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_4529_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_4530_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_4531_divide__right__mono__neg,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ ( divide_divide_real @ A2 @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_4532_divide__right__mono__neg,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ ( divide_divide_rat @ A2 @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_4533_less__iff__diff__less__0,axiom,
    ( ord_less_real
    = ( ^ [A4: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).

% less_iff_diff_less_0
thf(fact_4534_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A4: rat,B4: rat] : ( ord_less_rat @ ( minus_minus_rat @ A4 @ B4 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_4535_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A4: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_4536_divide__strict__right__mono__neg,axiom,
    ! [B2: real,A2: real,C: real] :
      ( ( ord_less_real @ B2 @ A2 )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_4537_divide__strict__right__mono__neg,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ A2 @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_4538_divide__strict__right__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).

% divide_strict_right_mono
thf(fact_4539_divide__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( divide_divide_rat @ A2 @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ) ).

% divide_strict_right_mono
thf(fact_4540_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_4541_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_4542_divide__less__cancel,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ A2 @ B2 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_real @ B2 @ A2 ) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
thf(fact_4543_divide__less__cancel,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A2 @ C ) @ ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A2 @ B2 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B2 @ A2 ) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
thf(fact_4544_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_4545_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_4546_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_4547_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_4548_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_4549_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_4550_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_4551_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_4552_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_4553_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_4554_unat__minus__one,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( W != zero_z3563351764282998399l_num1 )
     => ( ( semiri7341220984566936280m1_nat @ ( minus_4019991460397169231l_num1 @ W @ one_on7727431528512463931l_num1 ) )
        = ( minus_minus_nat @ ( semiri7341220984566936280m1_nat @ W ) @ one_one_nat ) ) ) ).

% unat_minus_one
thf(fact_4555_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,N: real,J: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
     => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
       => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
         => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
           => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_4556_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,N: rat,J: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
     => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
       => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
         => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
           => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_4557_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
         => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_4558_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N: int,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
     => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
       => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
         => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
           => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_4559_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,N: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
     => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_4560_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,N: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
     => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_4561_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
     => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_4562_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,N: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
     => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_4563_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( ( ( minus_minus_nat @ B2 @ A2 )
            = C )
          = ( B2
            = ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_4564_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_4565_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_4566_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 )
        = ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_4567_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C )
        = ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_4568_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A2 )
        = ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_4569_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_4570_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_4571_le__add__diff,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 ) ) ) ).

% le_add_diff
thf(fact_4572_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_4573_le__diff__eq,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ A2 @ ( minus_minus_real @ C @ B2 ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B2 ) @ C ) ) ).

% le_diff_eq
thf(fact_4574_le__diff__eq,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( minus_minus_rat @ C @ B2 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C ) ) ).

% le_diff_eq
thf(fact_4575_le__diff__eq,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ ( minus_minus_int @ C @ B2 ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ C ) ) ).

% le_diff_eq
thf(fact_4576_diff__le__eq,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ ( minus_minus_real @ A2 @ B2 ) @ C )
      = ( ord_less_eq_real @ A2 @ ( plus_plus_real @ C @ B2 ) ) ) ).

% diff_le_eq
thf(fact_4577_diff__le__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C )
      = ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ C @ B2 ) ) ) ).

% diff_le_eq
thf(fact_4578_diff__le__eq,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
      = ( ord_less_eq_int @ A2 @ ( plus_plus_int @ C @ B2 ) ) ) ).

% diff_le_eq
thf(fact_4579_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_4580_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_4581_diff__less__eq,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_real @ ( minus_minus_real @ A2 @ B2 ) @ C )
      = ( ord_less_real @ A2 @ ( plus_plus_real @ C @ B2 ) ) ) ).

% diff_less_eq
thf(fact_4582_diff__less__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C )
      = ( ord_less_rat @ A2 @ ( plus_plus_rat @ C @ B2 ) ) ) ).

% diff_less_eq
thf(fact_4583_diff__less__eq,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
      = ( ord_less_int @ A2 @ ( plus_plus_int @ C @ B2 ) ) ) ).

% diff_less_eq
thf(fact_4584_less__diff__eq,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( ord_less_real @ A2 @ ( minus_minus_real @ C @ B2 ) )
      = ( ord_less_real @ ( plus_plus_real @ A2 @ B2 ) @ C ) ) ).

% less_diff_eq
thf(fact_4585_less__diff__eq,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ ( minus_minus_rat @ C @ B2 ) )
      = ( ord_less_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C ) ) ).

% less_diff_eq
thf(fact_4586_less__diff__eq,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_int @ A2 @ ( minus_minus_int @ C @ B2 ) )
      = ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ C ) ) ).

% less_diff_eq
thf(fact_4587_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_4588_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_4589_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_4590_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_4591_divide__numeral__1,axiom,
    ! [A2: real] :
      ( ( divide_divide_real @ A2 @ ( numeral_numeral_real @ one ) )
      = A2 ) ).

% divide_numeral_1
thf(fact_4592_divide__numeral__1,axiom,
    ! [A2: rat] :
      ( ( divide_divide_rat @ A2 @ ( numeral_numeral_rat @ one ) )
      = A2 ) ).

% divide_numeral_1
thf(fact_4593_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_4594_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_4595_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_4596_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_4597_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_4598_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_4599_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_4600_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_4601_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_4602_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_4603_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A4: complex,B4: complex] : ( plus_plus_complex @ A4 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4604_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_uint32
    = ( ^ [A4: uint32,B4: uint32] : ( plus_plus_uint32 @ A4 @ ( uminus_uminus_uint32 @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4605_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A4: real,B4: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4606_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A4: rat,B4: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4607_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_4608_diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A4: complex,B4: complex] : ( plus_plus_complex @ A4 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_4609_diff__conv__add__uminus,axiom,
    ( minus_minus_uint32
    = ( ^ [A4: uint32,B4: uint32] : ( plus_plus_uint32 @ A4 @ ( uminus_uminus_uint32 @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_4610_diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A4: real,B4: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_4611_diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A4: rat,B4: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_4612_diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_4613_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_4614_group__cancel_Osub2,axiom,
    ! [B: uint32,K: uint32,B2: uint32,A2: uint32] :
      ( ( B
        = ( plus_plus_uint32 @ K @ B2 ) )
     => ( ( minus_minus_uint32 @ A2 @ B )
        = ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ K ) @ ( minus_minus_uint32 @ A2 @ B2 ) ) ) ) ).

% group_cancel.sub2
thf(fact_4615_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_4616_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_4617_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_4618_of__int__floor__le,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X ) ).

% of_int_floor_le
thf(fact_4619_of__int__floor__le,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X ) ).

% of_int_floor_le
thf(fact_4620_power__one__over,axiom,
    ! [A2: complex,N: nat] :
      ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A2 ) @ N )
      = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A2 @ N ) ) ) ).

% power_one_over
thf(fact_4621_power__one__over,axiom,
    ! [A2: real,N: nat] :
      ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A2 ) @ N )
      = ( divide_divide_real @ one_one_real @ ( power_power_real @ A2 @ N ) ) ) ).

% power_one_over
thf(fact_4622_power__one__over,axiom,
    ! [A2: rat,N: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) @ N )
      = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A2 @ N ) ) ) ).

% power_one_over
thf(fact_4623_dvd__power__le,axiom,
    ! [X: uint32,Y: uint32,N: nat,M: nat] :
      ( ( dvd_dvd_uint32 @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_uint32 @ ( power_power_uint32 @ X @ N ) @ ( power_power_uint32 @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4624_dvd__power__le,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1,N: nat,M: nat] :
      ( ( dvd_dv6812691276156420380l_num1 @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dv6812691276156420380l_num1 @ ( power_2184487114949457152l_num1 @ X @ N ) @ ( power_2184487114949457152l_num1 @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4625_dvd__power__le,axiom,
    ! [X: nat,Y: nat,N: nat,M: nat] :
      ( ( dvd_dvd_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4626_dvd__power__le,axiom,
    ! [X: real,Y: real,N: nat,M: nat] :
      ( ( dvd_dvd_real @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4627_dvd__power__le,axiom,
    ! [X: int,Y: int,N: nat,M: nat] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4628_dvd__power__le,axiom,
    ! [X: complex,Y: complex,N: nat,M: nat] :
      ( ( dvd_dvd_complex @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4629_dvd__power__le,axiom,
    ! [X: code_integer,Y: code_integer,N: nat,M: nat] :
      ( ( dvd_dvd_Code_integer @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4630_power__le__dvd,axiom,
    ! [A2: uint32,N: nat,B2: uint32,M: nat] :
      ( ( dvd_dvd_uint32 @ ( power_power_uint32 @ A2 @ N ) @ B2 )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_uint32 @ ( power_power_uint32 @ A2 @ M ) @ B2 ) ) ) ).

% power_le_dvd
thf(fact_4631_power__le__dvd,axiom,
    ! [A2: word_N3645301735248828278l_num1,N: nat,B2: word_N3645301735248828278l_num1,M: nat] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( power_2184487114949457152l_num1 @ A2 @ N ) @ B2 )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dv6812691276156420380l_num1 @ ( power_2184487114949457152l_num1 @ A2 @ M ) @ B2 ) ) ) ).

% power_le_dvd
thf(fact_4632_power__le__dvd,axiom,
    ! [A2: nat,N: nat,B2: nat,M: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A2 @ N ) @ B2 )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_nat @ ( power_power_nat @ A2 @ M ) @ B2 ) ) ) ).

% power_le_dvd
thf(fact_4633_power__le__dvd,axiom,
    ! [A2: real,N: nat,B2: real,M: nat] :
      ( ( dvd_dvd_real @ ( power_power_real @ A2 @ N ) @ B2 )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_real @ ( power_power_real @ A2 @ M ) @ B2 ) ) ) ).

% power_le_dvd
thf(fact_4634_power__le__dvd,axiom,
    ! [A2: int,N: nat,B2: int,M: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A2 @ N ) @ B2 )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_int @ ( power_power_int @ A2 @ M ) @ B2 ) ) ) ).

% power_le_dvd
thf(fact_4635_power__le__dvd,axiom,
    ! [A2: complex,N: nat,B2: complex,M: nat] :
      ( ( dvd_dvd_complex @ ( power_power_complex @ A2 @ N ) @ B2 )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_complex @ ( power_power_complex @ A2 @ M ) @ B2 ) ) ) ).

% power_le_dvd
thf(fact_4636_power__le__dvd,axiom,
    ! [A2: code_integer,N: nat,B2: code_integer,M: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A2 @ N ) @ B2 )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A2 @ M ) @ B2 ) ) ) ).

% power_le_dvd
thf(fact_4637_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A2: uint32] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_uint32 @ ( power_power_uint32 @ A2 @ M ) @ ( power_power_uint32 @ A2 @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_4638_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A2: word_N3645301735248828278l_num1] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dv6812691276156420380l_num1 @ ( power_2184487114949457152l_num1 @ A2 @ M ) @ ( power_2184487114949457152l_num1 @ A2 @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_4639_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A2: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_4640_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A2: real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_real @ ( power_power_real @ A2 @ M ) @ ( power_power_real @ A2 @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_4641_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A2: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_4642_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A2: complex] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_complex @ ( power_power_complex @ A2 @ M ) @ ( power_power_complex @ A2 @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_4643_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A2: code_integer] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A2 @ M ) @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_4644_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_4645_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_4646_nat__dvd__not__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).

% nat_dvd_not_less
thf(fact_4647_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
          = zero_z3403309356797280102nteger )
        | ( ord_less_eq_nat @ M @ N ) ) ) ).

% even_mask_div_iff
thf(fact_4648_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( divide1791077408188789448l_num1 @ ( minus_4019991460397169231l_num1 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ M ) @ one_on7727431528512463931l_num1 ) @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N )
          = zero_z3563351764282998399l_num1 )
        | ( ord_less_eq_nat @ M @ N ) ) ) ).

% even_mask_div_iff
thf(fact_4649_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( divide_divide_uint32 @ ( minus_minus_uint32 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ M ) @ one_one_uint32 ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N )
          = zero_zero_uint32 )
        | ( ord_less_eq_nat @ M @ N ) ) ) ).

% even_mask_div_iff
thf(fact_4650_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          = zero_zero_nat )
        | ( ord_less_eq_nat @ M @ N ) ) ) ).

% even_mask_div_iff
thf(fact_4651_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
          = zero_zero_int )
        | ( ord_less_eq_nat @ M @ N ) ) ) ).

% even_mask_div_iff
thf(fact_4652_word__numeral__alt,axiom,
    ( numera7442385471795722001l_num1
    = ( ^ [B4: num] : ( ring_17408606157368542149l_num1 @ ( numeral_numeral_int @ B4 ) ) ) ) ).

% word_numeral_alt
thf(fact_4653_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_4654_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_4655_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_4656_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_4657_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_4658_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_4659_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_4660_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_4661_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_4662_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_4663_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_4664_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_4665_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_4666_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_4667_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_4668_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_4669_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_4670_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_4671_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_4672_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_4673_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_4674_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_4675_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_4676_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_4677_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_4678_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_4679_set__minus__singleton__eq,axiom,
    ! [X: complex,X5: set_complex] :
      ( ~ ( member_complex @ X @ X5 )
     => ( ( minus_811609699411566653omplex @ X5 @ ( insert_complex @ X @ bot_bot_set_complex ) )
        = X5 ) ) ).

% set_minus_singleton_eq
thf(fact_4680_set__minus__singleton__eq,axiom,
    ! [X: real,X5: set_real] :
      ( ~ ( member_real @ X @ X5 )
     => ( ( minus_minus_set_real @ X5 @ ( insert_real @ X @ bot_bot_set_real ) )
        = X5 ) ) ).

% set_minus_singleton_eq
thf(fact_4681_set__minus__singleton__eq,axiom,
    ! [X: $o,X5: set_o] :
      ( ~ ( member_o @ X @ X5 )
     => ( ( minus_minus_set_o @ X5 @ ( insert_o @ X @ bot_bot_set_o ) )
        = X5 ) ) ).

% set_minus_singleton_eq
thf(fact_4682_set__minus__singleton__eq,axiom,
    ! [X: int,X5: set_int] :
      ( ~ ( member_int @ X @ X5 )
     => ( ( minus_minus_set_int @ X5 @ ( insert_int @ X @ bot_bot_set_int ) )
        = X5 ) ) ).

% set_minus_singleton_eq
thf(fact_4683_set__minus__singleton__eq,axiom,
    ! [X: nat,X5: set_nat] :
      ( ~ ( member_nat @ X @ X5 )
     => ( ( minus_minus_set_nat @ X5 @ ( insert_nat @ X @ bot_bot_set_nat ) )
        = X5 ) ) ).

% set_minus_singleton_eq
thf(fact_4684_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).

% diff_less
thf(fact_4685_subset__Diff__insert,axiom,
    ! [A: set_o,B: set_o,X: $o,C2: set_o] :
      ( ( ord_less_eq_set_o @ A @ ( minus_minus_set_o @ B @ ( insert_o @ X @ C2 ) ) )
      = ( ( ord_less_eq_set_o @ A @ ( minus_minus_set_o @ B @ C2 ) )
        & ~ ( member_o @ X @ A ) ) ) ).

% subset_Diff_insert
thf(fact_4686_subset__Diff__insert,axiom,
    ! [A: set_real,B: set_real,X: real,C2: set_real] :
      ( ( ord_less_eq_set_real @ A @ ( minus_minus_set_real @ B @ ( insert_real @ X @ C2 ) ) )
      = ( ( ord_less_eq_set_real @ A @ ( minus_minus_set_real @ B @ C2 ) )
        & ~ ( member_real @ X @ A ) ) ) ).

% subset_Diff_insert
thf(fact_4687_subset__Diff__insert,axiom,
    ! [A: set_int,B: set_int,X: int,C2: set_int] :
      ( ( ord_less_eq_set_int @ A @ ( minus_minus_set_int @ B @ ( insert_int @ X @ C2 ) ) )
      = ( ( ord_less_eq_set_int @ A @ ( minus_minus_set_int @ B @ C2 ) )
        & ~ ( member_int @ X @ A ) ) ) ).

% subset_Diff_insert
thf(fact_4688_subset__Diff__insert,axiom,
    ! [A: set_complex,B: set_complex,X: complex,C2: set_complex] :
      ( ( ord_le211207098394363844omplex @ A @ ( minus_811609699411566653omplex @ B @ ( insert_complex @ X @ C2 ) ) )
      = ( ( ord_le211207098394363844omplex @ A @ ( minus_811609699411566653omplex @ B @ C2 ) )
        & ~ ( member_complex @ X @ A ) ) ) ).

% subset_Diff_insert
thf(fact_4689_subset__Diff__insert,axiom,
    ! [A: set_nat,B: set_nat,X: nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ ( insert_nat @ X @ C2 ) ) )
      = ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ C2 ) )
        & ~ ( member_nat @ X @ A ) ) ) ).

% subset_Diff_insert
thf(fact_4690_diff__add__0,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_4691_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).

% less_diff_conv
thf(fact_4692_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ ( ord_less_nat @ M @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_4693_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_nat @ M @ N ) ) ) ) ).

% less_diff_iff
thf(fact_4694_diff__less__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C @ A2 )
       => ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).

% diff_less_mono
thf(fact_4695_uint__sub__if__size,axiom,
    ! [Y: word_N3645301735248828278l_num1,X: word_N3645301735248828278l_num1] :
      ( ( ( ord_less_eq_int @ ( semiri7338730514057886004m1_int @ Y ) @ ( semiri7338730514057886004m1_int @ X ) )
       => ( ( semiri7338730514057886004m1_int @ ( minus_4019991460397169231l_num1 @ X @ Y ) )
          = ( minus_minus_int @ ( semiri7338730514057886004m1_int @ X ) @ ( semiri7338730514057886004m1_int @ Y ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( semiri7338730514057886004m1_int @ Y ) @ ( semiri7338730514057886004m1_int @ X ) )
       => ( ( semiri7338730514057886004m1_int @ ( minus_4019991460397169231l_num1 @ X @ Y ) )
          = ( plus_plus_int @ ( minus_minus_int @ ( semiri7338730514057886004m1_int @ X ) @ ( semiri7338730514057886004m1_int @ Y ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( size_s8261804613246490634l_num1 @ X ) ) ) ) ) ) ).

% uint_sub_if_size
thf(fact_4696_wi__hom__add,axiom,
    ! [A2: int,B2: int] :
      ( ( plus_p361126936061061375l_num1 @ ( ring_17408606157368542149l_num1 @ A2 ) @ ( ring_17408606157368542149l_num1 @ B2 ) )
      = ( ring_17408606157368542149l_num1 @ ( plus_plus_int @ A2 @ B2 ) ) ) ).

% wi_hom_add
thf(fact_4697_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
      = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).

% le_diff_conv
thf(fact_4698_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_4699_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
        = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_4700_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
        = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_4701_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( minus_minus_nat @ J @ I )
          = K )
        = ( J
          = ( plus_plus_nat @ K @ I ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_4702_Diff__partition,axiom,
    ! [A: set_complex,B: set_complex] :
      ( ( ord_le211207098394363844omplex @ A @ B )
     => ( ( sup_sup_set_complex @ A @ ( minus_811609699411566653omplex @ B @ A ) )
        = B ) ) ).

% Diff_partition
thf(fact_4703_Diff__partition,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( sup_sup_set_int @ A @ ( minus_minus_set_int @ B @ A ) )
        = B ) ) ).

% Diff_partition
thf(fact_4704_Diff__partition,axiom,
    ! [A: set_real,B: set_real] :
      ( ( ord_less_eq_set_real @ A @ B )
     => ( ( sup_sup_set_real @ A @ ( minus_minus_set_real @ B @ A ) )
        = B ) ) ).

% Diff_partition
thf(fact_4705_Diff__partition,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
        = B ) ) ).

% Diff_partition
thf(fact_4706_Diff__subset__conv,axiom,
    ! [A: set_complex,B: set_complex,C2: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( minus_811609699411566653omplex @ A @ B ) @ C2 )
      = ( ord_le211207098394363844omplex @ A @ ( sup_sup_set_complex @ B @ C2 ) ) ) ).

% Diff_subset_conv
thf(fact_4707_Diff__subset__conv,axiom,
    ! [A: set_int,B: set_int,C2: set_int] :
      ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A @ B ) @ C2 )
      = ( ord_less_eq_set_int @ A @ ( sup_sup_set_int @ B @ C2 ) ) ) ).

% Diff_subset_conv
thf(fact_4708_Diff__subset__conv,axiom,
    ! [A: set_real,B: set_real,C2: set_real] :
      ( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A @ B ) @ C2 )
      = ( ord_less_eq_set_real @ A @ ( sup_sup_set_real @ B @ C2 ) ) ) ).

% Diff_subset_conv
thf(fact_4709_Diff__subset__conv,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ C2 )
      = ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).

% Diff_subset_conv
thf(fact_4710_minus__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( minus_minus_int @ zero_zero_int @ L )
      = ( uminus_uminus_int @ L ) ) ).

% minus_int_code(2)
thf(fact_4711_wi__hom__neg,axiom,
    ! [A2: int] :
      ( ( uminus8244633308260627903l_num1 @ ( ring_17408606157368542149l_num1 @ A2 ) )
      = ( ring_17408606157368542149l_num1 @ ( uminus_uminus_int @ A2 ) ) ) ).

% wi_hom_neg
thf(fact_4712_int__le__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_eq_int @ I @ K )
     => ( ( P @ K )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ I3 @ K )
             => ( ( P @ I3 )
               => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_le_induct
thf(fact_4713_int__less__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_int @ I @ K )
     => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
       => ( ! [I3: int] :
              ( ( ord_less_int @ I3 @ K )
             => ( ( P @ I3 )
               => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_less_induct
thf(fact_4714_le__step__down__word,axiom,
    ! [I: word_N3645301735248828278l_num1,N: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ I @ N )
     => ( ( I != N )
       => ( ord_le3335648743751981014l_num1 @ I @ ( minus_4019991460397169231l_num1 @ N @ one_on7727431528512463931l_num1 ) ) ) ) ).

% le_step_down_word
thf(fact_4715_le__step__down__word__2,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ X @ Y )
     => ( ( X != Y )
       => ( ord_le3335648743751981014l_num1 @ X @ ( minus_4019991460397169231l_num1 @ Y @ one_on7727431528512463931l_num1 ) ) ) ) ).

% le_step_down_word_2
thf(fact_4716_ucast__eq,axiom,
    ( semiri1312839663145358974l_num1
    = ( ^ [W2: word_N3645301735248828278l_num1] : ( ring_17408606157368542149l_num1 @ ( semiri7338730514057886004m1_int @ W2 ) ) ) ) ).

% ucast_eq
thf(fact_4717_word__of__int__power__hom,axiom,
    ! [A2: int,N: nat] :
      ( ( power_2184487114949457152l_num1 @ ( ring_17408606157368542149l_num1 @ A2 ) @ N )
      = ( ring_17408606157368542149l_num1 @ ( power_power_int @ A2 @ N ) ) ) ).

% word_of_int_power_hom
thf(fact_4718_word__less__cases,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ X @ Y )
     => ( ( X
          = ( minus_4019991460397169231l_num1 @ Y @ one_on7727431528512463931l_num1 ) )
        | ( ord_le750835935415966154l_num1 @ X @ ( minus_4019991460397169231l_num1 @ Y @ one_on7727431528512463931l_num1 ) ) ) ) ).

% word_less_cases
thf(fact_4719_minus__real__def,axiom,
    ( minus_minus_real
    = ( ^ [X2: real,Y2: real] : ( plus_plus_real @ X2 @ ( uminus_uminus_real @ Y2 ) ) ) ) ).

% minus_real_def
thf(fact_4720_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_4721_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_4722_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_4723_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_4724_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_4725_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_4726_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_4727_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_4728_even__numeral,axiom,
    ! [N: num] : ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( numera9087168376688890119uint32 @ ( bit0 @ N ) ) ) ).

% even_numeral
thf(fact_4729_even__numeral,axiom,
    ! [N: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ).

% even_numeral
thf(fact_4730_even__numeral,axiom,
    ! [N: num] : ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( numera7442385471795722001l_num1 @ ( bit0 @ N ) ) ) ).

% even_numeral
thf(fact_4731_even__numeral,axiom,
    ! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).

% even_numeral
thf(fact_4732_even__numeral,axiom,
    ! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).

% even_numeral
thf(fact_4733_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_4734_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_4735_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_4736_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_4737_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_4738_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_4739_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_4740_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_4741_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_4742_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_4743_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_4744_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_4745_divide__le__cancel,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A2 @ B2 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).

% divide_le_cancel
thf(fact_4746_divide__le__cancel,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A2 @ C ) @ ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A2 @ B2 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).

% divide_le_cancel
thf(fact_4747_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_4748_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_4749_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_4750_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_4751_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_4752_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_4753_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_4754_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_4755_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_4756_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_4757_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_4758_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_4759_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_4760_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_4761_is__unit__power__iff,axiom,
    ! [A2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A2 @ N ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A2 @ one_one_nat )
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_4762_is__unit__power__iff,axiom,
    ! [A2: int,N: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A2 @ N ) @ one_one_int )
      = ( ( dvd_dvd_int @ A2 @ one_one_int )
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_4763_is__unit__power__iff,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A2 @ N ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_4764_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_4765_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_4766_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_4767_le__floor__iff,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_eq_int @ Z @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).

% le_floor_iff
thf(fact_4768_le__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_eq_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ).

% le_floor_iff
thf(fact_4769_floor__less__iff,axiom,
    ! [X: real,Z: int] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X ) @ Z )
      = ( ord_less_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).

% floor_less_iff
thf(fact_4770_floor__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
      = ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ).

% floor_less_iff
thf(fact_4771_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_4772_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_4773_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_4774_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_4775_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_4776_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_4777_floor__add__int,axiom,
    ! [X: real,Z: int] :
      ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ Z )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ) ).

% floor_add_int
thf(fact_4778_floor__add__int,axiom,
    ! [X: rat,Z: int] :
      ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ) ).

% floor_add_int
thf(fact_4779_int__add__floor,axiom,
    ! [Z: int,X: real] :
      ( ( plus_plus_int @ Z @ ( archim6058952711729229775r_real @ X ) )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ) ).

% int_add_floor
thf(fact_4780_int__add__floor,axiom,
    ! [Z: int,X: rat] :
      ( ( plus_plus_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ) ).

% int_add_floor
thf(fact_4781_dvd__imp__le,axiom,
    ! [K: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat @ K @ N ) ) ) ).

% dvd_imp_le
thf(fact_4782_floor__power,axiom,
    ! [X: real,N: nat] :
      ( ( X
        = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) )
     => ( ( archim6058952711729229775r_real @ ( power_power_real @ X @ N ) )
        = ( power_power_int @ ( archim6058952711729229775r_real @ X ) @ N ) ) ) ).

% floor_power
thf(fact_4783_floor__power,axiom,
    ! [X: rat,N: nat] :
      ( ( X
        = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) )
     => ( ( archim3151403230148437115or_rat @ ( power_power_rat @ X @ N ) )
        = ( power_power_int @ ( archim3151403230148437115or_rat @ X ) @ N ) ) ) ).

% floor_power
thf(fact_4784_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_4785_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_4786_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_4787_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_4788_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_4789_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_4790_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_4791_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_4792_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_4793_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 ) )
        & ! [D4: nat] :
            ( ( A2
              = ( plus_plus_nat @ B2 @ D4 ) )
           => ( P @ D4 ) ) ) ) ).

% nat_diff_split
thf(fact_4794_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 ) )
            | ? [D4: nat] :
                ( ( A2
                  = ( plus_plus_nat @ B2 @ D4 ) )
                & ~ ( P @ D4 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_4795_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).

% less_diff_conv2
thf(fact_4796_word__add__def,axiom,
    ( plus_p361126936061061375l_num1
    = ( ^ [A4: word_N3645301735248828278l_num1,B4: word_N3645301735248828278l_num1] : ( ring_17408606157368542149l_num1 @ ( plus_plus_int @ ( semiri7338730514057886004m1_int @ A4 ) @ ( semiri7338730514057886004m1_int @ B4 ) ) ) ) ) ).

% word_add_def
thf(fact_4797_word__minus__def,axiom,
    ( uminus8244633308260627903l_num1
    = ( ^ [A4: word_N3645301735248828278l_num1] : ( ring_17408606157368542149l_num1 @ ( uminus_uminus_int @ ( semiri7338730514057886004m1_int @ A4 ) ) ) ) ) ).

% word_minus_def
thf(fact_4798_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_4799_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_4800_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_4801_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_4802_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_4803_less__1__helper,axiom,
    ! [N: int,M: int] :
      ( ( ord_less_eq_int @ N @ M )
     => ( ord_less_int @ ( minus_minus_int @ N @ one_one_int ) @ M ) ) ).

% less_1_helper
thf(fact_4804_int__induct,axiom,
    ! [P: int > $o,K: int,I: int] :
      ( ( P @ K )
     => ( ! [I3: int] :
            ( ( ord_less_eq_int @ K @ I3 )
           => ( ( P @ I3 )
             => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ I3 @ K )
             => ( ( P @ I3 )
               => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_induct
thf(fact_4805_no__ulen__sub,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ ( minus_4019991460397169231l_num1 @ X @ Y ) @ X )
      = ( ord_less_eq_int @ ( semiri7338730514057886004m1_int @ Y ) @ ( semiri7338730514057886004m1_int @ X ) ) ) ).

% no_ulen_sub
thf(fact_4806_word__arith__power__alt,axiom,
    ( power_2184487114949457152l_num1
    = ( ^ [A4: word_N3645301735248828278l_num1,N4: nat] : ( ring_17408606157368542149l_num1 @ ( power_power_int @ ( semiri7338730514057886004m1_int @ A4 ) @ N4 ) ) ) ) ).

% word_arith_power_alt
thf(fact_4807_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_4808_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_4809_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_4810_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_4811_word__le__sub1,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( X != zero_z3563351764282998399l_num1 )
     => ( ( ord_le3335648743751981014l_num1 @ one_on7727431528512463931l_num1 @ X )
        = ( ord_le3335648743751981014l_num1 @ zero_z3563351764282998399l_num1 @ ( minus_4019991460397169231l_num1 @ X @ one_on7727431528512463931l_num1 ) ) ) ) ).

% word_le_sub1
thf(fact_4812_word__sub__1__le,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( X != zero_z3563351764282998399l_num1 )
     => ( ord_le3335648743751981014l_num1 @ ( minus_4019991460397169231l_num1 @ X @ one_on7727431528512463931l_num1 ) @ X ) ) ).

% word_sub_1_le
thf(fact_4813_word__must__wrap,axiom,
    ! [X: word_N3645301735248828278l_num1,N: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ X @ ( minus_4019991460397169231l_num1 @ N @ one_on7727431528512463931l_num1 ) )
     => ( ( ord_le3335648743751981014l_num1 @ N @ X )
       => ( N = zero_z3563351764282998399l_num1 ) ) ) ).

% word_must_wrap
thf(fact_4814_word__minus__one__le__leq,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ ( minus_4019991460397169231l_num1 @ X @ one_on7727431528512463931l_num1 ) @ Y )
     => ( ord_le3335648743751981014l_num1 @ X @ Y ) ) ).

% word_minus_one_le_leq
thf(fact_4815_word__le__minus__one__leq,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ X @ Y )
     => ( ord_le3335648743751981014l_num1 @ X @ ( minus_4019991460397169231l_num1 @ Y @ one_on7727431528512463931l_num1 ) ) ) ).

% word_le_minus_one_leq
thf(fact_4816_plus__minus__not__NULL__ab,axiom,
    ! [X: word_N3645301735248828278l_num1,Ab: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ X @ ( minus_4019991460397169231l_num1 @ Ab @ C ) )
     => ( ( ord_le3335648743751981014l_num1 @ C @ Ab )
       => ( ( C != zero_z3563351764282998399l_num1 )
         => ( ( plus_p361126936061061375l_num1 @ X @ C )
           != zero_z3563351764282998399l_num1 ) ) ) ) ).

% plus_minus_not_NULL_ab
thf(fact_4817_gt0__iff__gem1,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ X )
      = ( ord_le750835935415966154l_num1 @ ( minus_4019991460397169231l_num1 @ X @ one_on7727431528512463931l_num1 ) @ X ) ) ).

% gt0_iff_gem1
thf(fact_4818_word__less__sub1,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( X != zero_z3563351764282998399l_num1 )
     => ( ( ord_le750835935415966154l_num1 @ one_on7727431528512463931l_num1 @ X )
        = ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ ( minus_4019991460397169231l_num1 @ X @ one_on7727431528512463931l_num1 ) ) ) ) ).

% word_less_sub1
thf(fact_4819_sub__wrap,axiom,
    ! [X: word_N3645301735248828278l_num1,Z: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ X @ ( minus_4019991460397169231l_num1 @ X @ Z ) )
      = ( ( Z = zero_z3563351764282998399l_num1 )
        | ( ord_le750835935415966154l_num1 @ X @ Z ) ) ) ).

% sub_wrap
thf(fact_4820_word__diff__less,axiom,
    ! [N: word_N3645301735248828278l_num1,M: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ N )
     => ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ M )
       => ( ( ord_le3335648743751981014l_num1 @ N @ M )
         => ( ord_le750835935415966154l_num1 @ ( minus_4019991460397169231l_num1 @ M @ N ) @ M ) ) ) ) ).

% word_diff_less
thf(fact_4821_uint__plus__if__size,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( ( ord_less_int @ ( plus_plus_int @ ( semiri7338730514057886004m1_int @ X ) @ ( semiri7338730514057886004m1_int @ Y ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( size_s8261804613246490634l_num1 @ X ) ) )
       => ( ( semiri7338730514057886004m1_int @ ( plus_p361126936061061375l_num1 @ X @ Y ) )
          = ( plus_plus_int @ ( semiri7338730514057886004m1_int @ X ) @ ( semiri7338730514057886004m1_int @ Y ) ) ) )
      & ( ~ ( ord_less_int @ ( plus_plus_int @ ( semiri7338730514057886004m1_int @ X ) @ ( semiri7338730514057886004m1_int @ Y ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( size_s8261804613246490634l_num1 @ X ) ) )
       => ( ( semiri7338730514057886004m1_int @ ( plus_p361126936061061375l_num1 @ X @ Y ) )
          = ( minus_minus_int @ ( plus_plus_int @ ( semiri7338730514057886004m1_int @ X ) @ ( semiri7338730514057886004m1_int @ Y ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( size_s8261804613246490634l_num1 @ X ) ) ) ) ) ) ).

% uint_plus_if_size
thf(fact_4822_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_4823_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_4824_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_4825_powr__realpow,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N ) )
        = ( power_power_real @ X @ N ) ) ) ).

% powr_realpow
thf(fact_4826_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_4827_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_4828_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_4829_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_4830_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_4831_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_4832_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_4833_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_4834_floor__exists,axiom,
    ! [X: real] :
    ? [Z3: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X )
      & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_4835_floor__exists,axiom,
    ! [X: rat] :
    ? [Z3: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_4836_floor__exists1,axiom,
    ! [X: real] :
    ? [X3: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X3 ) @ X )
      & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X3 @ one_one_int ) ) )
      & ! [Y5: int] :
          ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y5 ) @ X )
            & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
         => ( Y5 = X3 ) ) ) ).

% floor_exists1
thf(fact_4837_floor__exists1,axiom,
    ! [X: rat] :
    ? [X3: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X3 ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X3 @ one_one_int ) ) )
      & ! [Y5: int] :
          ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y5 ) @ X )
            & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
         => ( Y5 = X3 ) ) ) ).

% floor_exists1
thf(fact_4838_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_17408606157368542149l_num1 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_4839_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_4840_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_uint32 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_4841_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_4842_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_4843_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_4844_even__zero,axiom,
    dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ zero_zero_uint32 ).

% even_zero
thf(fact_4845_even__zero,axiom,
    dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).

% even_zero
thf(fact_4846_even__zero,axiom,
    dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ zero_z3563351764282998399l_num1 ).

% even_zero
thf(fact_4847_even__zero,axiom,
    dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).

% even_zero
thf(fact_4848_even__zero,axiom,
    dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).

% even_zero
thf(fact_4849_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_4850_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_4851_odd__even__add,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A2 )
     => ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B2 )
       => ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_uint32 @ A2 @ B2 ) ) ) ) ).

% odd_even_add
thf(fact_4852_odd__even__add,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
     => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 )
       => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) ) ) ) ).

% odd_even_add
thf(fact_4853_odd__even__add,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A2 )
     => ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ B2 )
       => ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( plus_p361126936061061375l_num1 @ A2 @ B2 ) ) ) ) ).

% odd_even_add
thf(fact_4854_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_4855_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_4856_odd__one,axiom,
    ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ one_one_uint32 ) ).

% odd_one
thf(fact_4857_odd__one,axiom,
    ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).

% odd_one
thf(fact_4858_odd__one,axiom,
    ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ one_on7727431528512463931l_num1 ) ).

% odd_one
thf(fact_4859_odd__one,axiom,
    ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).

% odd_one
thf(fact_4860_odd__one,axiom,
    ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).

% odd_one
thf(fact_4861_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_4862_even__minus,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( uminus8244633308260627903l_num1 @ A2 ) )
      = ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A2 ) ) ).

% even_minus
thf(fact_4863_even__minus,axiom,
    ! [A2: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( uminus_uminus_uint32 @ A2 ) )
      = ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A2 ) ) ).

% even_minus
thf(fact_4864_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_4865_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_4866_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_4867_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_4868_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_4869_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_4870_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_4871_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_4872_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ N ) @ ( ring_18347121197199848620nteger @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_4873_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_4874_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_4875_dvd__power__iff,axiom,
    ! [X: nat,M: nat,N: nat] :
      ( ( X != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N ) )
        = ( ( dvd_dvd_nat @ X @ one_one_nat )
          | ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_4876_dvd__power__iff,axiom,
    ! [X: int,M: nat,N: nat] :
      ( ( X != zero_zero_int )
     => ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N ) )
        = ( ( dvd_dvd_int @ X @ one_one_int )
          | ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_4877_dvd__power__iff,axiom,
    ! [X: code_integer,M: nat,N: nat] :
      ( ( X != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ N ) )
        = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
          | ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_4878_dvd__power,axiom,
    ! [N: nat,X: uint32] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_uint32 ) )
     => ( dvd_dvd_uint32 @ X @ ( power_power_uint32 @ X @ N ) ) ) ).

% dvd_power
thf(fact_4879_dvd__power,axiom,
    ! [N: nat,X: word_N3645301735248828278l_num1] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_on7727431528512463931l_num1 ) )
     => ( dvd_dv6812691276156420380l_num1 @ X @ ( power_2184487114949457152l_num1 @ X @ N ) ) ) ).

% dvd_power
thf(fact_4880_dvd__power,axiom,
    ! [N: nat,X: rat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_rat ) )
     => ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N ) ) ) ).

% dvd_power
thf(fact_4881_dvd__power,axiom,
    ! [N: nat,X: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_nat ) )
     => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).

% dvd_power
thf(fact_4882_dvd__power,axiom,
    ! [N: nat,X: real] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_real ) )
     => ( dvd_dvd_real @ X @ ( power_power_real @ X @ N ) ) ) ).

% dvd_power
thf(fact_4883_dvd__power,axiom,
    ! [N: nat,X: int] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_int ) )
     => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).

% dvd_power
thf(fact_4884_dvd__power,axiom,
    ! [N: nat,X: complex] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_complex ) )
     => ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N ) ) ) ).

% dvd_power
thf(fact_4885_dvd__power,axiom,
    ! [N: nat,X: code_integer] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_Code_integer ) )
     => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N ) ) ) ).

% dvd_power
thf(fact_4886_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_4887_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_4888_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_4889_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_4890_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_4891_int__le__real__less,axiom,
    ( ord_less_eq_int
    = ( ^ [N4: int,M3: int] : ( ord_less_real @ ( ring_1_of_int_real @ N4 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M3 ) @ one_one_real ) ) ) ) ).

% int_le_real_less
thf(fact_4892_int__less__real__le,axiom,
    ( ord_less_int
    = ( ^ [N4: int,M3: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N4 ) @ one_one_real ) @ ( ring_1_of_int_real @ M3 ) ) ) ) ).

% int_less_real_le
thf(fact_4893_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
     => ( ( ord_less_nat @ one_one_nat @ I )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_dvd_imp_le
thf(fact_4894_word__neg__numeral__alt,axiom,
    ! [B2: num] :
      ( ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ B2 ) )
      = ( ring_17408606157368542149l_num1 @ ( uminus_uminus_int @ ( numeral_numeral_int @ B2 ) ) ) ) ).

% word_neg_numeral_alt
thf(fact_4895_ceiling__altdef,axiom,
    ( archim7802044766580827645g_real
    = ( ^ [X2: real] :
          ( if_int
          @ ( X2
            = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X2 ) ) )
          @ ( archim6058952711729229775r_real @ X2 )
          @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X2 ) @ one_one_int ) ) ) ) ).

% ceiling_altdef
thf(fact_4896_ceiling__altdef,axiom,
    ( archim2889992004027027881ng_rat
    = ( ^ [X2: rat] :
          ( if_int
          @ ( X2
            = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X2 ) ) )
          @ ( archim3151403230148437115or_rat @ X2 )
          @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X2 ) @ one_one_int ) ) ) ) ).

% ceiling_altdef
thf(fact_4897_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_4898_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_4899_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_4900_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_4901_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_4902_measure__unat,axiom,
    ! [P4: word_N3645301735248828278l_num1] :
      ( ( P4 != zero_z3563351764282998399l_num1 )
     => ( ord_less_nat @ ( semiri7341220984566936280m1_nat @ ( minus_4019991460397169231l_num1 @ P4 @ one_on7727431528512463931l_num1 ) ) @ ( semiri7341220984566936280m1_nat @ P4 ) ) ) ).

% measure_unat
thf(fact_4903_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_4904_floor__eq,axiom,
    ! [N: int,X: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
     => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X )
          = N ) ) ) ).

% floor_eq
thf(fact_4905_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_4906_word__sub__plus__one__nonzero,axiom,
    ! [N7: word_N3645301735248828278l_num1,N: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ N7 @ N )
     => ( ( N7 != zero_z3563351764282998399l_num1 )
       => ( ( plus_p361126936061061375l_num1 @ ( minus_4019991460397169231l_num1 @ N @ N7 ) @ one_on7727431528512463931l_num1 )
         != zero_z3563351764282998399l_num1 ) ) ) ).

% word_sub_plus_one_nonzero
thf(fact_4907_word__leq__minus__one__le,axiom,
    ! [Y: word_N3645301735248828278l_num1,X: word_N3645301735248828278l_num1] :
      ( ( Y != zero_z3563351764282998399l_num1 )
     => ( ( ord_le3335648743751981014l_num1 @ X @ ( minus_4019991460397169231l_num1 @ Y @ one_on7727431528512463931l_num1 ) )
       => ( ord_le750835935415966154l_num1 @ X @ Y ) ) ) ).

% word_leq_minus_one_le
thf(fact_4908_word__leq__le__minus__one,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ X @ Y )
     => ( ( X != zero_z3563351764282998399l_num1 )
       => ( ord_le750835935415966154l_num1 @ ( minus_4019991460397169231l_num1 @ X @ one_on7727431528512463931l_num1 ) @ Y ) ) ) ).

% word_leq_le_minus_one
thf(fact_4909_le__m1__iff__lt,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ X )
      = ( ( ord_le3335648743751981014l_num1 @ Y @ ( minus_4019991460397169231l_num1 @ X @ one_on7727431528512463931l_num1 ) )
        = ( ord_le750835935415966154l_num1 @ Y @ X ) ) ) ).

% le_m1_iff_lt
thf(fact_4910_less__1__simp,axiom,
    ! [N: word_N3645301735248828278l_num1,M: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ ( minus_4019991460397169231l_num1 @ N @ one_on7727431528512463931l_num1 ) @ M )
      = ( ( ord_le3335648743751981014l_num1 @ N @ M )
        & ( N != zero_z3563351764282998399l_num1 ) ) ) ).

% less_1_simp
thf(fact_4911_word__less__nowrapI,axiom,
    ! [X: word_N3645301735248828278l_num1,Z: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ X @ ( minus_4019991460397169231l_num1 @ Z @ K ) )
     => ( ( ord_le3335648743751981014l_num1 @ K @ Z )
       => ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ K )
         => ( ord_le750835935415966154l_num1 @ X @ ( plus_p361126936061061375l_num1 @ X @ K ) ) ) ) ) ).

% word_less_nowrapI
thf(fact_4912_plus__minus__not__NULL,axiom,
    ! [X: word_N3645301735248828278l_num1,Ab: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ X @ ( minus_4019991460397169231l_num1 @ Ab @ C ) )
     => ( ( ord_le3335648743751981014l_num1 @ C @ Ab )
       => ( ( C != zero_z3563351764282998399l_num1 )
         => ( ( plus_p361126936061061375l_num1 @ X @ C )
           != zero_z3563351764282998399l_num1 ) ) ) ) ).

% plus_minus_not_NULL
thf(fact_4913_word__less__nowrapI_H,axiom,
    ! [X: word_N3645301735248828278l_num1,Z: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ X @ ( minus_4019991460397169231l_num1 @ Z @ K ) )
     => ( ( ord_le3335648743751981014l_num1 @ K @ Z )
       => ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ K )
         => ( ord_le750835935415966154l_num1 @ X @ ( plus_p361126936061061375l_num1 @ X @ K ) ) ) ) ) ).

% word_less_nowrapI'
thf(fact_4914_ceiling__diff__floor__le__1,axiom,
    ! [X: real] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ ( archim6058952711729229775r_real @ X ) ) @ one_one_int ) ).

% ceiling_diff_floor_le_1
thf(fact_4915_ceiling__diff__floor__le__1,axiom,
    ! [X: rat] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim3151403230148437115or_rat @ X ) ) @ one_one_int ) ).

% ceiling_diff_floor_le_1
thf(fact_4916_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_4917_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_4918_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_4919_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_4920_floor__unique,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X )
     => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X )
          = Z ) ) ) ).

% floor_unique
thf(fact_4921_floor__unique,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X )
     => ( ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) )
       => ( ( archim3151403230148437115or_rat @ X )
          = Z ) ) ) ).

% floor_unique
thf(fact_4922_floor__eq__iff,axiom,
    ! [X: real,A2: int] :
      ( ( ( archim6058952711729229775r_real @ X )
        = A2 )
      = ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A2 ) @ X )
        & ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ A2 ) @ one_one_real ) ) ) ) ).

% floor_eq_iff
thf(fact_4923_floor__eq__iff,axiom,
    ! [X: rat,A2: int] :
      ( ( ( archim3151403230148437115or_rat @ X )
        = A2 )
      = ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A2 ) @ X )
        & ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A2 ) @ one_one_rat ) ) ) ) ).

% floor_eq_iff
thf(fact_4924_floor__split,axiom,
    ! [P: int > $o,T: real] :
      ( ( P @ ( archim6058952711729229775r_real @ T ) )
      = ( ! [I4: int] :
            ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I4 ) @ T )
              & ( ord_less_real @ T @ ( plus_plus_real @ ( ring_1_of_int_real @ I4 ) @ one_one_real ) ) )
           => ( P @ I4 ) ) ) ) ).

% floor_split
thf(fact_4925_floor__split,axiom,
    ! [P: int > $o,T: rat] :
      ( ( P @ ( archim3151403230148437115or_rat @ T ) )
      = ( ! [I4: int] :
            ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I4 ) @ T )
              & ( ord_less_rat @ T @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I4 ) @ one_one_rat ) ) )
           => ( P @ I4 ) ) ) ) ).

% floor_split
thf(fact_4926_less__floor__iff,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_int @ Z @ ( archim6058952711729229775r_real @ X ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X ) ) ).

% less_floor_iff
thf(fact_4927_less__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).

% less_floor_iff
thf(fact_4928_floor__le__iff,axiom,
    ! [X: real,Z: int] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ Z )
      = ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).

% floor_le_iff
thf(fact_4929_floor__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
      = ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).

% floor_le_iff
thf(fact_4930_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_4931_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_4932_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_4933_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_4934_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_4935_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_4936_floor__correct,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X ) ) @ X )
      & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ one_one_int ) ) ) ) ).

% floor_correct
thf(fact_4937_floor__correct,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ) ) ).

% floor_correct
thf(fact_4938_div__exp__eq,axiom,
    ! [A2: code_integer,M: nat,N: nat] :
      ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( divide6298287555418463151nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).

% div_exp_eq
thf(fact_4939_div__exp__eq,axiom,
    ! [A2: word_N3645301735248828278l_num1,M: nat,N: nat] :
      ( ( divide1791077408188789448l_num1 @ ( divide1791077408188789448l_num1 @ A2 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ M ) ) @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N ) )
      = ( divide1791077408188789448l_num1 @ A2 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).

% div_exp_eq
thf(fact_4940_div__exp__eq,axiom,
    ! [A2: uint32,M: nat,N: nat] :
      ( ( divide_divide_uint32 @ ( divide_divide_uint32 @ A2 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N ) )
      = ( divide_divide_uint32 @ A2 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).

% div_exp_eq
thf(fact_4941_div__exp__eq,axiom,
    ! [A2: nat,M: nat,N: 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 ) ) @ N ) )
      = ( divide_divide_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).

% div_exp_eq
thf(fact_4942_div__exp__eq,axiom,
    ! [A2: int,M: nat,N: 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 ) ) @ N ) )
      = ( divide_divide_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).

% div_exp_eq
thf(fact_4943_power__mono__odd,axiom,
    ! [N: nat,A2: real,B2: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ A2 @ B2 )
       => ( ord_less_eq_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ).

% power_mono_odd
thf(fact_4944_power__mono__odd,axiom,
    ! [N: nat,A2: code_integer,B2: code_integer] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ ( power_8256067586552552935nteger @ B2 @ N ) ) ) ) ).

% power_mono_odd
thf(fact_4945_power__mono__odd,axiom,
    ! [N: nat,A2: rat,B2: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_rat @ A2 @ B2 )
       => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ).

% power_mono_odd
thf(fact_4946_power__mono__odd,axiom,
    ! [N: nat,A2: int,B2: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_int @ A2 @ B2 )
       => ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).

% power_mono_odd
thf(fact_4947_inverse__of__nat__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( N != zero_zero_nat )
       => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_4948_inverse__of__nat__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( N != zero_zero_nat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_4949_odd__pos,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% odd_pos
thf(fact_4950_uminus__power__if,axiom,
    ! [N: nat,A2: code_integer] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A2 ) @ N )
          = ( power_8256067586552552935nteger @ A2 @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A2 ) @ N )
          = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_4951_uminus__power__if,axiom,
    ! [N: nat,A2: complex] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A2 ) @ N )
          = ( power_power_complex @ A2 @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A2 ) @ N )
          = ( uminus1482373934393186551omplex @ ( power_power_complex @ A2 @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_4952_uminus__power__if,axiom,
    ! [N: nat,A2: uint32] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ A2 ) @ N )
          = ( power_power_uint32 @ A2 @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ A2 ) @ N )
          = ( uminus_uminus_uint32 @ ( power_power_uint32 @ A2 @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_4953_uminus__power__if,axiom,
    ! [N: nat,A2: real] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_real @ ( uminus_uminus_real @ A2 ) @ N )
          = ( power_power_real @ A2 @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_real @ ( uminus_uminus_real @ A2 ) @ N )
          = ( uminus_uminus_real @ ( power_power_real @ A2 @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_4954_uminus__power__if,axiom,
    ! [N: nat,A2: rat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ A2 ) @ N )
          = ( power_power_rat @ A2 @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ A2 ) @ N )
          = ( uminus_uminus_rat @ ( power_power_rat @ A2 @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_4955_uminus__power__if,axiom,
    ! [N: nat,A2: int] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_int @ ( uminus_uminus_int @ A2 ) @ N )
          = ( power_power_int @ A2 @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_int @ ( uminus_uminus_int @ A2 ) @ N )
          = ( uminus_uminus_int @ ( power_power_int @ A2 @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_4956_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
       != zero_z3403309356797280102nteger ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_4957_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N )
       != zero_z3563351764282998399l_num1 )
     => ( ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
       != zero_z3563351764282998399l_num1 ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_4958_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
       != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_4959_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
       != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_4960_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
        = ( ord_less_eq_nat @ M @ N ) ) ) ).

% dvd_power_iff_le
thf(fact_4961_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).

% diff_le_diff_pow
thf(fact_4962_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_2184487114949457152l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_2184487114949457152l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4963_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4964_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4965_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4966_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4967_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4968_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_4969_floor__eq2,axiom,
    ! [N: int,X: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X )
     => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X )
          = N ) ) ) ).

% floor_eq2
thf(fact_4970_zero__le__even__power,axiom,
    ! [N: nat,A2: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ N ) ) ) ).

% zero_le_even_power
thf(fact_4971_zero__le__even__power,axiom,
    ! [N: nat,A2: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ).

% zero_le_even_power
thf(fact_4972_zero__le__even__power,axiom,
    ! [N: nat,A2: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ N ) ) ) ).

% zero_le_even_power
thf(fact_4973_zero__le__even__power,axiom,
    ! [N: nat,A2: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ N ) ) ) ).

% zero_le_even_power
thf(fact_4974_zero__le__odd__power,axiom,
    ! [N: nat,A2: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ N ) )
        = ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ) ).

% zero_le_odd_power
thf(fact_4975_zero__le__odd__power,axiom,
    ! [N: nat,A2: code_integer] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A2 @ N ) )
        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ) ).

% zero_le_odd_power
thf(fact_4976_zero__le__odd__power,axiom,
    ! [N: nat,A2: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ N ) )
        = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ) ).

% zero_le_odd_power
thf(fact_4977_zero__le__odd__power,axiom,
    ! [N: nat,A2: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ N ) )
        = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ) ).

% zero_le_odd_power
thf(fact_4978_zero__le__power__eq,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ) ) ).

% zero_le_power_eq
thf(fact_4979_zero__le__power__eq,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A2 @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ) ) ).

% zero_le_power_eq
thf(fact_4980_zero__le__power__eq,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ) ) ).

% zero_le_power_eq
thf(fact_4981_zero__le__power__eq,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ) ) ).

% zero_le_power_eq
thf(fact_4982_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_2184487114949457152l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ N )
          = one_on7727431528512463931l_num1 ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_2184487114949457152l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ N )
          = ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ) ) ).

% minus_one_power_iff
thf(fact_4983_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
          = one_one_Code_integer ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% minus_one_power_iff
thf(fact_4984_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
          = one_one_complex ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% minus_one_power_iff
thf(fact_4985_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N )
          = one_one_uint32 ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N )
          = ( uminus_uminus_uint32 @ one_one_uint32 ) ) ) ) ).

% minus_one_power_iff
thf(fact_4986_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
          = one_one_real ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% minus_one_power_iff
thf(fact_4987_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
          = one_one_rat ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% minus_one_power_iff
thf(fact_4988_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
          = one_one_int ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% minus_one_power_iff
thf(fact_4989_word__of__int__2p,axiom,
    ! [N: nat] :
      ( ( ring_17408606157368542149l_num1 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N ) ) ).

% word_of_int_2p
thf(fact_4990_ceiling__log2__div2,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).

% ceiling_log2_div2
thf(fact_4991_ceiling__eq,axiom,
    ! [N: int,X: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
     => ( ( ord_less_eq_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
       => ( ( archim7802044766580827645g_real @ X )
          = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).

% ceiling_eq
thf(fact_4992_ceiling__eq,axiom,
    ! [N: int,X: rat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ N ) @ X )
     => ( ( ord_less_eq_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N ) @ one_one_rat ) )
       => ( ( archim2889992004027027881ng_rat @ X )
          = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).

% ceiling_eq
thf(fact_4993_zero__less__power__eq,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A2 @ N ) )
      = ( ( N = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( A2 != zero_z3403309356797280102nteger ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ) ) ).

% zero_less_power_eq
thf(fact_4994_zero__less__power__eq,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A2 @ N ) )
      = ( ( N = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( A2 != zero_zero_real ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_real @ zero_zero_real @ A2 ) ) ) ) ).

% zero_less_power_eq
thf(fact_4995_zero__less__power__eq,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ N ) )
      = ( ( N = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( A2 != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_rat @ zero_zero_rat @ A2 ) ) ) ) ).

% zero_less_power_eq
thf(fact_4996_zero__less__power__eq,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A2 @ N ) )
      = ( ( N = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( A2 != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_int @ zero_zero_int @ A2 ) ) ) ) ).

% zero_less_power_eq
thf(fact_4997_odd__word__imp__even__next,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri7341220984566936280m1_nat @ X ) )
     => ( ( ( plus_p361126936061061375l_num1 @ X @ one_on7727431528512463931l_num1 )
          = zero_z3563351764282998399l_num1 )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri7341220984566936280m1_nat @ ( plus_p361126936061061375l_num1 @ X @ one_on7727431528512463931l_num1 ) ) ) ) ) ).

% odd_word_imp_even_next
thf(fact_4998_even__word__imp__odd__next,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri7341220984566936280m1_nat @ X ) )
     => ( ( ( plus_p361126936061061375l_num1 @ X @ one_on7727431528512463931l_num1 )
          = zero_z3563351764282998399l_num1 )
        | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri7341220984566936280m1_nat @ ( plus_p361126936061061375l_num1 @ X @ one_on7727431528512463931l_num1 ) ) ) ) ) ).

% even_word_imp_odd_next
thf(fact_4999_uint__range__size,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( semiri7338730514057886004m1_int @ W ) )
      & ( ord_less_int @ ( semiri7338730514057886004m1_int @ W ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( size_s8261804613246490634l_num1 @ W ) ) ) ) ).

% uint_range_size
thf(fact_5000_power__le__zero__eq,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ A2 @ N ) @ zero_zero_real )
      = ( ( ord_less_nat @ zero_zero_nat @ N )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( ord_less_eq_real @ A2 @ zero_zero_real ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( A2 = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_5001_power__le__zero__eq,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ zero_z3403309356797280102nteger )
      = ( ( ord_less_nat @ zero_zero_nat @ N )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( A2 = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_5002_power__le__zero__eq,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N ) @ zero_zero_rat )
      = ( ( ord_less_nat @ zero_zero_nat @ N )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( ord_less_eq_rat @ A2 @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( A2 = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_5003_power__le__zero__eq,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ N )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( ord_less_eq_int @ A2 @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( A2 = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_5004_no__plus__overflow__uint__size,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ X @ ( plus_p361126936061061375l_num1 @ X @ Y ) )
      = ( ord_less_int @ ( plus_plus_int @ ( semiri7338730514057886004m1_int @ X ) @ ( semiri7338730514057886004m1_int @ Y ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( size_s8261804613246490634l_num1 @ X ) ) ) ) ).

% no_plus_overflow_uint_size
thf(fact_5005_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_5006_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_5007_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_5008_div__eq__dividend__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ( divide_divide_nat @ M @ N )
          = M )
        = ( N = one_one_nat ) ) ) ).

% div_eq_dividend_iff
thf(fact_5009_pow__divides__pow__iff,axiom,
    ! [N: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B2 @ N ) )
        = ( dvd_dvd_nat @ A2 @ B2 ) ) ) ).

% pow_divides_pow_iff
thf(fact_5010_pow__divides__pow__iff,axiom,
    ! [N: nat,A2: int,B2: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B2 @ N ) )
        = ( dvd_dvd_int @ A2 @ B2 ) ) ) ).

% pow_divides_pow_iff
thf(fact_5011_div__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( divide_divide_nat @ M @ N )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_5012_VEBT__internal_Olog__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 ) ) ) ) ) ) ).

% VEBT_internal.log_ceil_idem
thf(fact_5013_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_5014_less__two__pow__divI,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ord_less_nat @ X @ ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).

% less_two_pow_divI
thf(fact_5015_less__two__pow__divD,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) )
     => ( ( ord_less_eq_nat @ M @ N )
        & ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% less_two_pow_divD
thf(fact_5016_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_5017_Diff__iff,axiom,
    ! [C: real,A: set_real,B: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A @ B ) )
      = ( ( member_real @ C @ A )
        & ~ ( member_real @ C @ B ) ) ) ).

% Diff_iff
thf(fact_5018_Diff__iff,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A @ B ) )
      = ( ( member_int @ C @ A )
        & ~ ( member_int @ C @ B ) ) ) ).

% Diff_iff
thf(fact_5019_Diff__iff,axiom,
    ! [C: complex,A: set_complex,B: set_complex] :
      ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A @ B ) )
      = ( ( member_complex @ C @ A )
        & ~ ( member_complex @ C @ B ) ) ) ).

% Diff_iff
thf(fact_5020_Diff__iff,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
      = ( ( member_nat @ C @ A )
        & ~ ( member_nat @ C @ B ) ) ) ).

% Diff_iff
thf(fact_5021_DiffI,axiom,
    ! [C: real,A: set_real,B: set_real] :
      ( ( member_real @ C @ A )
     => ( ~ ( member_real @ C @ B )
       => ( member_real @ C @ ( minus_minus_set_real @ A @ B ) ) ) ) ).

% DiffI
thf(fact_5022_DiffI,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( ( member_int @ C @ A )
     => ( ~ ( member_int @ C @ B )
       => ( member_int @ C @ ( minus_minus_set_int @ A @ B ) ) ) ) ).

% DiffI
thf(fact_5023_DiffI,axiom,
    ! [C: complex,A: set_complex,B: set_complex] :
      ( ( member_complex @ C @ A )
     => ( ~ ( member_complex @ C @ B )
       => ( member_complex @ C @ ( minus_811609699411566653omplex @ A @ B ) ) ) ) ).

% DiffI
thf(fact_5024_DiffI,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C @ A )
     => ( ~ ( member_nat @ C @ B )
       => ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).

% DiffI
thf(fact_5025_div__of__0__id,axiom,
    ! [N: word_N3645301735248828278l_num1] :
      ( ( divide1791077408188789448l_num1 @ zero_z3563351764282998399l_num1 @ N )
      = zero_z3563351764282998399l_num1 ) ).

% div_of_0_id
thf(fact_5026_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_5027_int__dvd__int__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( dvd_dvd_nat @ M @ N ) ) ).

% int_dvd_int_iff
thf(fact_5028_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L @ K )
       => ( ( divide_divide_int @ K @ L )
          = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
thf(fact_5029_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L )
       => ( ( divide_divide_int @ K @ L )
          = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
thf(fact_5030_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_5031_word__div__no,axiom,
    ! [A2: num,B2: num] :
      ( ( divide1791077408188789448l_num1 @ ( numera7442385471795722001l_num1 @ A2 ) @ ( numera7442385471795722001l_num1 @ B2 ) )
      = ( ring_17408606157368542149l_num1 @ ( divide_divide_int @ ( semiri7338730514057886004m1_int @ ( numera7442385471795722001l_num1 @ A2 ) ) @ ( semiri7338730514057886004m1_int @ ( numera7442385471795722001l_num1 @ B2 ) ) ) ) ) ).

% word_div_no
thf(fact_5032_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_5033_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_5034_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_5035_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_5036_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_5037_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_5038_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_5039_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_5040_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_5041_real__of__int__div4,axiom,
    ! [N: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) ) ).

% real_of_int_div4
thf(fact_5042_word__div__def,axiom,
    ( divide1791077408188789448l_num1
    = ( ^ [A4: word_N3645301735248828278l_num1,B4: word_N3645301735248828278l_num1] : ( ring_17408606157368542149l_num1 @ ( divide_divide_int @ ( semiri7338730514057886004m1_int @ A4 ) @ ( semiri7338730514057886004m1_int @ B4 ) ) ) ) ) ).

% word_div_def
thf(fact_5043_uint__div__distrib,axiom,
    ! [V: word_N3645301735248828278l_num1,W: word_N3645301735248828278l_num1] :
      ( ( semiri7338730514057886004m1_int @ ( divide1791077408188789448l_num1 @ V @ W ) )
      = ( divide_divide_int @ ( semiri7338730514057886004m1_int @ V ) @ ( semiri7338730514057886004m1_int @ W ) ) ) ).

% uint_div_distrib
thf(fact_5044_uint__div,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( semiri7338730514057886004m1_int @ ( divide1791077408188789448l_num1 @ X @ Y ) )
      = ( divide_divide_int @ ( semiri7338730514057886004m1_int @ X ) @ ( semiri7338730514057886004m1_int @ Y ) ) ) ).

% uint_div
thf(fact_5045_set__diff__eq,axiom,
    ( minus_minus_set_real
    = ( ^ [A3: set_real,B3: set_real] :
          ( collect_real
          @ ^ [X2: real] :
              ( ( member_real @ X2 @ A3 )
              & ~ ( member_real @ X2 @ B3 ) ) ) ) ) ).

% set_diff_eq
thf(fact_5046_set__diff__eq,axiom,
    ( minus_811609699411566653omplex
    = ( ^ [A3: set_complex,B3: set_complex] :
          ( collect_complex
          @ ^ [X2: complex] :
              ( ( member_complex @ X2 @ A3 )
              & ~ ( member_complex @ X2 @ B3 ) ) ) ) ) ).

% set_diff_eq
thf(fact_5047_set__diff__eq,axiom,
    ( minus_1052850069191792384nt_int
    = ( ^ [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ^ [X2: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X2 @ A3 )
              & ~ ( member5262025264175285858nt_int @ X2 @ B3 ) ) ) ) ) ).

% set_diff_eq
thf(fact_5048_set__diff__eq,axiom,
    ( minus_minus_set_int
    = ( ^ [A3: set_int,B3: set_int] :
          ( collect_int
          @ ^ [X2: int] :
              ( ( member_int @ X2 @ A3 )
              & ~ ( member_int @ X2 @ B3 ) ) ) ) ) ).

% set_diff_eq
thf(fact_5049_set__diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( collect_nat
          @ ^ [X2: nat] :
              ( ( member_nat @ X2 @ A3 )
              & ~ ( member_nat @ X2 @ B3 ) ) ) ) ) ).

% set_diff_eq
thf(fact_5050_word__of__int__Ex,axiom,
    ! [X: word_N3645301735248828278l_num1] :
    ? [Y4: int] :
      ( X
      = ( ring_17408606157368542149l_num1 @ Y4 ) ) ).

% word_of_int_Ex
thf(fact_5051_wi__hom__sub,axiom,
    ! [A2: int,B2: int] :
      ( ( minus_4019991460397169231l_num1 @ ( ring_17408606157368542149l_num1 @ A2 ) @ ( ring_17408606157368542149l_num1 @ B2 ) )
      = ( ring_17408606157368542149l_num1 @ ( minus_minus_int @ A2 @ B2 ) ) ) ).

% wi_hom_sub
thf(fact_5052_zdvd__zdiffD,axiom,
    ! [K: int,M: int,N: int] :
      ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
     => ( ( dvd_dvd_int @ K @ N )
       => ( dvd_dvd_int @ K @ M ) ) ) ).

% zdvd_zdiffD
thf(fact_5053_minus__set__def,axiom,
    ( minus_minus_set_real
    = ( ^ [A3: set_real,B3: set_real] :
          ( collect_real
          @ ( minus_minus_real_o
            @ ^ [X2: real] : ( member_real @ X2 @ A3 )
            @ ^ [X2: real] : ( member_real @ X2 @ B3 ) ) ) ) ) ).

% minus_set_def
thf(fact_5054_minus__set__def,axiom,
    ( minus_811609699411566653omplex
    = ( ^ [A3: set_complex,B3: set_complex] :
          ( collect_complex
          @ ( minus_8727706125548526216plex_o
            @ ^ [X2: complex] : ( member_complex @ X2 @ A3 )
            @ ^ [X2: complex] : ( member_complex @ X2 @ B3 ) ) ) ) ) ).

% minus_set_def
thf(fact_5055_minus__set__def,axiom,
    ( minus_1052850069191792384nt_int
    = ( ^ [A3: set_Pr958786334691620121nt_int,B3: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ( minus_711738161318947805_int_o
            @ ^ [X2: product_prod_int_int] : ( member5262025264175285858nt_int @ X2 @ A3 )
            @ ^ [X2: product_prod_int_int] : ( member5262025264175285858nt_int @ X2 @ B3 ) ) ) ) ) ).

% minus_set_def
thf(fact_5056_minus__set__def,axiom,
    ( minus_minus_set_int
    = ( ^ [A3: set_int,B3: set_int] :
          ( collect_int
          @ ( minus_minus_int_o
            @ ^ [X2: int] : ( member_int @ X2 @ A3 )
            @ ^ [X2: int] : ( member_int @ X2 @ B3 ) ) ) ) ) ).

% minus_set_def
thf(fact_5057_minus__set__def,axiom,
    ( minus_minus_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( collect_nat
          @ ( minus_minus_nat_o
            @ ^ [X2: nat] : ( member_nat @ X2 @ A3 )
            @ ^ [X2: nat] : ( member_nat @ X2 @ B3 ) ) ) ) ) ).

% minus_set_def
thf(fact_5058_DiffD2,axiom,
    ! [C: real,A: set_real,B: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A @ B ) )
     => ~ ( member_real @ C @ B ) ) ).

% DiffD2
thf(fact_5059_DiffD2,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A @ B ) )
     => ~ ( member_int @ C @ B ) ) ).

% DiffD2
thf(fact_5060_DiffD2,axiom,
    ! [C: complex,A: set_complex,B: set_complex] :
      ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A @ B ) )
     => ~ ( member_complex @ C @ B ) ) ).

% DiffD2
thf(fact_5061_DiffD2,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
     => ~ ( member_nat @ C @ B ) ) ).

% DiffD2
thf(fact_5062_DiffD1,axiom,
    ! [C: real,A: set_real,B: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A @ B ) )
     => ( member_real @ C @ A ) ) ).

% DiffD1
thf(fact_5063_DiffD1,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A @ B ) )
     => ( member_int @ C @ A ) ) ).

% DiffD1
thf(fact_5064_DiffD1,axiom,
    ! [C: complex,A: set_complex,B: set_complex] :
      ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A @ B ) )
     => ( member_complex @ C @ A ) ) ).

% DiffD1
thf(fact_5065_DiffD1,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
     => ( member_nat @ C @ A ) ) ).

% DiffD1
thf(fact_5066_DiffE,axiom,
    ! [C: real,A: set_real,B: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A @ B ) )
     => ~ ( ( member_real @ C @ A )
         => ( member_real @ C @ B ) ) ) ).

% DiffE
thf(fact_5067_DiffE,axiom,
    ! [C: int,A: set_int,B: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A @ B ) )
     => ~ ( ( member_int @ C @ A )
         => ( member_int @ C @ B ) ) ) ).

% DiffE
thf(fact_5068_DiffE,axiom,
    ! [C: complex,A: set_complex,B: set_complex] :
      ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A @ B ) )
     => ~ ( ( member_complex @ C @ A )
         => ( member_complex @ C @ B ) ) ) ).

% DiffE
thf(fact_5069_DiffE,axiom,
    ! [C: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
     => ~ ( ( member_nat @ C @ A )
         => ( member_nat @ C @ B ) ) ) ).

% DiffE
thf(fact_5070_real__of__int__div,axiom,
    ! [D: int,N: int] :
      ( ( dvd_dvd_int @ D @ N )
     => ( ( ring_1_of_int_real @ ( divide_divide_int @ N @ D ) )
        = ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ D ) ) ) ) ).

% real_of_int_div
thf(fact_5071_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).

% diff_commute
thf(fact_5072_int__div__sub__1,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_eq_int @ one_one_int @ M )
     => ( ( ( dvd_dvd_int @ M @ N )
         => ( ( divide_divide_int @ ( minus_minus_int @ N @ one_one_int ) @ M )
            = ( minus_minus_int @ ( divide_divide_int @ N @ M ) @ one_one_int ) ) )
        & ( ~ ( dvd_dvd_int @ M @ N )
         => ( ( divide_divide_int @ ( minus_minus_int @ N @ one_one_int ) @ M )
            = ( divide_divide_int @ N @ M ) ) ) ) ) ).

% int_div_sub_1
thf(fact_5073_real__of__int__div2,axiom,
    ! [N: int,X: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) ) ).

% real_of_int_div2
thf(fact_5074_real__of__int__div3,axiom,
    ! [N: int,X: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) @ one_one_real ) ).

% real_of_int_div3
thf(fact_5075_length__induct,axiom,
    ! [P: list_o > $o,Xs: list_o] :
      ( ! [Xs2: list_o] :
          ( ! [Ys2: list_o] :
              ( ( ord_less_nat @ ( size_size_list_o @ Ys2 ) @ ( size_size_list_o @ Xs2 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_5076_word__div__1,axiom,
    ! [N: word_N3645301735248828278l_num1] :
      ( ( divide1791077408188789448l_num1 @ N @ one_on7727431528512463931l_num1 )
      = N ) ).

% word_div_1
thf(fact_5077_div__by__0__word,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( divide1791077408188789448l_num1 @ X @ zero_z3563351764282998399l_num1 )
      = zero_z3563351764282998399l_num1 ) ).

% div_by_0_word
thf(fact_5078_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_5079_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_5080_zdvd__antisym__nonneg,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ N )
       => ( ( dvd_dvd_int @ M @ N )
         => ( ( dvd_dvd_int @ N @ M )
           => ( M = N ) ) ) ) ) ).

% zdvd_antisym_nonneg
thf(fact_5081_zdvd__not__zless,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_int @ M @ N )
       => ~ ( dvd_dvd_int @ N @ M ) ) ) ).

% zdvd_not_zless
thf(fact_5082_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_5083_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_5084_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_5085_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_5086_div__word__self,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( W != zero_z3563351764282998399l_num1 )
     => ( ( divide1791077408188789448l_num1 @ W @ W )
        = one_on7727431528512463931l_num1 ) ) ).

% div_word_self
thf(fact_5087_word__div__lt__eq__0,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ X @ Y )
     => ( ( divide1791077408188789448l_num1 @ X @ Y )
        = zero_z3563351764282998399l_num1 ) ) ).

% word_div_lt_eq_0
thf(fact_5088_word__less__div,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( ( divide1791077408188789448l_num1 @ X @ Y )
        = zero_z3563351764282998399l_num1 )
     => ( ( Y = zero_z3563351764282998399l_num1 )
        | ( ord_le750835935415966154l_num1 @ X @ Y ) ) ) ).

% word_less_div
thf(fact_5089_word__div__less,axiom,
    ! [W: word_N3645301735248828278l_num1,V: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ W @ V )
     => ( ( divide1791077408188789448l_num1 @ W @ V )
        = zero_z3563351764282998399l_num1 ) ) ).

% word_div_less
thf(fact_5090_length__pos__if__in__set,axiom,
    ! [X: real,Xs: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_5091_length__pos__if__in__set,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_5092_length__pos__if__in__set,axiom,
    ! [X: complex,Xs: list_complex] :
      ( ( member_complex @ X @ ( set_complex2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_5093_length__pos__if__in__set,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_5094_length__pos__if__in__set,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( member_o @ X @ ( set_o2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_5095_zdvd__imp__le,axiom,
    ! [Z: int,N: int] :
      ( ( dvd_dvd_int @ Z @ N )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int @ Z @ N ) ) ) ).

% zdvd_imp_le
thf(fact_5096_zdiv__mono1,axiom,
    ! [A2: int,A6: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ A6 )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B2 ) @ ( divide_divide_int @ A6 @ B2 ) ) ) ) ).

% zdiv_mono1
thf(fact_5097_zdiv__mono2,axiom,
    ! [A2: int,B7: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B7 )
       => ( ( ord_less_eq_int @ B7 @ B2 )
         => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B2 ) @ ( divide_divide_int @ A2 @ B7 ) ) ) ) ) ).

% zdiv_mono2
thf(fact_5098_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_5099_zdiv__mono1__neg,axiom,
    ! [A2: int,A6: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ A6 )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A6 @ B2 ) @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).

% zdiv_mono1_neg
thf(fact_5100_zdiv__mono2__neg,axiom,
    ! [A2: int,B7: int,B2: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B7 )
       => ( ( ord_less_eq_int @ B7 @ B2 )
         => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B7 ) @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ) ).

% zdiv_mono2_neg
thf(fact_5101_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
      = ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ K )
          & ( ord_less_eq_int @ zero_zero_int @ L ) )
        | ( ( ord_less_int @ K @ zero_zero_int )
          & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).

% div_int_pos_iff
thf(fact_5102_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ L @ K )
     => ( ( ord_less_int @ zero_zero_int @ L )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).

% div_positive_int
thf(fact_5103_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_5104_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_5105_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_5106_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_5107_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_5108_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_5109_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_5110_real__of__nat__div4,axiom,
    ! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).

% real_of_nat_div4
thf(fact_5111_div__less__dividend__word,axiom,
    ! [X: word_N3645301735248828278l_num1,N: word_N3645301735248828278l_num1] :
      ( ( X != zero_z3563351764282998399l_num1 )
     => ( ( N != one_on7727431528512463931l_num1 )
       => ( ord_le750835935415966154l_num1 @ ( divide1791077408188789448l_num1 @ X @ N ) @ X ) ) ) ).

% div_less_dividend_word
thf(fact_5112_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_5113_real__of__nat__div,axiom,
    ! [D: nat,N: nat] :
      ( ( dvd_dvd_nat @ D @ N )
     => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
        = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).

% real_of_nat_div
thf(fact_5114_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_5115_num_Osize_I4_J,axiom,
    ( ( size_size_num @ one )
    = zero_zero_nat ) ).

% num.size(4)
thf(fact_5116_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).

% even_diff_iff
thf(fact_5117_verit__less__mono__div__int2,axiom,
    ! [A: int,B: int,N: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
       => ( ord_less_eq_int @ ( divide_divide_int @ B @ N ) @ ( divide_divide_int @ A @ N ) ) ) ) ).

% verit_less_mono_div_int2
thf(fact_5118_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_5119_real__of__nat__div2,axiom,
    ! [N: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) ) ).

% real_of_nat_div2
thf(fact_5120_real__of__nat__div3,axiom,
    ! [N: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) @ one_one_real ) ).

% real_of_nat_div3
thf(fact_5121_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_5122_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_5123_word__div__sub,axiom,
    ! [Y: word_N3645301735248828278l_num1,X: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ Y @ X )
     => ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ Y )
       => ( ( divide1791077408188789448l_num1 @ ( minus_4019991460397169231l_num1 @ X @ Y ) @ Y )
          = ( minus_4019991460397169231l_num1 @ ( divide1791077408188789448l_num1 @ X @ Y ) @ one_on7727431528512463931l_num1 ) ) ) ) ).

% word_div_sub
thf(fact_5124_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_5125_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
       => ( ( divide_divide_int @ K @ L )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% div_pos_neg_trivial
thf(fact_5126_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ( ord_less_eq_int @ L @ K )
       => ( ( divide_divide_int @ K @ L )
          = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).

% div_pos_geq
thf(fact_5127_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_5128_log__base__pow,axiom,
    ! [A2: real,N: nat,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( log @ ( power_power_real @ A2 @ N ) @ X )
        = ( divide_divide_real @ ( log @ A2 @ X ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log_base_pow
thf(fact_5129_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_5130_minus__1__div__exp__eq__int,axiom,
    ! [N: nat] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% minus_1_div_exp_eq_int
thf(fact_5131_word__less__two__pow__divD,axiom,
    ! [X: word_N3645301735248828278l_num1,N: nat,M: nat] :
      ( ( ord_le750835935415966154l_num1 @ X @ ( divide1791077408188789448l_num1 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N ) @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ M ) ) )
     => ( ( ord_less_eq_nat @ M @ N )
        & ( ord_le750835935415966154l_num1 @ X @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% word_less_two_pow_divD
thf(fact_5132_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A2: nat,B2: nat] :
      ( ! [A5: nat,B6: nat] :
          ( ( P @ A5 @ B6 )
          = ( P @ B6 @ A5 ) )
     => ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
       => ( ! [A5: nat,B6: nat] :
              ( ( P @ A5 @ B6 )
             => ( P @ A5 @ ( plus_plus_nat @ A5 @ B6 ) ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% Euclid_induct
thf(fact_5133_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
     => ( A2 = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
thf(fact_5134_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_5135_gcd__nat_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
      = ( A2 = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
thf(fact_5136_gcd__nat_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
        & ( zero_zero_nat != A2 ) ) ).

% gcd_nat.extremum_strict
thf(fact_5137_gcd__nat_Oextremum,axiom,
    ! [A2: nat] : ( dvd_dvd_nat @ A2 @ zero_zero_nat ) ).

% gcd_nat.extremum
thf(fact_5138_powr__neg__numeral,axiom,
    ! [X: real,N: num] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
        = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ) ).

% powr_neg_numeral
thf(fact_5139_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( divide_divide_nat @ M @ N )
        = zero_zero_nat )
      = ( ( ord_less_nat @ M @ N )
        | ( N = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
thf(fact_5140_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
      = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_5141_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N ) )
      = ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_5142_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4939895301339042750nteger @ ( divide_divide_nat @ M @ N ) )
      = ( divide6298287555418463151nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_5143_dvd__pos__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ M @ N )
       => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).

% dvd_pos_nat
thf(fact_5144_of__nat__dvd__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( dvd_dvd_nat @ M @ N ) ) ).

% of_nat_dvd_iff
thf(fact_5145_of__nat__dvd__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( dvd_dvd_nat @ M @ N ) ) ).

% of_nat_dvd_iff
thf(fact_5146_of__nat__dvd__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
      = ( dvd_dvd_nat @ M @ N ) ) ).

% of_nat_dvd_iff
thf(fact_5147_numeral__Bit0__div__2,axiom,
    ! [N: num] :
      ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( numeral_numeral_nat @ N ) ) ).

% numeral_Bit0_div_2
thf(fact_5148_numeral__Bit0__div__2,axiom,
    ! [N: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( numeral_numeral_int @ N ) ) ).

% numeral_Bit0_div_2
thf(fact_5149_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_5150_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_5151_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_5152_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_5153_div__greater__zero__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
      = ( ( ord_less_eq_nat @ N @ M )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% div_greater_zero_iff
thf(fact_5154_div__le__mono2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).

% div_le_mono2
thf(fact_5155_div__less__dividend,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).

% div_less_dividend
thf(fact_5156_power__diff__power__eq,axiom,
    ! [A2: code_integer,N: nat,M: nat] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( ( ord_less_eq_nat @ N @ M )
         => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A2 @ M ) @ ( power_8256067586552552935nteger @ A2 @ N ) )
            = ( power_8256067586552552935nteger @ A2 @ ( minus_minus_nat @ M @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M )
         => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A2 @ M ) @ ( power_8256067586552552935nteger @ A2 @ N ) )
            = ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A2 @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_5157_power__diff__power__eq,axiom,
    ! [A2: nat,N: nat,M: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) )
            = ( power_power_nat @ A2 @ ( minus_minus_nat @ M @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) )
            = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A2 @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_5158_power__diff__power__eq,axiom,
    ! [A2: int,N: nat,M: nat] :
      ( ( A2 != zero_zero_int )
     => ( ( ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) )
            = ( power_power_int @ A2 @ ( minus_minus_nat @ M @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) )
            = ( divide_divide_int @ one_one_int @ ( power_power_int @ A2 @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_5159_power__sub,axiom,
    ! [N: nat,M: nat,A2: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( ord_less_nat @ zero_zero_nat @ A2 )
       => ( ( power_power_nat @ A2 @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) ) ) ) ) ).

% power_sub
thf(fact_5160_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_5161_two__pow__div__gt__le,axiom,
    ! [V: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ V @ ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% two_pow_div_gt_le
thf(fact_5162_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_5163_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_5164_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_5165_VEBT__internal_Opow__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 ) ) ).

% VEBT_internal.pow_sum
thf(fact_5166_artanh__def,axiom,
    ( artanh_real
    = ( ^ [X2: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% artanh_def
thf(fact_5167_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_5168_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_5169_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_5170_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_5171_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_5172_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_5173_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_5174_ln__one,axiom,
    ( ( ln_ln_real @ one_one_real )
    = zero_zero_real ) ).

% ln_one
thf(fact_5175_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_5176_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_5177_neq__if__length__neq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs )
       != ( size_size_list_o @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_5178_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_o] :
      ( ( size_size_list_o @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_5179_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_5180_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_5181_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_5182_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_5183_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_5184_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_5185_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_5186_pinf_I1_J,axiom,
    ! [P: real > $o,P5: real > $o,Q: real > $o,Q2: real > $o] :
      ( ? [Z5: real] :
        ! [X3: real] :
          ( ( ord_less_real @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( ord_less_real @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: real] :
          ! [X6: real] :
            ( ( ord_less_real @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P5 @ X6 )
                & ( Q2 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_5187_pinf_I1_J,axiom,
    ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q2: rat > $o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: rat] :
          ! [X6: rat] :
            ( ( ord_less_rat @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P5 @ X6 )
                & ( Q2 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_5188_pinf_I1_J,axiom,
    ! [P: num > $o,P5: num > $o,Q: num > $o,Q2: num > $o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( ( ord_less_num @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: num] :
          ! [X6: num] :
            ( ( ord_less_num @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P5 @ X6 )
                & ( Q2 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_5189_pinf_I1_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P5 @ X6 )
                & ( Q2 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_5190_pinf_I1_J,axiom,
    ! [P: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( ( ord_less_int @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: int] :
          ! [X6: int] :
            ( ( ord_less_int @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P5 @ X6 )
                & ( Q2 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_5191_pinf_I2_J,axiom,
    ! [P: real > $o,P5: real > $o,Q: real > $o,Q2: real > $o] :
      ( ? [Z5: real] :
        ! [X3: real] :
          ( ( ord_less_real @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( ord_less_real @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: real] :
          ! [X6: real] :
            ( ( ord_less_real @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P5 @ X6 )
                | ( Q2 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_5192_pinf_I2_J,axiom,
    ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q2: rat > $o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: rat] :
          ! [X6: rat] :
            ( ( ord_less_rat @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P5 @ X6 )
                | ( Q2 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_5193_pinf_I2_J,axiom,
    ! [P: num > $o,P5: num > $o,Q: num > $o,Q2: num > $o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( ( ord_less_num @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: num] :
          ! [X6: num] :
            ( ( ord_less_num @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P5 @ X6 )
                | ( Q2 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_5194_pinf_I2_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P5 @ X6 )
                | ( Q2 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_5195_pinf_I2_J,axiom,
    ! [P: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( ( ord_less_int @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: int] :
          ! [X6: int] :
            ( ( ord_less_int @ Z3 @ X6 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P5 @ X6 )
                | ( Q2 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_5196_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(3)
thf(fact_5197_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(3)
thf(fact_5198_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(3)
thf(fact_5199_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(3)
thf(fact_5200_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(3)
thf(fact_5201_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(4)
thf(fact_5202_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(4)
thf(fact_5203_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(4)
thf(fact_5204_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(4)
thf(fact_5205_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z3 @ X6 )
     => ( X6 != T ) ) ).

% pinf(4)
thf(fact_5206_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z3 @ X6 )
     => ~ ( ord_less_real @ X6 @ T ) ) ).

% pinf(5)
thf(fact_5207_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z3 @ X6 )
     => ~ ( ord_less_rat @ X6 @ T ) ) ).

% pinf(5)
thf(fact_5208_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z3 @ X6 )
     => ~ ( ord_less_num @ X6 @ T ) ) ).

% pinf(5)
thf(fact_5209_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ~ ( ord_less_nat @ X6 @ T ) ) ).

% pinf(5)
thf(fact_5210_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z3 @ X6 )
     => ~ ( ord_less_int @ X6 @ T ) ) ).

% pinf(5)
thf(fact_5211_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z3 @ X6 )
     => ( ord_less_real @ T @ X6 ) ) ).

% pinf(7)
thf(fact_5212_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z3 @ X6 )
     => ( ord_less_rat @ T @ X6 ) ) ).

% pinf(7)
thf(fact_5213_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z3 @ X6 )
     => ( ord_less_num @ T @ X6 ) ) ).

% pinf(7)
thf(fact_5214_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ( ord_less_nat @ T @ X6 ) ) ).

% pinf(7)
thf(fact_5215_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z3 @ X6 )
     => ( ord_less_int @ T @ X6 ) ) ).

% pinf(7)
thf(fact_5216_minf_I1_J,axiom,
    ! [P: real > $o,P5: real > $o,Q: real > $o,Q2: real > $o] :
      ( ? [Z5: real] :
        ! [X3: real] :
          ( ( ord_less_real @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( ord_less_real @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: real] :
          ! [X6: real] :
            ( ( ord_less_real @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P5 @ X6 )
                & ( Q2 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_5217_minf_I1_J,axiom,
    ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q2: rat > $o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: rat] :
          ! [X6: rat] :
            ( ( ord_less_rat @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P5 @ X6 )
                & ( Q2 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_5218_minf_I1_J,axiom,
    ! [P: num > $o,P5: num > $o,Q: num > $o,Q2: num > $o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( ( ord_less_num @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: num] :
          ! [X6: num] :
            ( ( ord_less_num @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P5 @ X6 )
                & ( Q2 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_5219_minf_I1_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P5 @ X6 )
                & ( Q2 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_5220_minf_I1_J,axiom,
    ! [P: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( ( ord_less_int @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: int] :
          ! [X6: int] :
            ( ( ord_less_int @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P5 @ X6 )
                & ( Q2 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_5221_minf_I2_J,axiom,
    ! [P: real > $o,P5: real > $o,Q: real > $o,Q2: real > $o] :
      ( ? [Z5: real] :
        ! [X3: real] :
          ( ( ord_less_real @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( ord_less_real @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: real] :
          ! [X6: real] :
            ( ( ord_less_real @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P5 @ X6 )
                | ( Q2 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_5222_minf_I2_J,axiom,
    ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q2: rat > $o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: rat] :
          ! [X6: rat] :
            ( ( ord_less_rat @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P5 @ X6 )
                | ( Q2 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_5223_minf_I2_J,axiom,
    ! [P: num > $o,P5: num > $o,Q: num > $o,Q2: num > $o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( ( ord_less_num @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: num] :
          ! [X6: num] :
            ( ( ord_less_num @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P5 @ X6 )
                | ( Q2 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_5224_minf_I2_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P5 @ X6 )
                | ( Q2 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_5225_minf_I2_J,axiom,
    ! [P: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( ( ord_less_int @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P5 @ X3 ) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q2 @ X3 ) ) )
       => ? [Z3: int] :
          ! [X6: int] :
            ( ( ord_less_int @ X6 @ Z3 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P5 @ X6 )
                | ( Q2 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_5226_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(3)
thf(fact_5227_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(3)
thf(fact_5228_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(3)
thf(fact_5229_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(3)
thf(fact_5230_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(3)
thf(fact_5231_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(4)
thf(fact_5232_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(4)
thf(fact_5233_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(4)
thf(fact_5234_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(4)
thf(fact_5235_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z3 )
     => ( X6 != T ) ) ).

% minf(4)
thf(fact_5236_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z3 )
     => ( ord_less_real @ X6 @ T ) ) ).

% minf(5)
thf(fact_5237_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z3 )
     => ( ord_less_rat @ X6 @ T ) ) ).

% minf(5)
thf(fact_5238_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z3 )
     => ( ord_less_num @ X6 @ T ) ) ).

% minf(5)
thf(fact_5239_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ( ord_less_nat @ X6 @ T ) ) ).

% minf(5)
thf(fact_5240_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z3 )
     => ( ord_less_int @ X6 @ T ) ) ).

% minf(5)
thf(fact_5241_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z3 )
     => ~ ( ord_less_real @ T @ X6 ) ) ).

% minf(7)
thf(fact_5242_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z3 )
     => ~ ( ord_less_rat @ T @ X6 ) ) ).

% minf(7)
thf(fact_5243_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z3 )
     => ~ ( ord_less_num @ T @ X6 ) ) ).

% minf(7)
thf(fact_5244_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ~ ( ord_less_nat @ T @ X6 ) ) ).

% minf(7)
thf(fact_5245_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z3 )
     => ~ ( ord_less_int @ T @ X6 ) ) ).

% minf(7)
thf(fact_5246_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_5247_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_5248_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_5249_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_5250_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_5251_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_5252_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_5253_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_5254_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z3 @ X6 )
     => ~ ( ord_less_eq_real @ X6 @ T ) ) ).

% pinf(6)
thf(fact_5255_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z3 @ X6 )
     => ~ ( ord_less_eq_rat @ X6 @ T ) ) ).

% pinf(6)
thf(fact_5256_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z3 @ X6 )
     => ~ ( ord_less_eq_num @ X6 @ T ) ) ).

% pinf(6)
thf(fact_5257_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ~ ( ord_less_eq_nat @ X6 @ T ) ) ).

% pinf(6)
thf(fact_5258_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z3 @ X6 )
     => ~ ( ord_less_eq_int @ X6 @ T ) ) ).

% pinf(6)
thf(fact_5259_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z3 @ X6 )
     => ( ord_less_eq_real @ T @ X6 ) ) ).

% pinf(8)
thf(fact_5260_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z3 @ X6 )
     => ( ord_less_eq_rat @ T @ X6 ) ) ).

% pinf(8)
thf(fact_5261_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z3 @ X6 )
     => ( ord_less_eq_num @ T @ X6 ) ) ).

% pinf(8)
thf(fact_5262_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ( ord_less_eq_nat @ T @ X6 ) ) ).

% pinf(8)
thf(fact_5263_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z3 @ X6 )
     => ( ord_less_eq_int @ T @ X6 ) ) ).

% pinf(8)
thf(fact_5264_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z3 )
     => ( ord_less_eq_real @ X6 @ T ) ) ).

% minf(6)
thf(fact_5265_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z3 )
     => ( ord_less_eq_rat @ X6 @ T ) ) ).

% minf(6)
thf(fact_5266_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z3 )
     => ( ord_less_eq_num @ X6 @ T ) ) ).

% minf(6)
thf(fact_5267_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ( ord_less_eq_nat @ X6 @ T ) ) ).

% minf(6)
thf(fact_5268_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z3 )
     => ( ord_less_eq_int @ X6 @ T ) ) ).

% minf(6)
thf(fact_5269_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z3 )
     => ~ ( ord_less_eq_real @ T @ X6 ) ) ).

% minf(8)
thf(fact_5270_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z3 )
     => ~ ( ord_less_eq_rat @ T @ X6 ) ) ).

% minf(8)
thf(fact_5271_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z3 )
     => ~ ( ord_less_eq_num @ T @ X6 ) ) ).

% minf(8)
thf(fact_5272_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ~ ( ord_less_eq_nat @ T @ X6 ) ) ).

% minf(8)
thf(fact_5273_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z3 )
     => ~ ( ord_less_eq_int @ T @ X6 ) ) ).

% minf(8)
thf(fact_5274_imp__le__cong,axiom,
    ! [X: int,X7: int,P: $o,P5: $o] :
      ( ( X = X7 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
         => ( P = P5 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
           => P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
           => P5 ) ) ) ) ).

% imp_le_cong
thf(fact_5275_conj__le__cong,axiom,
    ! [X: int,X7: int,P: $o,P5: $o] :
      ( ( X = X7 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
         => ( P = P5 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
            & P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
            & P5 ) ) ) ) ).

% conj_le_cong
thf(fact_5276_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_5277_pinf_I9_J,axiom,
    ! [D: uint32,S2: uint32] :
    ? [Z3: uint32] :
    ! [X6: uint32] :
      ( ( ord_less_uint32 @ Z3 @ X6 )
     => ( ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ X6 @ S2 ) )
        = ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ X6 @ S2 ) ) ) ) ).

% pinf(9)
thf(fact_5278_pinf_I9_J,axiom,
    ! [D: word_N3645301735248828278l_num1,S2: word_N3645301735248828278l_num1] :
    ? [Z3: word_N3645301735248828278l_num1] :
    ! [X6: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ Z3 @ X6 )
     => ( ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ X6 @ S2 ) )
        = ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ X6 @ S2 ) ) ) ) ).

% pinf(9)
thf(fact_5279_pinf_I9_J,axiom,
    ! [D: code_integer,S2: code_integer] :
    ? [Z3: code_integer] :
    ! [X6: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z3 @ X6 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S2 ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S2 ) ) ) ) ).

% pinf(9)
thf(fact_5280_pinf_I9_J,axiom,
    ! [D: real,S2: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z3 @ X6 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S2 ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S2 ) ) ) ) ).

% pinf(9)
thf(fact_5281_pinf_I9_J,axiom,
    ! [D: rat,S2: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z3 @ X6 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S2 ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S2 ) ) ) ) ).

% pinf(9)
thf(fact_5282_pinf_I9_J,axiom,
    ! [D: nat,S2: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S2 ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S2 ) ) ) ) ).

% pinf(9)
thf(fact_5283_pinf_I9_J,axiom,
    ! [D: int,S2: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z3 @ X6 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S2 ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S2 ) ) ) ) ).

% pinf(9)
thf(fact_5284_pinf_I10_J,axiom,
    ! [D: uint32,S2: uint32] :
    ? [Z3: uint32] :
    ! [X6: uint32] :
      ( ( ord_less_uint32 @ Z3 @ X6 )
     => ( ( ~ ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ X6 @ S2 ) ) )
        = ( ~ ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ X6 @ S2 ) ) ) ) ) ).

% pinf(10)
thf(fact_5285_pinf_I10_J,axiom,
    ! [D: word_N3645301735248828278l_num1,S2: word_N3645301735248828278l_num1] :
    ? [Z3: word_N3645301735248828278l_num1] :
    ! [X6: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ Z3 @ X6 )
     => ( ( ~ ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ X6 @ S2 ) ) )
        = ( ~ ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ X6 @ S2 ) ) ) ) ) ).

% pinf(10)
thf(fact_5286_pinf_I10_J,axiom,
    ! [D: code_integer,S2: code_integer] :
    ? [Z3: code_integer] :
    ! [X6: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z3 @ X6 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S2 ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S2 ) ) ) ) ) ).

% pinf(10)
thf(fact_5287_pinf_I10_J,axiom,
    ! [D: real,S2: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z3 @ X6 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S2 ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S2 ) ) ) ) ) ).

% pinf(10)
thf(fact_5288_pinf_I10_J,axiom,
    ! [D: rat,S2: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z3 @ X6 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S2 ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S2 ) ) ) ) ) ).

% pinf(10)
thf(fact_5289_pinf_I10_J,axiom,
    ! [D: nat,S2: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z3 @ X6 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S2 ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S2 ) ) ) ) ) ).

% pinf(10)
thf(fact_5290_pinf_I10_J,axiom,
    ! [D: int,S2: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z3 @ X6 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S2 ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S2 ) ) ) ) ) ).

% pinf(10)
thf(fact_5291_minf_I9_J,axiom,
    ! [D: uint32,S2: uint32] :
    ? [Z3: uint32] :
    ! [X6: uint32] :
      ( ( ord_less_uint32 @ X6 @ Z3 )
     => ( ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ X6 @ S2 ) )
        = ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ X6 @ S2 ) ) ) ) ).

% minf(9)
thf(fact_5292_minf_I9_J,axiom,
    ! [D: word_N3645301735248828278l_num1,S2: word_N3645301735248828278l_num1] :
    ? [Z3: word_N3645301735248828278l_num1] :
    ! [X6: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ X6 @ Z3 )
     => ( ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ X6 @ S2 ) )
        = ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ X6 @ S2 ) ) ) ) ).

% minf(9)
thf(fact_5293_minf_I9_J,axiom,
    ! [D: code_integer,S2: code_integer] :
    ? [Z3: code_integer] :
    ! [X6: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X6 @ Z3 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S2 ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S2 ) ) ) ) ).

% minf(9)
thf(fact_5294_minf_I9_J,axiom,
    ! [D: real,S2: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z3 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S2 ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S2 ) ) ) ) ).

% minf(9)
thf(fact_5295_minf_I9_J,axiom,
    ! [D: rat,S2: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z3 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S2 ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S2 ) ) ) ) ).

% minf(9)
thf(fact_5296_minf_I9_J,axiom,
    ! [D: nat,S2: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S2 ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S2 ) ) ) ) ).

% minf(9)
thf(fact_5297_minf_I9_J,axiom,
    ! [D: int,S2: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z3 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S2 ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S2 ) ) ) ) ).

% minf(9)
thf(fact_5298_minf_I10_J,axiom,
    ! [D: uint32,S2: uint32] :
    ? [Z3: uint32] :
    ! [X6: uint32] :
      ( ( ord_less_uint32 @ X6 @ Z3 )
     => ( ( ~ ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ X6 @ S2 ) ) )
        = ( ~ ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ X6 @ S2 ) ) ) ) ) ).

% minf(10)
thf(fact_5299_minf_I10_J,axiom,
    ! [D: word_N3645301735248828278l_num1,S2: word_N3645301735248828278l_num1] :
    ? [Z3: word_N3645301735248828278l_num1] :
    ! [X6: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ X6 @ Z3 )
     => ( ( ~ ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ X6 @ S2 ) ) )
        = ( ~ ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ X6 @ S2 ) ) ) ) ) ).

% minf(10)
thf(fact_5300_minf_I10_J,axiom,
    ! [D: code_integer,S2: code_integer] :
    ? [Z3: code_integer] :
    ! [X6: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X6 @ Z3 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S2 ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S2 ) ) ) ) ) ).

% minf(10)
thf(fact_5301_minf_I10_J,axiom,
    ! [D: real,S2: real] :
    ? [Z3: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z3 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S2 ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S2 ) ) ) ) ) ).

% minf(10)
thf(fact_5302_minf_I10_J,axiom,
    ! [D: rat,S2: rat] :
    ? [Z3: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z3 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S2 ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S2 ) ) ) ) ) ).

% minf(10)
thf(fact_5303_minf_I10_J,axiom,
    ! [D: nat,S2: nat] :
    ? [Z3: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z3 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S2 ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S2 ) ) ) ) ) ).

% minf(10)
thf(fact_5304_minf_I10_J,axiom,
    ! [D: int,S2: int] :
    ? [Z3: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z3 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S2 ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S2 ) ) ) ) ) ).

% minf(10)
thf(fact_5305_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_5306_Bolzano,axiom,
    ! [A2: real,B2: real,P: real > real > $o] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ! [A5: real,B6: real,C4: real] :
            ( ( P @ A5 @ B6 )
           => ( ( P @ B6 @ C4 )
             => ( ( ord_less_eq_real @ A5 @ B6 )
               => ( ( ord_less_eq_real @ B6 @ C4 )
                 => ( P @ A5 @ C4 ) ) ) ) )
       => ( ! [X3: real] :
              ( ( ord_less_eq_real @ A2 @ X3 )
             => ( ( ord_less_eq_real @ X3 @ B2 )
               => ? [D5: real] :
                    ( ( ord_less_real @ zero_zero_real @ D5 )
                    & ! [A5: real,B6: real] :
                        ( ( ( ord_less_eq_real @ A5 @ X3 )
                          & ( ord_less_eq_real @ X3 @ B6 )
                          & ( ord_less_real @ ( minus_minus_real @ B6 @ A5 ) @ D5 ) )
                       => ( P @ A5 @ B6 ) ) ) ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% Bolzano
thf(fact_5307_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_5308_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_5309_even__word__def,axiom,
    ( even_w9054469088133485505l_num1
    = ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ).

% even_word_def
thf(fact_5310_arcosh__def,axiom,
    ( arcosh_real
    = ( ^ [X2: real] : ( ln_ln_real @ ( plus_plus_real @ X2 @ ( powr_real @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% arcosh_def
thf(fact_5311_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_5312_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_5313_mult__cancel__right,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( ( times_times_real @ A2 @ C )
        = ( times_times_real @ B2 @ C ) )
      = ( ( C = zero_zero_real )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
thf(fact_5314_mult__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ( times_times_rat @ A2 @ C )
        = ( times_times_rat @ B2 @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
thf(fact_5315_mult__cancel__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ( times_times_nat @ A2 @ C )
        = ( times_times_nat @ B2 @ C ) )
      = ( ( C = zero_zero_nat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
thf(fact_5316_mult__cancel__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ( times_times_int @ A2 @ C )
        = ( times_times_int @ B2 @ C ) )
      = ( ( C = zero_zero_int )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
thf(fact_5317_mult__cancel__left,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( ( times_times_real @ C @ A2 )
        = ( times_times_real @ C @ B2 ) )
      = ( ( C = zero_zero_real )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
thf(fact_5318_mult__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ( times_times_rat @ C @ A2 )
        = ( times_times_rat @ C @ B2 ) )
      = ( ( C = zero_zero_rat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
thf(fact_5319_mult__cancel__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ( times_times_nat @ C @ A2 )
        = ( times_times_nat @ C @ B2 ) )
      = ( ( C = zero_zero_nat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
thf(fact_5320_mult__cancel__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ( times_times_int @ C @ A2 )
        = ( times_times_int @ C @ B2 ) )
      = ( ( C = zero_zero_int )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
thf(fact_5321_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_5322_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_5323_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_5324_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_5325_mult__zero__right,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ A2 @ zero_z3563351764282998399l_num1 )
      = zero_z3563351764282998399l_num1 ) ).

% mult_zero_right
thf(fact_5326_mult__zero__right,axiom,
    ! [A2: real] :
      ( ( times_times_real @ A2 @ zero_zero_real )
      = zero_zero_real ) ).

% mult_zero_right
thf(fact_5327_mult__zero__right,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ A2 @ zero_zero_rat )
      = zero_zero_rat ) ).

% mult_zero_right
thf(fact_5328_mult__zero__right,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ A2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_5329_mult__zero__right,axiom,
    ! [A2: int] :
      ( ( times_times_int @ A2 @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_5330_mult__zero__left,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ zero_z3563351764282998399l_num1 @ A2 )
      = zero_z3563351764282998399l_num1 ) ).

% mult_zero_left
thf(fact_5331_mult__zero__left,axiom,
    ! [A2: real] :
      ( ( times_times_real @ zero_zero_real @ A2 )
      = zero_zero_real ) ).

% mult_zero_left
thf(fact_5332_mult__zero__left,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A2 )
      = zero_zero_rat ) ).

% mult_zero_left
thf(fact_5333_mult__zero__left,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_5334_mult__zero__left,axiom,
    ! [A2: int] :
      ( ( times_times_int @ zero_zero_int @ A2 )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_5335_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ M ) @ ( numera7442385471795722001l_num1 @ N ) )
      = ( numera7442385471795722001l_num1 @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_5336_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_5337_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_5338_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_5339_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_5340_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ V ) @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ W ) @ Z ) )
      = ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_5341_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_5342_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_5343_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_5344_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_5345_mult__1,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ one_on7727431528512463931l_num1 @ A2 )
      = A2 ) ).

% mult_1
thf(fact_5346_mult__1,axiom,
    ! [A2: real] :
      ( ( times_times_real @ one_one_real @ A2 )
      = A2 ) ).

% mult_1
thf(fact_5347_mult__1,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ one_one_rat @ A2 )
      = A2 ) ).

% mult_1
thf(fact_5348_mult__1,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ one_one_nat @ A2 )
      = A2 ) ).

% mult_1
thf(fact_5349_mult__1,axiom,
    ! [A2: int] :
      ( ( times_times_int @ one_one_int @ A2 )
      = A2 ) ).

% mult_1
thf(fact_5350_mult_Oright__neutral,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ A2 @ one_on7727431528512463931l_num1 )
      = A2 ) ).

% mult.right_neutral
thf(fact_5351_mult_Oright__neutral,axiom,
    ! [A2: real] :
      ( ( times_times_real @ A2 @ one_one_real )
      = A2 ) ).

% mult.right_neutral
thf(fact_5352_mult_Oright__neutral,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ A2 @ one_one_rat )
      = A2 ) ).

% mult.right_neutral
thf(fact_5353_mult_Oright__neutral,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ A2 @ one_one_nat )
      = A2 ) ).

% mult.right_neutral
thf(fact_5354_mult_Oright__neutral,axiom,
    ! [A2: int] :
      ( ( times_times_int @ A2 @ one_one_int )
      = A2 ) ).

% mult.right_neutral
thf(fact_5355_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_5356_mult__minus__left,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ A2 ) @ B2 )
      = ( uminus_uminus_uint32 @ ( times_times_uint32 @ A2 @ B2 ) ) ) ).

% mult_minus_left
thf(fact_5357_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_5358_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_5359_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_5360_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_5361_minus__mult__minus,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ A2 ) @ ( uminus_uminus_uint32 @ B2 ) )
      = ( times_times_uint32 @ A2 @ B2 ) ) ).

% minus_mult_minus
thf(fact_5362_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_5363_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_5364_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_5365_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_5366_mult__minus__right,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( times_times_uint32 @ A2 @ ( uminus_uminus_uint32 @ B2 ) )
      = ( uminus_uminus_uint32 @ ( times_times_uint32 @ A2 @ B2 ) ) ) ).

% mult_minus_right
thf(fact_5367_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_5368_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_5369_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_5370_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% of_nat_mult
thf(fact_5371_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_mult
thf(fact_5372_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% of_nat_mult
thf(fact_5373_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_mult
thf(fact_5374_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4939895301339042750nteger @ ( times_times_nat @ M @ N ) )
      = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% of_nat_mult
thf(fact_5375_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).

% of_nat_mult
thf(fact_5376_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_5377_of__int__mult,axiom,
    ! [W: int,Z: int] :
      ( ( ring_17408606157368542149l_num1 @ ( times_times_int @ W @ Z ) )
      = ( times_7065122842183080059l_num1 @ ( ring_17408606157368542149l_num1 @ W ) @ ( ring_17408606157368542149l_num1 @ Z ) ) ) ).

% of_int_mult
thf(fact_5378_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_5379_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_5380_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_5381_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_5382_tanh__0,axiom,
    ( ( tanh_real @ zero_zero_real )
    = zero_zero_real ) ).

% tanh_0
thf(fact_5383_tanh__real__zero__iff,axiom,
    ! [X: real] :
      ( ( ( tanh_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% tanh_real_zero_iff
thf(fact_5384_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_5385_round__of__int,axiom,
    ! [N: int] :
      ( ( archim8280529875227126926d_real @ ( ring_1_of_int_real @ N ) )
      = N ) ).

% round_of_int
thf(fact_5386_round__of__int,axiom,
    ! [N: int] :
      ( ( archim7778729529865785530nd_rat @ ( ring_1_of_int_rat @ N ) )
      = N ) ).

% round_of_int
thf(fact_5387_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_5388_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_5389_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_5390_mult__cancel__right2,axiom,
    ! [A2: real,C: real] :
      ( ( ( times_times_real @ A2 @ C )
        = C )
      = ( ( C = zero_zero_real )
        | ( A2 = one_one_real ) ) ) ).

% mult_cancel_right2
thf(fact_5391_mult__cancel__right2,axiom,
    ! [A2: rat,C: rat] :
      ( ( ( times_times_rat @ A2 @ C )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A2 = one_one_rat ) ) ) ).

% mult_cancel_right2
thf(fact_5392_mult__cancel__right2,axiom,
    ! [A2: int,C: int] :
      ( ( ( times_times_int @ A2 @ C )
        = C )
      = ( ( C = zero_zero_int )
        | ( A2 = one_one_int ) ) ) ).

% mult_cancel_right2
thf(fact_5393_mult__cancel__right1,axiom,
    ! [C: real,B2: real] :
      ( ( C
        = ( times_times_real @ B2 @ C ) )
      = ( ( C = zero_zero_real )
        | ( B2 = one_one_real ) ) ) ).

% mult_cancel_right1
thf(fact_5394_mult__cancel__right1,axiom,
    ! [C: rat,B2: rat] :
      ( ( C
        = ( times_times_rat @ B2 @ C ) )
      = ( ( C = zero_zero_rat )
        | ( B2 = one_one_rat ) ) ) ).

% mult_cancel_right1
thf(fact_5395_mult__cancel__right1,axiom,
    ! [C: int,B2: int] :
      ( ( C
        = ( times_times_int @ B2 @ C ) )
      = ( ( C = zero_zero_int )
        | ( B2 = one_one_int ) ) ) ).

% mult_cancel_right1
thf(fact_5396_mult__cancel__left2,axiom,
    ! [C: real,A2: real] :
      ( ( ( times_times_real @ C @ A2 )
        = C )
      = ( ( C = zero_zero_real )
        | ( A2 = one_one_real ) ) ) ).

% mult_cancel_left2
thf(fact_5397_mult__cancel__left2,axiom,
    ! [C: rat,A2: rat] :
      ( ( ( times_times_rat @ C @ A2 )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A2 = one_one_rat ) ) ) ).

% mult_cancel_left2
thf(fact_5398_mult__cancel__left2,axiom,
    ! [C: int,A2: int] :
      ( ( ( times_times_int @ C @ A2 )
        = C )
      = ( ( C = zero_zero_int )
        | ( A2 = one_one_int ) ) ) ).

% mult_cancel_left2
thf(fact_5399_mult__cancel__left1,axiom,
    ! [C: real,B2: real] :
      ( ( C
        = ( times_times_real @ C @ B2 ) )
      = ( ( C = zero_zero_real )
        | ( B2 = one_one_real ) ) ) ).

% mult_cancel_left1
thf(fact_5400_mult__cancel__left1,axiom,
    ! [C: rat,B2: rat] :
      ( ( C
        = ( times_times_rat @ C @ B2 ) )
      = ( ( C = zero_zero_rat )
        | ( B2 = one_one_rat ) ) ) ).

% mult_cancel_left1
thf(fact_5401_mult__cancel__left1,axiom,
    ! [C: int,B2: int] :
      ( ( C
        = ( times_times_int @ C @ B2 ) )
      = ( ( C = zero_zero_int )
        | ( B2 = one_one_int ) ) ) ).

% mult_cancel_left1
thf(fact_5402_distrib__right__numeral,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1,V: num] :
      ( ( times_7065122842183080059l_num1 @ ( plus_p361126936061061375l_num1 @ A2 @ B2 ) @ ( numera7442385471795722001l_num1 @ V ) )
      = ( plus_p361126936061061375l_num1 @ ( times_7065122842183080059l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ V ) ) @ ( times_7065122842183080059l_num1 @ B2 @ ( numera7442385471795722001l_num1 @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_5403_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_5404_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_5405_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_5406_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_5407_distrib__left__numeral,axiom,
    ! [V: num,B2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ V ) @ ( plus_p361126936061061375l_num1 @ B2 @ C ) )
      = ( plus_p361126936061061375l_num1 @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ V ) @ B2 ) @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_5408_distrib__left__numeral,axiom,
    ! [V: num,B2: real,C: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B2 @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B2 ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_5409_distrib__left__numeral,axiom,
    ! [V: num,B2: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B2 @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B2 ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_5410_distrib__left__numeral,axiom,
    ! [V: num,B2: nat,C: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B2 @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_5411_distrib__left__numeral,axiom,
    ! [V: num,B2: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B2 @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_5412_div__mult__mult1__if,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ( C = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) )
          = zero_zero_nat ) )
      & ( ( C != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) )
          = ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).

% div_mult_mult1_if
thf(fact_5413_div__mult__mult1__if,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ( C = zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
          = zero_zero_int ) )
      & ( ( C != zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
          = ( divide_divide_int @ A2 @ B2 ) ) ) ) ).

% div_mult_mult1_if
thf(fact_5414_div__mult__mult2,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) )
        = ( divide_divide_nat @ A2 @ B2 ) ) ) ).

% div_mult_mult2
thf(fact_5415_div__mult__mult2,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) )
        = ( divide_divide_int @ A2 @ B2 ) ) ) ).

% div_mult_mult2
thf(fact_5416_div__mult__mult1,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) )
        = ( divide_divide_nat @ A2 @ B2 ) ) ) ).

% div_mult_mult1
thf(fact_5417_div__mult__mult1,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
        = ( divide_divide_int @ A2 @ B2 ) ) ) ).

% div_mult_mult1
thf(fact_5418_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( ( C = zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) )
          = zero_zero_real ) )
      & ( ( C != zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) )
          = ( divide_divide_real @ A2 @ B2 ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_5419_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ( C = zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
          = zero_zero_rat ) )
      & ( ( C != zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
          = ( divide_divide_rat @ A2 @ B2 ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_5420_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) )
        = ( divide_divide_real @ A2 @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_5421_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
        = ( divide_divide_rat @ A2 @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_5422_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ B2 @ C ) )
        = ( divide_divide_real @ A2 @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_5423_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ B2 @ C ) )
        = ( divide_divide_rat @ A2 @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_5424_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ C ) )
        = ( divide_divide_real @ A2 @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_5425_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) )
        = ( divide_divide_rat @ A2 @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_5426_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ C @ B2 ) )
        = ( divide_divide_real @ A2 @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_5427_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ C @ B2 ) )
        = ( divide_divide_rat @ A2 @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_5428_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_5429_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_5430_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_5431_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_5432_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_5433_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_5434_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_5435_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_5436_left__diff__distrib__numeral,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1,V: num] :
      ( ( times_7065122842183080059l_num1 @ ( minus_4019991460397169231l_num1 @ A2 @ B2 ) @ ( numera7442385471795722001l_num1 @ V ) )
      = ( minus_4019991460397169231l_num1 @ ( times_7065122842183080059l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ V ) ) @ ( times_7065122842183080059l_num1 @ B2 @ ( numera7442385471795722001l_num1 @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_5437_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_5438_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_5439_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_5440_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ V ) @ ( minus_4019991460397169231l_num1 @ B2 @ C ) )
      = ( minus_4019991460397169231l_num1 @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ V ) @ B2 ) @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_5441_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: real,C: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B2 @ C ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B2 ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_5442_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B2 @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B2 ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_5443_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_5444_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_7065122842183080059l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ M ) ) @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ N ) ) )
      = ( numera7442385471795722001l_num1 @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_5445_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_5446_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ M ) ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ N ) ) )
      = ( numera9087168376688890119uint32 @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_5447_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_5448_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_5449_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_5450_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_7065122842183080059l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ M ) ) @ ( numera7442385471795722001l_num1 @ N ) )
      = ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_5451_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_5452_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ M ) ) @ ( numera9087168376688890119uint32 @ N ) )
      = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_5453_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_5454_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_5455_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_5456_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ M ) @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ N ) ) )
      = ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_5457_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_5458_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_uint32 @ ( numera9087168376688890119uint32 @ M ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ N ) ) )
      = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_5459_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_5460_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_5461_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_5462_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ V ) ) @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ W ) @ Y ) )
      = ( times_7065122842183080059l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_5463_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_5464_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: uint32] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ W ) @ Y ) )
      = ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_5465_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_5466_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_5467_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_5468_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ V ) @ ( times_7065122842183080059l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ W ) ) @ Y ) )
      = ( times_7065122842183080059l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_5469_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_5470_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: uint32] :
      ( ( times_times_uint32 @ ( numera9087168376688890119uint32 @ V ) @ ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ W ) ) @ Y ) )
      = ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_5471_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_5472_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_5473_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_5474_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ V ) ) @ ( times_7065122842183080059l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ W ) ) @ Y ) )
      = ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_5475_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_5476_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: uint32] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) @ ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ W ) ) @ Y ) )
      = ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_5477_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_5478_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_5479_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_5480_mult__minus1,axiom,
    ! [Z: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ Z )
      = ( uminus8244633308260627903l_num1 @ Z ) ) ).

% mult_minus1
thf(fact_5481_mult__minus1,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1
thf(fact_5482_mult__minus1,axiom,
    ! [Z: uint32] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ Z )
      = ( uminus_uminus_uint32 @ Z ) ) ).

% mult_minus1
thf(fact_5483_mult__minus1,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
      = ( uminus_uminus_real @ Z ) ) ).

% mult_minus1
thf(fact_5484_mult__minus1,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1
thf(fact_5485_mult__minus1,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1
thf(fact_5486_mult__minus1__right,axiom,
    ! [Z: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ Z @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
      = ( uminus8244633308260627903l_num1 @ Z ) ) ).

% mult_minus1_right
thf(fact_5487_mult__minus1__right,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1_right
thf(fact_5488_mult__minus1__right,axiom,
    ! [Z: uint32] :
      ( ( times_times_uint32 @ Z @ ( uminus_uminus_uint32 @ one_one_uint32 ) )
      = ( uminus_uminus_uint32 @ Z ) ) ).

% mult_minus1_right
thf(fact_5489_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_5490_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_5491_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_5492_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A2 ) @ ( times_3573771949741848930nteger @ C @ B2 ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( dvd_dvd_Code_integer @ A2 @ B2 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_5493_dvd__mult__cancel__left,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) )
      = ( ( C = zero_zero_real )
        | ( dvd_dvd_real @ A2 @ B2 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_5494_dvd__mult__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
      = ( ( C = zero_zero_rat )
        | ( dvd_dvd_rat @ A2 @ B2 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_5495_dvd__mult__cancel__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A2 @ B2 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_5496_dvd__mult__cancel__right,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B2 @ C ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( dvd_dvd_Code_integer @ A2 @ B2 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_5497_dvd__mult__cancel__right,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ C ) )
      = ( ( C = zero_zero_real )
        | ( dvd_dvd_real @ A2 @ B2 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_5498_dvd__mult__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) )
      = ( ( C = zero_zero_rat )
        | ( dvd_dvd_rat @ A2 @ B2 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_5499_dvd__mult__cancel__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A2 @ B2 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_5500_dvd__times__left__cancel__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ ( times_3573771949741848930nteger @ A2 @ C ) )
        = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_5501_dvd__times__left__cancel__iff,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_nat @ A2 @ C ) )
        = ( dvd_dvd_nat @ B2 @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_5502_dvd__times__left__cancel__iff,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( A2 != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C ) )
        = ( dvd_dvd_int @ B2 @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_5503_dvd__times__right__cancel__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B2 @ A2 ) @ ( times_3573771949741848930nteger @ C @ A2 ) )
        = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_5504_dvd__times__right__cancel__iff,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ A2 ) @ ( times_times_nat @ C @ A2 ) )
        = ( dvd_dvd_nat @ B2 @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_5505_dvd__times__right__cancel__iff,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( A2 != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ B2 @ A2 ) @ ( times_times_int @ C @ A2 ) )
        = ( dvd_dvd_int @ B2 @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_5506_dvd__add__times__triv__right__iff,axiom,
    ! [A2: uint32,B2: uint32,C: uint32] :
      ( ( dvd_dvd_uint32 @ A2 @ ( plus_plus_uint32 @ B2 @ ( times_times_uint32 @ C @ A2 ) ) )
      = ( dvd_dvd_uint32 @ A2 @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_5507_dvd__add__times__triv__right__iff,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A2 @ ( plus_p361126936061061375l_num1 @ B2 @ ( times_7065122842183080059l_num1 @ C @ A2 ) ) )
      = ( dvd_dv6812691276156420380l_num1 @ A2 @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_5508_dvd__add__times__triv__right__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ ( plus_p5714425477246183910nteger @ B2 @ ( times_3573771949741848930nteger @ C @ A2 ) ) )
      = ( dvd_dvd_Code_integer @ A2 @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_5509_dvd__add__times__triv__right__iff,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( dvd_dvd_real @ A2 @ ( plus_plus_real @ B2 @ ( times_times_real @ C @ A2 ) ) )
      = ( dvd_dvd_real @ A2 @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_5510_dvd__add__times__triv__right__iff,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( dvd_dvd_rat @ A2 @ ( plus_plus_rat @ B2 @ ( times_times_rat @ C @ A2 ) ) )
      = ( dvd_dvd_rat @ A2 @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_5511_dvd__add__times__triv__right__iff,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B2 @ ( times_times_nat @ C @ A2 ) ) )
      = ( dvd_dvd_nat @ A2 @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_5512_dvd__add__times__triv__right__iff,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B2 @ ( times_times_int @ C @ A2 ) ) )
      = ( dvd_dvd_int @ A2 @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_5513_dvd__add__times__triv__left__iff,axiom,
    ! [A2: uint32,C: uint32,B2: uint32] :
      ( ( dvd_dvd_uint32 @ A2 @ ( plus_plus_uint32 @ ( times_times_uint32 @ C @ A2 ) @ B2 ) )
      = ( dvd_dvd_uint32 @ A2 @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_5514_dvd__add__times__triv__left__iff,axiom,
    ! [A2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A2 @ ( plus_p361126936061061375l_num1 @ ( times_7065122842183080059l_num1 @ C @ A2 ) @ B2 ) )
      = ( dvd_dv6812691276156420380l_num1 @ A2 @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_5515_dvd__add__times__triv__left__iff,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A2 ) @ B2 ) )
      = ( dvd_dvd_Code_integer @ A2 @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_5516_dvd__add__times__triv__left__iff,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( dvd_dvd_real @ A2 @ ( plus_plus_real @ ( times_times_real @ C @ A2 ) @ B2 ) )
      = ( dvd_dvd_real @ A2 @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_5517_dvd__add__times__triv__left__iff,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( dvd_dvd_rat @ A2 @ ( plus_plus_rat @ ( times_times_rat @ C @ A2 ) @ B2 ) )
      = ( dvd_dvd_rat @ A2 @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_5518_dvd__add__times__triv__left__iff,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ ( times_times_nat @ C @ A2 ) @ B2 ) )
      = ( dvd_dvd_nat @ A2 @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_5519_dvd__add__times__triv__left__iff,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ ( times_times_int @ C @ A2 ) @ B2 ) )
      = ( dvd_dvd_int @ A2 @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_5520_unit__prod,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ one_one_Code_integer ) ) ) ).

% unit_prod
thf(fact_5521_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_5522_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_5523_dvd__div__mult__self,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ B2 )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B2 @ A2 ) @ A2 )
        = B2 ) ) ).

% dvd_div_mult_self
thf(fact_5524_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_5525_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_5526_dvd__mult__div__cancel,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ B2 )
     => ( ( times_3573771949741848930nteger @ A2 @ ( divide6298287555418463151nteger @ B2 @ A2 ) )
        = B2 ) ) ).

% dvd_mult_div_cancel
thf(fact_5527_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_5528_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_5529_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_5530_round__0,axiom,
    ( ( archim8280529875227126926d_real @ zero_zero_real )
    = zero_zero_int ) ).

% round_0
thf(fact_5531_round__0,axiom,
    ( ( archim7778729529865785530nd_rat @ zero_zero_rat )
    = zero_zero_int ) ).

% round_0
thf(fact_5532_round__numeral,axiom,
    ! [N: num] :
      ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% round_numeral
thf(fact_5533_round__numeral,axiom,
    ! [N: num] :
      ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% round_numeral
thf(fact_5534_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_5535_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_5536_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_5537_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_5538_round__1,axiom,
    ( ( archim8280529875227126926d_real @ one_one_real )
    = one_one_int ) ).

% round_1
thf(fact_5539_round__1,axiom,
    ( ( archim7778729529865785530nd_rat @ one_one_rat )
    = one_one_int ) ).

% round_1
thf(fact_5540_round__of__nat,axiom,
    ! [N: nat] :
      ( ( archim8280529875227126926d_real @ ( semiri5074537144036343181t_real @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% round_of_nat
thf(fact_5541_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_5542_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_5543_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_5544_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_5545_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_5546_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_5547_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_5548_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_5549_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_5550_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_5551_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_5552_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_5553_div__mult__self4,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ C ) @ A2 ) @ B2 )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).

% div_mult_self4
thf(fact_5554_div__mult__self4,axiom,
    ! [B2: int,C: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B2 @ C ) @ A2 ) @ B2 )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).

% div_mult_self4
thf(fact_5555_div__mult__self3,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B2 ) @ A2 ) @ B2 )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).

% div_mult_self3
thf(fact_5556_div__mult__self3,axiom,
    ! [B2: int,C: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B2 ) @ A2 ) @ B2 )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).

% div_mult_self3
thf(fact_5557_div__mult__self2,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) @ B2 )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).

% div_mult_self2
thf(fact_5558_div__mult__self2,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ ( times_times_int @ B2 @ C ) ) @ B2 )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).

% div_mult_self2
thf(fact_5559_div__mult__self1,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ C @ B2 ) ) @ B2 )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).

% div_mult_self1
thf(fact_5560_div__mult__self1,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ ( times_times_int @ C @ B2 ) ) @ B2 )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).

% div_mult_self1
thf(fact_5561_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_5562_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_5563_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_5564_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_5565_left__minus__one__mult__self,axiom,
    ! [N: nat,A2: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( power_2184487114949457152l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ N ) @ ( times_7065122842183080059l_num1 @ ( power_2184487114949457152l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ N ) @ A2 ) )
      = A2 ) ).

% left_minus_one_mult_self
thf(fact_5566_left__minus__one__mult__self,axiom,
    ! [N: nat,A2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A2 ) )
      = A2 ) ).

% left_minus_one_mult_self
thf(fact_5567_left__minus__one__mult__self,axiom,
    ! [N: nat,A2: complex] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A2 ) )
      = A2 ) ).

% left_minus_one_mult_self
thf(fact_5568_left__minus__one__mult__self,axiom,
    ! [N: nat,A2: uint32] :
      ( ( times_times_uint32 @ ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N ) @ ( times_times_uint32 @ ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N ) @ A2 ) )
      = A2 ) ).

% left_minus_one_mult_self
thf(fact_5569_left__minus__one__mult__self,axiom,
    ! [N: nat,A2: real] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ A2 ) )
      = A2 ) ).

% left_minus_one_mult_self
thf(fact_5570_left__minus__one__mult__self,axiom,
    ! [N: nat,A2: rat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ A2 ) )
      = A2 ) ).

% left_minus_one_mult_self
thf(fact_5571_left__minus__one__mult__self,axiom,
    ! [N: nat,A2: int] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A2 ) )
      = A2 ) ).

% left_minus_one_mult_self
thf(fact_5572_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_7065122842183080059l_num1 @ ( power_2184487114949457152l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ N ) @ ( power_2184487114949457152l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ N ) )
      = one_on7727431528512463931l_num1 ) ).

% minus_one_mult_self
thf(fact_5573_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) )
      = one_one_Code_integer ) ).

% minus_one_mult_self
thf(fact_5574_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
      = one_one_complex ) ).

% minus_one_mult_self
thf(fact_5575_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_uint32 @ ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N ) @ ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N ) )
      = one_one_uint32 ) ).

% minus_one_mult_self
thf(fact_5576_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
      = one_one_real ) ).

% minus_one_mult_self
thf(fact_5577_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) )
      = one_one_rat ) ).

% minus_one_mult_self
thf(fact_5578_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
      = one_one_int ) ).

% minus_one_mult_self
thf(fact_5579_unit__div__mult__self,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B2 @ A2 ) @ A2 )
        = B2 ) ) ).

% unit_div_mult_self
thf(fact_5580_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_5581_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_5582_unit__mult__div__div,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A2 ) )
        = ( divide6298287555418463151nteger @ B2 @ A2 ) ) ) ).

% unit_mult_div_div
thf(fact_5583_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_5584_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_5585_power__add__numeral,axiom,
    ! [A2: complex,M: num,N: num] :
      ( ( times_times_complex @ ( power_power_complex @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A2 @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_complex @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_5586_power__add__numeral,axiom,
    ! [A2: code_integer,M: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ N ) ) )
      = ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_5587_power__add__numeral,axiom,
    ! [A2: real,M: num,N: num] :
      ( ( times_times_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_5588_power__add__numeral,axiom,
    ! [A2: rat,M: num,N: num] :
      ( ( times_times_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_5589_power__add__numeral,axiom,
    ! [A2: nat,M: num,N: num] :
      ( ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_5590_power__add__numeral,axiom,
    ! [A2: int,M: num,N: num] :
      ( ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_5591_power__add__numeral2,axiom,
    ! [A2: complex,M: num,N: num,B2: complex] :
      ( ( times_times_complex @ ( power_power_complex @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A2 @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
      = ( times_times_complex @ ( power_power_complex @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_5592_power__add__numeral2,axiom,
    ! [A2: code_integer,M: num,N: num,B2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_5593_power__add__numeral2,axiom,
    ! [A2: real,M: num,N: num,B2: real] :
      ( ( times_times_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
      = ( times_times_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_5594_power__add__numeral2,axiom,
    ! [A2: rat,M: num,N: num,B2: rat] :
      ( ( times_times_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
      = ( times_times_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_5595_power__add__numeral2,axiom,
    ! [A2: nat,M: num,N: num,B2: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
      = ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_5596_power__add__numeral2,axiom,
    ! [A2: int,M: num,N: num,B2: int] :
      ( ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
      = ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_5597_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_5598_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_5599_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_5600_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_5601_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_5602_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_5603_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_5604_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_5605_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_5606_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_5607_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_5608_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_5609_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_5610_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_5611_even__mult__iff,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( times_times_uint32 @ A2 @ B2 ) )
      = ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A2 )
        | ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_mult_iff
thf(fact_5612_even__mult__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A2 @ B2 ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
        | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_mult_iff
thf(fact_5613_even__mult__iff,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( times_7065122842183080059l_num1 @ A2 @ B2 ) )
      = ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A2 )
        | ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_mult_iff
thf(fact_5614_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_5615_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_5616_round__neg__numeral,axiom,
    ! [N: num] :
      ( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% round_neg_numeral
thf(fact_5617_round__neg__numeral,axiom,
    ! [N: num] :
      ( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% round_neg_numeral
thf(fact_5618_odd__two__times__div__two__succ,axiom,
    ! [A2: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
     => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
        = A2 ) ) ).

% odd_two_times_div_two_succ
thf(fact_5619_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_5620_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_5621_mult__commute__abs,axiom,
    ! [C: real] :
      ( ( ^ [X2: real] : ( times_times_real @ X2 @ C ) )
      = ( times_times_real @ C ) ) ).

% mult_commute_abs
thf(fact_5622_mult__commute__abs,axiom,
    ! [C: rat] :
      ( ( ^ [X2: rat] : ( times_times_rat @ X2 @ C ) )
      = ( times_times_rat @ C ) ) ).

% mult_commute_abs
thf(fact_5623_mult__commute__abs,axiom,
    ! [C: nat] :
      ( ( ^ [X2: nat] : ( times_times_nat @ X2 @ C ) )
      = ( times_times_nat @ C ) ) ).

% mult_commute_abs
thf(fact_5624_mult__commute__abs,axiom,
    ! [C: int] :
      ( ( ^ [X2: int] : ( times_times_int @ X2 @ C ) )
      = ( times_times_int @ C ) ) ).

% mult_commute_abs
thf(fact_5625_mult_Oassoc,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A2 @ B2 ) @ C )
      = ( times_times_real @ A2 @ ( times_times_real @ B2 @ C ) ) ) ).

% mult.assoc
thf(fact_5626_mult_Oassoc,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A2 @ B2 ) @ C )
      = ( times_times_rat @ A2 @ ( times_times_rat @ B2 @ C ) ) ) ).

% mult.assoc
thf(fact_5627_mult_Oassoc,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
      = ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).

% mult.assoc
thf(fact_5628_mult_Oassoc,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A2 @ B2 ) @ C )
      = ( times_times_int @ A2 @ ( times_times_int @ B2 @ C ) ) ) ).

% mult.assoc
thf(fact_5629_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_real
    = ( ^ [A4: real,B4: real] : ( times_times_real @ B4 @ A4 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_5630_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_rat
    = ( ^ [A4: rat,B4: rat] : ( times_times_rat @ B4 @ A4 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_5631_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A4: nat,B4: nat] : ( times_times_nat @ B4 @ A4 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_5632_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A4: int,B4: int] : ( times_times_int @ B4 @ A4 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_5633_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: real,A2: real,C: real] :
      ( ( times_times_real @ B2 @ ( times_times_real @ A2 @ C ) )
      = ( times_times_real @ A2 @ ( times_times_real @ B2 @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_5634_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( times_times_rat @ B2 @ ( times_times_rat @ A2 @ C ) )
      = ( times_times_rat @ A2 @ ( times_times_rat @ B2 @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_5635_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( times_times_nat @ B2 @ ( times_times_nat @ A2 @ C ) )
      = ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_5636_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( times_times_int @ B2 @ ( times_times_int @ A2 @ C ) )
      = ( times_times_int @ A2 @ ( times_times_int @ B2 @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_5637_mult__right__cancel,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A2 @ C )
          = ( times_times_real @ B2 @ C ) )
        = ( A2 = B2 ) ) ) ).

% mult_right_cancel
thf(fact_5638_mult__right__cancel,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A2 @ C )
          = ( times_times_rat @ B2 @ C ) )
        = ( A2 = B2 ) ) ) ).

% mult_right_cancel
thf(fact_5639_mult__right__cancel,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ A2 @ C )
          = ( times_times_nat @ B2 @ C ) )
        = ( A2 = B2 ) ) ) ).

% mult_right_cancel
thf(fact_5640_mult__right__cancel,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ A2 @ C )
          = ( times_times_int @ B2 @ C ) )
        = ( A2 = B2 ) ) ) ).

% mult_right_cancel
thf(fact_5641_mult__left__cancel,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ C @ A2 )
          = ( times_times_real @ C @ B2 ) )
        = ( A2 = B2 ) ) ) ).

% mult_left_cancel
thf(fact_5642_mult__left__cancel,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ C @ A2 )
          = ( times_times_rat @ C @ B2 ) )
        = ( A2 = B2 ) ) ) ).

% mult_left_cancel
thf(fact_5643_mult__left__cancel,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ C @ A2 )
          = ( times_times_nat @ C @ B2 ) )
        = ( A2 = B2 ) ) ) ).

% mult_left_cancel
thf(fact_5644_mult__left__cancel,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ C @ A2 )
          = ( times_times_int @ C @ B2 ) )
        = ( A2 = B2 ) ) ) ).

% mult_left_cancel
thf(fact_5645_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_5646_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_5647_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_5648_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_5649_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_5650_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_5651_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_5652_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_5653_mult__not__zero,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( ( times_7065122842183080059l_num1 @ A2 @ B2 )
       != zero_z3563351764282998399l_num1 )
     => ( ( A2 != zero_z3563351764282998399l_num1 )
        & ( B2 != zero_z3563351764282998399l_num1 ) ) ) ).

% mult_not_zero
thf(fact_5654_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_5655_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_5656_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_5657_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_5658_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A2 @ B2 ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_5659_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_5660_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_5661_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( times_times_real @ A2 @ ( plus_plus_real @ B2 @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A2 @ B2 ) @ ( times_times_real @ A2 @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_5662_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ A2 @ ( plus_plus_rat @ B2 @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A2 @ B2 ) @ ( times_times_rat @ A2 @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_5663_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( times_times_int @ A2 @ ( plus_plus_int @ B2 @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_5664_comm__semiring__class_Odistrib,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A2 @ B2 ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_5665_comm__semiring__class_Odistrib,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_5666_comm__semiring__class_Odistrib,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_5667_comm__semiring__class_Odistrib,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_5668_distrib__left,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( times_times_real @ A2 @ ( plus_plus_real @ B2 @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A2 @ B2 ) @ ( times_times_real @ A2 @ C ) ) ) ).

% distrib_left
thf(fact_5669_distrib__left,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ A2 @ ( plus_plus_rat @ B2 @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A2 @ B2 ) @ ( times_times_rat @ A2 @ C ) ) ) ).

% distrib_left
thf(fact_5670_distrib__left,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_nat @ A2 @ C ) ) ) ).

% distrib_left
thf(fact_5671_distrib__left,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( times_times_int @ A2 @ ( plus_plus_int @ B2 @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C ) ) ) ).

% distrib_left
thf(fact_5672_distrib__right,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A2 @ B2 ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).

% distrib_right
thf(fact_5673_distrib__right,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).

% distrib_right
thf(fact_5674_distrib__right,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).

% distrib_right
thf(fact_5675_distrib__right,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).

% distrib_right
thf(fact_5676_combine__common__factor,axiom,
    ! [A2: real,E2: real,B2: real,C: real] :
      ( ( plus_plus_real @ ( times_times_real @ A2 @ E2 ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E2 ) @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A2 @ B2 ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_5677_combine__common__factor,axiom,
    ! [A2: rat,E2: rat,B2: rat,C: rat] :
      ( ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E2 ) @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A2 @ B2 ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_5678_combine__common__factor,axiom,
    ! [A2: nat,E2: nat,B2: nat,C: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A2 @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B2 @ E2 ) @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_5679_combine__common__factor,axiom,
    ! [A2: int,E2: int,B2: int,C: int] :
      ( ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E2 ) @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_5680_mult_Ocomm__neutral,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ A2 @ one_on7727431528512463931l_num1 )
      = A2 ) ).

% mult.comm_neutral
thf(fact_5681_mult_Ocomm__neutral,axiom,
    ! [A2: real] :
      ( ( times_times_real @ A2 @ one_one_real )
      = A2 ) ).

% mult.comm_neutral
thf(fact_5682_mult_Ocomm__neutral,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ A2 @ one_one_rat )
      = A2 ) ).

% mult.comm_neutral
thf(fact_5683_mult_Ocomm__neutral,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ A2 @ one_one_nat )
      = A2 ) ).

% mult.comm_neutral
thf(fact_5684_mult_Ocomm__neutral,axiom,
    ! [A2: int] :
      ( ( times_times_int @ A2 @ one_one_int )
      = A2 ) ).

% mult.comm_neutral
thf(fact_5685_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ one_on7727431528512463931l_num1 @ A2 )
      = A2 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_5686_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_5687_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_5688_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_5689_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_5690_left__diff__distrib,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( times_times_real @ ( minus_minus_real @ A2 @ B2 ) @ C )
      = ( minus_minus_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).

% left_diff_distrib
thf(fact_5691_left__diff__distrib,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C )
      = ( minus_minus_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).

% left_diff_distrib
thf(fact_5692_left__diff__distrib,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
      = ( minus_minus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).

% left_diff_distrib
thf(fact_5693_right__diff__distrib,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( times_times_real @ A2 @ ( minus_minus_real @ B2 @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A2 @ B2 ) @ ( times_times_real @ A2 @ C ) ) ) ).

% right_diff_distrib
thf(fact_5694_right__diff__distrib,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ A2 @ ( minus_minus_rat @ B2 @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A2 @ B2 ) @ ( times_times_rat @ A2 @ C ) ) ) ).

% right_diff_distrib
thf(fact_5695_right__diff__distrib,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( times_times_int @ A2 @ ( minus_minus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C ) ) ) ).

% right_diff_distrib
thf(fact_5696_left__diff__distrib_H,axiom,
    ! [B2: real,C: real,A2: real] :
      ( ( times_times_real @ ( minus_minus_real @ B2 @ C ) @ A2 )
      = ( minus_minus_real @ ( times_times_real @ B2 @ A2 ) @ ( times_times_real @ C @ A2 ) ) ) ).

% left_diff_distrib'
thf(fact_5697_left__diff__distrib_H,axiom,
    ! [B2: rat,C: rat,A2: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ B2 @ C ) @ A2 )
      = ( minus_minus_rat @ ( times_times_rat @ B2 @ A2 ) @ ( times_times_rat @ C @ A2 ) ) ) ).

% left_diff_distrib'
thf(fact_5698_left__diff__distrib_H,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ B2 @ C ) @ A2 )
      = ( minus_minus_nat @ ( times_times_nat @ B2 @ A2 ) @ ( times_times_nat @ C @ A2 ) ) ) ).

% left_diff_distrib'
thf(fact_5699_left__diff__distrib_H,axiom,
    ! [B2: int,C: int,A2: int] :
      ( ( times_times_int @ ( minus_minus_int @ B2 @ C ) @ A2 )
      = ( minus_minus_int @ ( times_times_int @ B2 @ A2 ) @ ( times_times_int @ C @ A2 ) ) ) ).

% left_diff_distrib'
thf(fact_5700_right__diff__distrib_H,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( times_times_real @ A2 @ ( minus_minus_real @ B2 @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A2 @ B2 ) @ ( times_times_real @ A2 @ C ) ) ) ).

% right_diff_distrib'
thf(fact_5701_right__diff__distrib_H,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ A2 @ ( minus_minus_rat @ B2 @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A2 @ B2 ) @ ( times_times_rat @ A2 @ C ) ) ) ).

% right_diff_distrib'
thf(fact_5702_right__diff__distrib_H,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ A2 @ ( minus_minus_nat @ B2 @ C ) )
      = ( minus_minus_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_nat @ A2 @ C ) ) ) ).

% right_diff_distrib'
thf(fact_5703_right__diff__distrib_H,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( times_times_int @ A2 @ ( minus_minus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C ) ) ) ).

% right_diff_distrib'
thf(fact_5704_power__commutes,axiom,
    ! [A2: complex,N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ A2 @ N ) @ A2 )
      = ( times_times_complex @ A2 @ ( power_power_complex @ A2 @ N ) ) ) ).

% power_commutes
thf(fact_5705_power__commutes,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ A2 )
      = ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ).

% power_commutes
thf(fact_5706_power__commutes,axiom,
    ! [A2: real,N: nat] :
      ( ( times_times_real @ ( power_power_real @ A2 @ N ) @ A2 )
      = ( times_times_real @ A2 @ ( power_power_real @ A2 @ N ) ) ) ).

% power_commutes
thf(fact_5707_power__commutes,axiom,
    ! [A2: rat,N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ A2 @ N ) @ A2 )
      = ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N ) ) ) ).

% power_commutes
thf(fact_5708_power__commutes,axiom,
    ! [A2: nat,N: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A2 @ N ) @ A2 )
      = ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ).

% power_commutes
thf(fact_5709_power__commutes,axiom,
    ! [A2: int,N: nat] :
      ( ( times_times_int @ ( power_power_int @ A2 @ N ) @ A2 )
      = ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ).

% power_commutes
thf(fact_5710_power__mult__distrib,axiom,
    ! [A2: complex,B2: complex,N: nat] :
      ( ( power_power_complex @ ( times_times_complex @ A2 @ B2 ) @ N )
      = ( times_times_complex @ ( power_power_complex @ A2 @ N ) @ ( power_power_complex @ B2 @ N ) ) ) ).

% power_mult_distrib
thf(fact_5711_power__mult__distrib,axiom,
    ! [A2: code_integer,B2: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ N )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ ( power_8256067586552552935nteger @ B2 @ N ) ) ) ).

% power_mult_distrib
thf(fact_5712_power__mult__distrib,axiom,
    ! [A2: real,B2: real,N: nat] :
      ( ( power_power_real @ ( times_times_real @ A2 @ B2 ) @ N )
      = ( times_times_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ B2 @ N ) ) ) ).

% power_mult_distrib
thf(fact_5713_power__mult__distrib,axiom,
    ! [A2: rat,B2: rat,N: nat] :
      ( ( power_power_rat @ ( times_times_rat @ A2 @ B2 ) @ N )
      = ( times_times_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ).

% power_mult_distrib
thf(fact_5714_power__mult__distrib,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( ( power_power_nat @ ( times_times_nat @ A2 @ B2 ) @ N )
      = ( times_times_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ).

% power_mult_distrib
thf(fact_5715_power__mult__distrib,axiom,
    ! [A2: int,B2: int,N: nat] :
      ( ( power_power_int @ ( times_times_int @ A2 @ B2 ) @ N )
      = ( times_times_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B2 @ N ) ) ) ).

% power_mult_distrib
thf(fact_5716_power__commuting__commutes,axiom,
    ! [X: complex,Y: complex,N: nat] :
      ( ( ( times_times_complex @ X @ Y )
        = ( times_times_complex @ Y @ X ) )
     => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ Y )
        = ( times_times_complex @ Y @ ( power_power_complex @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_5717_power__commuting__commutes,axiom,
    ! [X: code_integer,Y: code_integer,N: nat] :
      ( ( ( times_3573771949741848930nteger @ X @ Y )
        = ( times_3573771949741848930nteger @ Y @ X ) )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ N ) @ Y )
        = ( times_3573771949741848930nteger @ Y @ ( power_8256067586552552935nteger @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_5718_power__commuting__commutes,axiom,
    ! [X: real,Y: real,N: nat] :
      ( ( ( times_times_real @ X @ Y )
        = ( times_times_real @ Y @ X ) )
     => ( ( times_times_real @ ( power_power_real @ X @ N ) @ Y )
        = ( times_times_real @ Y @ ( power_power_real @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_5719_power__commuting__commutes,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( ( times_times_rat @ X @ Y )
        = ( times_times_rat @ Y @ X ) )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ Y )
        = ( times_times_rat @ Y @ ( power_power_rat @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_5720_power__commuting__commutes,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = ( times_times_nat @ Y @ X ) )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y )
        = ( times_times_nat @ Y @ ( power_power_nat @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_5721_power__commuting__commutes,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( ( times_times_int @ X @ Y )
        = ( times_times_int @ Y @ X ) )
     => ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y )
        = ( times_times_int @ Y @ ( power_power_int @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_5722_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_5723_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_5724_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_5725_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_5726_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_5727_minus__mult__commute,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ A2 ) @ B2 )
      = ( times_times_uint32 @ A2 @ ( uminus_uminus_uint32 @ B2 ) ) ) ).

% minus_mult_commute
thf(fact_5728_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_5729_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_5730_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_5731_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_5732_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_5733_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_5734_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_5735_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_5736_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_5737_dvdE,axiom,
    ! [B2: uint32,A2: uint32] :
      ( ( dvd_dvd_uint32 @ B2 @ A2 )
     => ~ ! [K2: uint32] :
            ( A2
           != ( times_times_uint32 @ B2 @ K2 ) ) ) ).

% dvdE
thf(fact_5738_dvdE,axiom,
    ! [B2: word_N3645301735248828278l_num1,A2: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ B2 @ A2 )
     => ~ ! [K2: word_N3645301735248828278l_num1] :
            ( A2
           != ( times_7065122842183080059l_num1 @ B2 @ K2 ) ) ) ).

% dvdE
thf(fact_5739_dvdE,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ A2 )
     => ~ ! [K2: code_integer] :
            ( A2
           != ( times_3573771949741848930nteger @ B2 @ K2 ) ) ) ).

% dvdE
thf(fact_5740_dvdE,axiom,
    ! [B2: real,A2: real] :
      ( ( dvd_dvd_real @ B2 @ A2 )
     => ~ ! [K2: real] :
            ( A2
           != ( times_times_real @ B2 @ K2 ) ) ) ).

% dvdE
thf(fact_5741_dvdE,axiom,
    ! [B2: rat,A2: rat] :
      ( ( dvd_dvd_rat @ B2 @ A2 )
     => ~ ! [K2: rat] :
            ( A2
           != ( times_times_rat @ B2 @ K2 ) ) ) ).

% dvdE
thf(fact_5742_dvdE,axiom,
    ! [B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ B2 @ A2 )
     => ~ ! [K2: nat] :
            ( A2
           != ( times_times_nat @ B2 @ K2 ) ) ) ).

% dvdE
thf(fact_5743_dvdE,axiom,
    ! [B2: int,A2: int] :
      ( ( dvd_dvd_int @ B2 @ A2 )
     => ~ ! [K2: int] :
            ( A2
           != ( times_times_int @ B2 @ K2 ) ) ) ).

% dvdE
thf(fact_5744_dvdI,axiom,
    ! [A2: uint32,B2: uint32,K: uint32] :
      ( ( A2
        = ( times_times_uint32 @ B2 @ K ) )
     => ( dvd_dvd_uint32 @ B2 @ A2 ) ) ).

% dvdI
thf(fact_5745_dvdI,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1] :
      ( ( A2
        = ( times_7065122842183080059l_num1 @ B2 @ K ) )
     => ( dvd_dv6812691276156420380l_num1 @ B2 @ A2 ) ) ).

% dvdI
thf(fact_5746_dvdI,axiom,
    ! [A2: code_integer,B2: code_integer,K: code_integer] :
      ( ( A2
        = ( times_3573771949741848930nteger @ B2 @ K ) )
     => ( dvd_dvd_Code_integer @ B2 @ A2 ) ) ).

% dvdI
thf(fact_5747_dvdI,axiom,
    ! [A2: real,B2: real,K: real] :
      ( ( A2
        = ( times_times_real @ B2 @ K ) )
     => ( dvd_dvd_real @ B2 @ A2 ) ) ).

% dvdI
thf(fact_5748_dvdI,axiom,
    ! [A2: rat,B2: rat,K: rat] :
      ( ( A2
        = ( times_times_rat @ B2 @ K ) )
     => ( dvd_dvd_rat @ B2 @ A2 ) ) ).

% dvdI
thf(fact_5749_dvdI,axiom,
    ! [A2: nat,B2: nat,K: nat] :
      ( ( A2
        = ( times_times_nat @ B2 @ K ) )
     => ( dvd_dvd_nat @ B2 @ A2 ) ) ).

% dvdI
thf(fact_5750_dvdI,axiom,
    ! [A2: int,B2: int,K: int] :
      ( ( A2
        = ( times_times_int @ B2 @ K ) )
     => ( dvd_dvd_int @ B2 @ A2 ) ) ).

% dvdI
thf(fact_5751_dvd__def,axiom,
    ( dvd_dvd_uint32
    = ( ^ [B4: uint32,A4: uint32] :
        ? [K3: uint32] :
          ( A4
          = ( times_times_uint32 @ B4 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_5752_dvd__def,axiom,
    ( dvd_dv6812691276156420380l_num1
    = ( ^ [B4: word_N3645301735248828278l_num1,A4: word_N3645301735248828278l_num1] :
        ? [K3: word_N3645301735248828278l_num1] :
          ( A4
          = ( times_7065122842183080059l_num1 @ B4 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_5753_dvd__def,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [B4: code_integer,A4: code_integer] :
        ? [K3: code_integer] :
          ( A4
          = ( times_3573771949741848930nteger @ B4 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_5754_dvd__def,axiom,
    ( dvd_dvd_real
    = ( ^ [B4: real,A4: real] :
        ? [K3: real] :
          ( A4
          = ( times_times_real @ B4 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_5755_dvd__def,axiom,
    ( dvd_dvd_rat
    = ( ^ [B4: rat,A4: rat] :
        ? [K3: rat] :
          ( A4
          = ( times_times_rat @ B4 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_5756_dvd__def,axiom,
    ( dvd_dvd_nat
    = ( ^ [B4: nat,A4: nat] :
        ? [K3: nat] :
          ( A4
          = ( times_times_nat @ B4 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_5757_dvd__def,axiom,
    ( dvd_dvd_int
    = ( ^ [B4: int,A4: int] :
        ? [K3: int] :
          ( A4
          = ( times_times_int @ B4 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_5758_dvd__mult,axiom,
    ! [A2: uint32,C: uint32,B2: uint32] :
      ( ( dvd_dvd_uint32 @ A2 @ C )
     => ( dvd_dvd_uint32 @ A2 @ ( times_times_uint32 @ B2 @ C ) ) ) ).

% dvd_mult
thf(fact_5759_dvd__mult,axiom,
    ! [A2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A2 @ C )
     => ( dvd_dv6812691276156420380l_num1 @ A2 @ ( times_7065122842183080059l_num1 @ B2 @ C ) ) ) ).

% dvd_mult
thf(fact_5760_dvd__mult,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ C )
     => ( dvd_dvd_Code_integer @ A2 @ ( times_3573771949741848930nteger @ B2 @ C ) ) ) ).

% dvd_mult
thf(fact_5761_dvd__mult,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( dvd_dvd_real @ A2 @ C )
     => ( dvd_dvd_real @ A2 @ ( times_times_real @ B2 @ C ) ) ) ).

% dvd_mult
thf(fact_5762_dvd__mult,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( dvd_dvd_rat @ A2 @ C )
     => ( dvd_dvd_rat @ A2 @ ( times_times_rat @ B2 @ C ) ) ) ).

% dvd_mult
thf(fact_5763_dvd__mult,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ C )
     => ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).

% dvd_mult
thf(fact_5764_dvd__mult,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ C )
     => ( dvd_dvd_int @ A2 @ ( times_times_int @ B2 @ C ) ) ) ).

% dvd_mult
thf(fact_5765_dvd__mult2,axiom,
    ! [A2: uint32,B2: uint32,C: uint32] :
      ( ( dvd_dvd_uint32 @ A2 @ B2 )
     => ( dvd_dvd_uint32 @ A2 @ ( times_times_uint32 @ B2 @ C ) ) ) ).

% dvd_mult2
thf(fact_5766_dvd__mult2,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A2 @ B2 )
     => ( dvd_dv6812691276156420380l_num1 @ A2 @ ( times_7065122842183080059l_num1 @ B2 @ C ) ) ) ).

% dvd_mult2
thf(fact_5767_dvd__mult2,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ B2 )
     => ( dvd_dvd_Code_integer @ A2 @ ( times_3573771949741848930nteger @ B2 @ C ) ) ) ).

% dvd_mult2
thf(fact_5768_dvd__mult2,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( dvd_dvd_real @ A2 @ B2 )
     => ( dvd_dvd_real @ A2 @ ( times_times_real @ B2 @ C ) ) ) ).

% dvd_mult2
thf(fact_5769_dvd__mult2,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( dvd_dvd_rat @ A2 @ B2 )
     => ( dvd_dvd_rat @ A2 @ ( times_times_rat @ B2 @ C ) ) ) ).

% dvd_mult2
thf(fact_5770_dvd__mult2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A2 @ B2 )
     => ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ C ) ) ) ).

% dvd_mult2
thf(fact_5771_dvd__mult2,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A2 @ B2 )
     => ( dvd_dvd_int @ A2 @ ( times_times_int @ B2 @ C ) ) ) ).

% dvd_mult2
thf(fact_5772_dvd__mult__left,axiom,
    ! [A2: uint32,B2: uint32,C: uint32] :
      ( ( dvd_dvd_uint32 @ ( times_times_uint32 @ A2 @ B2 ) @ C )
     => ( dvd_dvd_uint32 @ A2 @ C ) ) ).

% dvd_mult_left
thf(fact_5773_dvd__mult__left,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( times_7065122842183080059l_num1 @ A2 @ B2 ) @ C )
     => ( dvd_dv6812691276156420380l_num1 @ A2 @ C ) ) ).

% dvd_mult_left
thf(fact_5774_dvd__mult__left,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C )
     => ( dvd_dvd_Code_integer @ A2 @ C ) ) ).

% dvd_mult_left
thf(fact_5775_dvd__mult__left,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A2 @ B2 ) @ C )
     => ( dvd_dvd_real @ A2 @ C ) ) ).

% dvd_mult_left
thf(fact_5776_dvd__mult__left,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A2 @ B2 ) @ C )
     => ( dvd_dvd_rat @ A2 @ C ) ) ).

% dvd_mult_left
thf(fact_5777_dvd__mult__left,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
     => ( dvd_dvd_nat @ A2 @ C ) ) ).

% dvd_mult_left
thf(fact_5778_dvd__mult__left,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ C )
     => ( dvd_dvd_int @ A2 @ C ) ) ).

% dvd_mult_left
thf(fact_5779_dvd__triv__left,axiom,
    ! [A2: uint32,B2: uint32] : ( dvd_dvd_uint32 @ A2 @ ( times_times_uint32 @ A2 @ B2 ) ) ).

% dvd_triv_left
thf(fact_5780_dvd__triv__left,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ A2 @ ( times_7065122842183080059l_num1 @ A2 @ B2 ) ) ).

% dvd_triv_left
thf(fact_5781_dvd__triv__left,axiom,
    ! [A2: code_integer,B2: code_integer] : ( dvd_dvd_Code_integer @ A2 @ ( times_3573771949741848930nteger @ A2 @ B2 ) ) ).

% dvd_triv_left
thf(fact_5782_dvd__triv__left,axiom,
    ! [A2: real,B2: real] : ( dvd_dvd_real @ A2 @ ( times_times_real @ A2 @ B2 ) ) ).

% dvd_triv_left
thf(fact_5783_dvd__triv__left,axiom,
    ! [A2: rat,B2: rat] : ( dvd_dvd_rat @ A2 @ ( times_times_rat @ A2 @ B2 ) ) ).

% dvd_triv_left
thf(fact_5784_dvd__triv__left,axiom,
    ! [A2: nat,B2: nat] : ( dvd_dvd_nat @ A2 @ ( times_times_nat @ A2 @ B2 ) ) ).

% dvd_triv_left
thf(fact_5785_dvd__triv__left,axiom,
    ! [A2: int,B2: int] : ( dvd_dvd_int @ A2 @ ( times_times_int @ A2 @ B2 ) ) ).

% dvd_triv_left
thf(fact_5786_mult__dvd__mono,axiom,
    ! [A2: uint32,B2: uint32,C: uint32,D: uint32] :
      ( ( dvd_dvd_uint32 @ A2 @ B2 )
     => ( ( dvd_dvd_uint32 @ C @ D )
       => ( dvd_dvd_uint32 @ ( times_times_uint32 @ A2 @ C ) @ ( times_times_uint32 @ B2 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_5787_mult__dvd__mono,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1,D: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A2 @ B2 )
     => ( ( dvd_dv6812691276156420380l_num1 @ C @ D )
       => ( dvd_dv6812691276156420380l_num1 @ ( times_7065122842183080059l_num1 @ A2 @ C ) @ ( times_7065122842183080059l_num1 @ B2 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_5788_mult__dvd__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer,D: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ B2 )
     => ( ( dvd_dvd_Code_integer @ C @ D )
       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B2 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_5789_mult__dvd__mono,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( ( dvd_dvd_real @ A2 @ B2 )
     => ( ( dvd_dvd_real @ C @ D )
       => ( dvd_dvd_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_5790_mult__dvd__mono,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( ( dvd_dvd_rat @ A2 @ B2 )
     => ( ( dvd_dvd_rat @ C @ D )
       => ( dvd_dvd_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_5791_mult__dvd__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( ( dvd_dvd_nat @ A2 @ B2 )
     => ( ( dvd_dvd_nat @ C @ D )
       => ( dvd_dvd_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_5792_mult__dvd__mono,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( ( dvd_dvd_int @ A2 @ B2 )
     => ( ( dvd_dvd_int @ C @ D )
       => ( dvd_dvd_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_5793_dvd__mult__right,axiom,
    ! [A2: uint32,B2: uint32,C: uint32] :
      ( ( dvd_dvd_uint32 @ ( times_times_uint32 @ A2 @ B2 ) @ C )
     => ( dvd_dvd_uint32 @ B2 @ C ) ) ).

% dvd_mult_right
thf(fact_5794_dvd__mult__right,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( times_7065122842183080059l_num1 @ A2 @ B2 ) @ C )
     => ( dvd_dv6812691276156420380l_num1 @ B2 @ C ) ) ).

% dvd_mult_right
thf(fact_5795_dvd__mult__right,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C )
     => ( dvd_dvd_Code_integer @ B2 @ C ) ) ).

% dvd_mult_right
thf(fact_5796_dvd__mult__right,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A2 @ B2 ) @ C )
     => ( dvd_dvd_real @ B2 @ C ) ) ).

% dvd_mult_right
thf(fact_5797_dvd__mult__right,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A2 @ B2 ) @ C )
     => ( dvd_dvd_rat @ B2 @ C ) ) ).

% dvd_mult_right
thf(fact_5798_dvd__mult__right,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
     => ( dvd_dvd_nat @ B2 @ C ) ) ).

% dvd_mult_right
thf(fact_5799_dvd__mult__right,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ C )
     => ( dvd_dvd_int @ B2 @ C ) ) ).

% dvd_mult_right
thf(fact_5800_dvd__triv__right,axiom,
    ! [A2: uint32,B2: uint32] : ( dvd_dvd_uint32 @ A2 @ ( times_times_uint32 @ B2 @ A2 ) ) ).

% dvd_triv_right
thf(fact_5801_dvd__triv__right,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ A2 @ ( times_7065122842183080059l_num1 @ B2 @ A2 ) ) ).

% dvd_triv_right
thf(fact_5802_dvd__triv__right,axiom,
    ! [A2: code_integer,B2: code_integer] : ( dvd_dvd_Code_integer @ A2 @ ( times_3573771949741848930nteger @ B2 @ A2 ) ) ).

% dvd_triv_right
thf(fact_5803_dvd__triv__right,axiom,
    ! [A2: real,B2: real] : ( dvd_dvd_real @ A2 @ ( times_times_real @ B2 @ A2 ) ) ).

% dvd_triv_right
thf(fact_5804_dvd__triv__right,axiom,
    ! [A2: rat,B2: rat] : ( dvd_dvd_rat @ A2 @ ( times_times_rat @ B2 @ A2 ) ) ).

% dvd_triv_right
thf(fact_5805_dvd__triv__right,axiom,
    ! [A2: nat,B2: nat] : ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ A2 ) ) ).

% dvd_triv_right
thf(fact_5806_dvd__triv__right,axiom,
    ! [A2: int,B2: int] : ( dvd_dvd_int @ A2 @ ( times_times_int @ B2 @ A2 ) ) ).

% dvd_triv_right
thf(fact_5807_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_5808_mult__of__int__commute,axiom,
    ! [X: int,Y: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( ring_17408606157368542149l_num1 @ X ) @ Y )
      = ( times_7065122842183080059l_num1 @ Y @ ( ring_17408606157368542149l_num1 @ X ) ) ) ).

% mult_of_int_commute
thf(fact_5809_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_5810_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_5811_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_5812_lambda__zero,axiom,
    ( ( ^ [H: word_N3645301735248828278l_num1] : zero_z3563351764282998399l_num1 )
    = ( times_7065122842183080059l_num1 @ zero_z3563351764282998399l_num1 ) ) ).

% lambda_zero
thf(fact_5813_lambda__zero,axiom,
    ( ( ^ [H: real] : zero_zero_real )
    = ( times_times_real @ zero_zero_real ) ) ).

% lambda_zero
thf(fact_5814_lambda__zero,axiom,
    ( ( ^ [H: rat] : zero_zero_rat )
    = ( times_times_rat @ zero_zero_rat ) ) ).

% lambda_zero
thf(fact_5815_lambda__zero,axiom,
    ( ( ^ [H: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_5816_lambda__zero,axiom,
    ( ( ^ [H: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_5817_lambda__one,axiom,
    ( ( ^ [X2: word_N3645301735248828278l_num1] : X2 )
    = ( times_7065122842183080059l_num1 @ one_on7727431528512463931l_num1 ) ) ).

% lambda_one
thf(fact_5818_lambda__one,axiom,
    ( ( ^ [X2: real] : X2 )
    = ( times_times_real @ one_one_real ) ) ).

% lambda_one
thf(fact_5819_lambda__one,axiom,
    ( ( ^ [X2: rat] : X2 )
    = ( times_times_rat @ one_one_rat ) ) ).

% lambda_one
thf(fact_5820_lambda__one,axiom,
    ( ( ^ [X2: nat] : X2 )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_5821_lambda__one,axiom,
    ( ( ^ [X2: int] : X2 )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_5822_le__mult__floor,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 @ ( times_times_int @ ( archim6058952711729229775r_real @ A2 ) @ ( archim6058952711729229775r_real @ B2 ) ) @ ( archim6058952711729229775r_real @ ( times_times_real @ A2 @ B2 ) ) ) ) ) ).

% le_mult_floor
thf(fact_5823_le__mult__floor,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 @ ( times_times_int @ ( archim3151403230148437115or_rat @ A2 ) @ ( archim3151403230148437115or_rat @ B2 ) ) @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ) ) ).

% le_mult_floor
thf(fact_5824_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_5825_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_5826_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_5827_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_5828_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_5829_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_5830_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_5831_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_5832_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_5833_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_5834_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_5835_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_5836_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_5837_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_5838_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_5839_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_5840_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_5841_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_5842_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_5843_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_5844_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_5845_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_5846_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_5847_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_5848_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_5849_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_5850_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_5851_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_5852_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_5853_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_5854_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_5855_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_5856_mult__right__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_5857_mult__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_5858_mult__right__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_5859_mult__right__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_5860_mult__right__mono__neg,axiom,
    ! [B2: real,A2: real,C: real] :
      ( ( ord_less_eq_real @ B2 @ A2 )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_5861_mult__right__mono__neg,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_5862_mult__right__mono__neg,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_5863_mult__left__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_5864_mult__left__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_5865_mult__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_5866_mult__left__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_5867_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_5868_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_5869_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_5870_mult__left__mono__neg,axiom,
    ! [B2: real,A2: real,C: real] :
      ( ( ord_less_eq_real @ B2 @ A2 )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) ) ) ) ).

% mult_left_mono_neg
thf(fact_5871_mult__left__mono__neg,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).

% mult_left_mono_neg
thf(fact_5872_mult__left__mono__neg,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% mult_left_mono_neg
thf(fact_5873_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_5874_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_5875_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_5876_zero__le__square,axiom,
    ! [A2: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ A2 ) ) ).

% zero_le_square
thf(fact_5877_zero__le__square,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ A2 ) ) ).

% zero_le_square
thf(fact_5878_zero__le__square,axiom,
    ! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ A2 ) ) ).

% zero_le_square
thf(fact_5879_mult__mono_H,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_5880_mult__mono_H,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_5881_mult__mono_H,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_5882_mult__mono_H,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_5883_mult__mono,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_5884_mult__mono,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_5885_mult__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_5886_mult__mono,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_5887_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_5888_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_5889_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_5890_not__square__less__zero,axiom,
    ! [A2: real] :
      ~ ( ord_less_real @ ( times_times_real @ A2 @ A2 ) @ zero_zero_real ) ).

% not_square_less_zero
thf(fact_5891_not__square__less__zero,axiom,
    ! [A2: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A2 @ A2 ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_5892_not__square__less__zero,axiom,
    ! [A2: int] :
      ~ ( ord_less_int @ ( times_times_int @ A2 @ A2 ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_5893_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_5894_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_5895_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_5896_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_5897_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_5898_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_5899_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_5900_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_5901_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_5902_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_5903_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_5904_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_5905_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_5906_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_5907_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_5908_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_5909_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_5910_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_5911_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_5912_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_5913_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_5914_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_5915_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_5916_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_5917_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_5918_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_5919_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_5920_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_5921_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_5922_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_5923_mult__less__cancel__left__neg,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) )
        = ( ord_less_real @ B2 @ A2 ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_5924_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
        = ( ord_less_rat @ B2 @ A2 ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_5925_mult__less__cancel__left__neg,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
        = ( ord_less_int @ B2 @ A2 ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_5926_mult__less__cancel__left__pos,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) )
        = ( ord_less_real @ A2 @ B2 ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_5927_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
        = ( ord_less_rat @ A2 @ B2 ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_5928_mult__less__cancel__left__pos,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
        = ( ord_less_int @ A2 @ B2 ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_5929_mult__strict__left__mono__neg,axiom,
    ! [B2: real,A2: real,C: real] :
      ( ( ord_less_real @ B2 @ A2 )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_5930_mult__strict__left__mono__neg,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_5931_mult__strict__left__mono__neg,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_5932_mult__strict__left__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_5933_mult__strict__left__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_5934_mult__strict__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_5935_mult__strict__left__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_5936_mult__less__cancel__left__disj,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A2 @ B2 ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_5937_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A2 @ B2 ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_5938_mult__less__cancel__left__disj,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A2 @ B2 ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_5939_mult__strict__right__mono__neg,axiom,
    ! [B2: real,A2: real,C: real] :
      ( ( ord_less_real @ B2 @ A2 )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_5940_mult__strict__right__mono__neg,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_5941_mult__strict__right__mono__neg,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_5942_mult__strict__right__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_5943_mult__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_5944_mult__strict__right__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_5945_mult__strict__right__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_5946_mult__less__cancel__right__disj,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A2 @ B2 ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_5947_mult__less__cancel__right__disj,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A2 @ B2 ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_5948_mult__less__cancel__right__disj,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A2 @ B2 ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_5949_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_5950_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_5951_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_5952_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_5953_mult__numeral__1,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ one ) @ A2 )
      = A2 ) ).

% mult_numeral_1
thf(fact_5954_mult__numeral__1,axiom,
    ! [A2: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A2 )
      = A2 ) ).

% mult_numeral_1
thf(fact_5955_mult__numeral__1,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A2 )
      = A2 ) ).

% mult_numeral_1
thf(fact_5956_mult__numeral__1,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A2 )
      = A2 ) ).

% mult_numeral_1
thf(fact_5957_mult__numeral__1,axiom,
    ! [A2: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A2 )
      = A2 ) ).

% mult_numeral_1
thf(fact_5958_mult__numeral__1__right,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ one ) )
      = A2 ) ).

% mult_numeral_1_right
thf(fact_5959_mult__numeral__1__right,axiom,
    ! [A2: real] :
      ( ( times_times_real @ A2 @ ( numeral_numeral_real @ one ) )
      = A2 ) ).

% mult_numeral_1_right
thf(fact_5960_mult__numeral__1__right,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ A2 @ ( numeral_numeral_rat @ one ) )
      = A2 ) ).

% mult_numeral_1_right
thf(fact_5961_mult__numeral__1__right,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ A2 @ ( numeral_numeral_nat @ one ) )
      = A2 ) ).

% mult_numeral_1_right
thf(fact_5962_mult__numeral__1__right,axiom,
    ! [A2: int] :
      ( ( times_times_int @ A2 @ ( numeral_numeral_int @ one ) )
      = A2 ) ).

% mult_numeral_1_right
thf(fact_5963_less__1__mult,axiom,
    ! [M: real,N: real] :
      ( ( ord_less_real @ one_one_real @ M )
     => ( ( ord_less_real @ one_one_real @ N )
       => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_5964_less__1__mult,axiom,
    ! [M: rat,N: rat] :
      ( ( ord_less_rat @ one_one_rat @ M )
     => ( ( ord_less_rat @ one_one_rat @ N )
       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_5965_less__1__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ M )
     => ( ( ord_less_nat @ one_one_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_5966_less__1__mult,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ one_one_int @ M )
     => ( ( ord_less_int @ one_one_int @ N )
       => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_5967_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_5968_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_5969_divide__eq__eq,axiom,
    ! [B2: real,C: real,A2: real] :
      ( ( ( divide_divide_real @ B2 @ C )
        = A2 )
      = ( ( ( C != zero_zero_real )
         => ( B2
            = ( times_times_real @ A2 @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% divide_eq_eq
thf(fact_5970_divide__eq__eq,axiom,
    ! [B2: rat,C: rat,A2: rat] :
      ( ( ( divide_divide_rat @ B2 @ C )
        = A2 )
      = ( ( ( C != zero_zero_rat )
         => ( B2
            = ( times_times_rat @ A2 @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
thf(fact_5971_eq__divide__eq,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( A2
        = ( divide_divide_real @ B2 @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ A2 @ C )
            = B2 ) )
        & ( ( C = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% eq_divide_eq
thf(fact_5972_eq__divide__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( A2
        = ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A2 @ C )
            = B2 ) )
        & ( ( C = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
thf(fact_5973_divide__eq__imp,axiom,
    ! [C: real,B2: real,A2: real] :
      ( ( C != zero_zero_real )
     => ( ( B2
          = ( times_times_real @ A2 @ C ) )
       => ( ( divide_divide_real @ B2 @ C )
          = A2 ) ) ) ).

% divide_eq_imp
thf(fact_5974_divide__eq__imp,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B2
          = ( times_times_rat @ A2 @ C ) )
       => ( ( divide_divide_rat @ B2 @ C )
          = A2 ) ) ) ).

% divide_eq_imp
thf(fact_5975_eq__divide__imp,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A2 @ C )
          = B2 )
       => ( A2
          = ( divide_divide_real @ B2 @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_5976_eq__divide__imp,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A2 @ C )
          = B2 )
       => ( A2
          = ( divide_divide_rat @ B2 @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_5977_nonzero__divide__eq__eq,axiom,
    ! [C: real,B2: real,A2: real] :
      ( ( C != zero_zero_real )
     => ( ( ( divide_divide_real @ B2 @ C )
          = A2 )
        = ( B2
          = ( times_times_real @ A2 @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_5978_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( divide_divide_rat @ B2 @ C )
          = A2 )
        = ( B2
          = ( times_times_rat @ A2 @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_5979_nonzero__eq__divide__eq,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( A2
          = ( divide_divide_real @ B2 @ C ) )
        = ( ( times_times_real @ A2 @ C )
          = B2 ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_5980_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A2
          = ( divide_divide_rat @ B2 @ C ) )
        = ( ( times_times_rat @ A2 @ C )
          = B2 ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_5981_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_5982_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_5983_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_5984_eq__add__iff2,axiom,
    ! [A2: real,E2: real,C: real,B2: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A2 @ E2 ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B2 @ E2 ) @ D ) )
      = ( C
        = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A2 ) @ E2 ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_5985_eq__add__iff2,axiom,
    ! [A2: rat,E2: rat,C: rat,B2: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B2 @ E2 ) @ D ) )
      = ( C
        = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A2 ) @ E2 ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_5986_eq__add__iff2,axiom,
    ! [A2: int,E2: int,C: int,B2: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B2 @ E2 ) @ D ) )
      = ( C
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A2 ) @ E2 ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_5987_eq__add__iff1,axiom,
    ! [A2: real,E2: real,C: real,B2: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A2 @ E2 ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B2 @ E2 ) @ D ) )
      = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A2 @ B2 ) @ E2 ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_5988_eq__add__iff1,axiom,
    ! [A2: rat,E2: rat,C: rat,B2: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B2 @ E2 ) @ D ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B2 ) @ E2 ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_5989_eq__add__iff1,axiom,
    ! [A2: int,E2: int,C: int,B2: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B2 @ E2 ) @ D ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ E2 ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_5990_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_5991_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_5992_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_5993_numeral__times__minus__swap,axiom,
    ! [W: num,X: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ W ) @ ( uminus8244633308260627903l_num1 @ X ) )
      = ( times_7065122842183080059l_num1 @ X @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5994_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_5995_numeral__times__minus__swap,axiom,
    ! [W: num,X: uint32] :
      ( ( times_times_uint32 @ ( numera9087168376688890119uint32 @ W ) @ ( uminus_uminus_uint32 @ X ) )
      = ( times_times_uint32 @ X @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5996_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_5997_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_5998_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_5999_left__right__inverse__power,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1,N: nat] :
      ( ( ( times_7065122842183080059l_num1 @ X @ Y )
        = one_on7727431528512463931l_num1 )
     => ( ( times_7065122842183080059l_num1 @ ( power_2184487114949457152l_num1 @ X @ N ) @ ( power_2184487114949457152l_num1 @ Y @ N ) )
        = one_on7727431528512463931l_num1 ) ) ).

% left_right_inverse_power
thf(fact_6000_left__right__inverse__power,axiom,
    ! [X: complex,Y: complex,N: nat] :
      ( ( ( times_times_complex @ X @ Y )
        = one_one_complex )
     => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) )
        = one_one_complex ) ) ).

% left_right_inverse_power
thf(fact_6001_left__right__inverse__power,axiom,
    ! [X: code_integer,Y: code_integer,N: nat] :
      ( ( ( times_3573771949741848930nteger @ X @ Y )
        = one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ N ) )
        = one_one_Code_integer ) ) ).

% left_right_inverse_power
thf(fact_6002_left__right__inverse__power,axiom,
    ! [X: real,Y: real,N: nat] :
      ( ( ( times_times_real @ X @ Y )
        = one_one_real )
     => ( ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) )
        = one_one_real ) ) ).

% left_right_inverse_power
thf(fact_6003_left__right__inverse__power,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( ( times_times_rat @ X @ Y )
        = one_one_rat )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y @ N ) )
        = one_one_rat ) ) ).

% left_right_inverse_power
thf(fact_6004_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = one_one_nat )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
        = one_one_nat ) ) ).

% left_right_inverse_power
thf(fact_6005_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( ( times_times_int @ X @ Y )
        = one_one_int )
     => ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
        = one_one_int ) ) ).

% left_right_inverse_power
thf(fact_6006_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_6007_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_6008_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_6009_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_6010_is__unit__mult__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
        & ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer ) ) ) ).

% is_unit_mult_iff
thf(fact_6011_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_6012_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_6013_dvd__mult__unit__iff,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A2 @ ( times_3573771949741848930nteger @ C @ B2 ) )
        = ( dvd_dvd_Code_integer @ A2 @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_6014_dvd__mult__unit__iff,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ C @ B2 ) )
        = ( dvd_dvd_nat @ A2 @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_6015_dvd__mult__unit__iff,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ A2 @ ( times_times_int @ C @ B2 ) )
        = ( dvd_dvd_int @ A2 @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_6016_mult__unit__dvd__iff,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C )
        = ( dvd_dvd_Code_integer @ A2 @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_6017_mult__unit__dvd__iff,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
        = ( dvd_dvd_nat @ A2 @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_6018_mult__unit__dvd__iff,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ C )
        = ( dvd_dvd_int @ A2 @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_6019_dvd__mult__unit__iff_H,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A2 @ ( times_3573771949741848930nteger @ B2 @ C ) )
        = ( dvd_dvd_Code_integer @ A2 @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_6020_dvd__mult__unit__iff_H,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ C ) )
        = ( dvd_dvd_nat @ A2 @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_6021_dvd__mult__unit__iff_H,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ A2 @ ( times_times_int @ B2 @ C ) )
        = ( dvd_dvd_int @ A2 @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_6022_mult__unit__dvd__iff_H,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C )
        = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_6023_mult__unit__dvd__iff_H,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ C )
        = ( dvd_dvd_nat @ B2 @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_6024_mult__unit__dvd__iff_H,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ C )
        = ( dvd_dvd_int @ B2 @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_6025_unit__mult__left__cancel,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
     => ( ( ( times_3573771949741848930nteger @ A2 @ B2 )
          = ( times_3573771949741848930nteger @ A2 @ C ) )
        = ( B2 = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_6026_unit__mult__left__cancel,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( ( ( times_times_nat @ A2 @ B2 )
          = ( times_times_nat @ A2 @ C ) )
        = ( B2 = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_6027_unit__mult__left__cancel,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( ( ( times_times_int @ A2 @ B2 )
          = ( times_times_int @ A2 @ C ) )
        = ( B2 = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_6028_unit__mult__right__cancel,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
     => ( ( ( times_3573771949741848930nteger @ B2 @ A2 )
          = ( times_3573771949741848930nteger @ C @ A2 ) )
        = ( B2 = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_6029_unit__mult__right__cancel,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( ( ( times_times_nat @ B2 @ A2 )
          = ( times_times_nat @ C @ A2 ) )
        = ( B2 = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_6030_unit__mult__right__cancel,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( ( ( times_times_int @ B2 @ A2 )
          = ( times_times_int @ C @ A2 ) )
        = ( B2 = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_6031_power__add,axiom,
    ! [A2: complex,M: nat,N: nat] :
      ( ( power_power_complex @ A2 @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_complex @ ( power_power_complex @ A2 @ M ) @ ( power_power_complex @ A2 @ N ) ) ) ).

% power_add
thf(fact_6032_power__add,axiom,
    ! [A2: code_integer,M: nat,N: nat] :
      ( ( power_8256067586552552935nteger @ A2 @ ( plus_plus_nat @ M @ N ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ M ) @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ).

% power_add
thf(fact_6033_power__add,axiom,
    ! [A2: real,M: nat,N: nat] :
      ( ( power_power_real @ A2 @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_real @ ( power_power_real @ A2 @ M ) @ ( power_power_real @ A2 @ N ) ) ) ).

% power_add
thf(fact_6034_power__add,axiom,
    ! [A2: rat,M: nat,N: nat] :
      ( ( power_power_rat @ A2 @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_rat @ ( power_power_rat @ A2 @ M ) @ ( power_power_rat @ A2 @ N ) ) ) ).

% power_add
thf(fact_6035_power__add,axiom,
    ! [A2: nat,M: nat,N: nat] :
      ( ( power_power_nat @ A2 @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) ) ) ).

% power_add
thf(fact_6036_power__add,axiom,
    ! [A2: int,M: nat,N: nat] :
      ( ( power_power_int @ A2 @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) ) ) ).

% power_add
thf(fact_6037_div__mult2__eq_H,axiom,
    ! [A2: int,M: nat,N: nat] :
      ( ( divide_divide_int @ A2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
      = ( divide_divide_int @ ( divide_divide_int @ A2 @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% div_mult2_eq'
thf(fact_6038_div__mult2__eq_H,axiom,
    ! [A2: nat,M: nat,N: nat] :
      ( ( divide_divide_nat @ A2 @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ A2 @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% div_mult2_eq'
thf(fact_6039_div__mult2__eq_H,axiom,
    ! [A2: code_integer,M: nat,N: nat] :
      ( ( divide6298287555418463151nteger @ A2 @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
      = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A2 @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% div_mult2_eq'
thf(fact_6040_dvd__div__mult,axiom,
    ! [C: code_integer,B2: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B2 )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B2 @ C ) @ A2 )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B2 @ A2 ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_6041_dvd__div__mult,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ C @ B2 )
     => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ C ) @ A2 )
        = ( divide_divide_nat @ ( times_times_nat @ B2 @ A2 ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_6042_dvd__div__mult,axiom,
    ! [C: int,B2: int,A2: int] :
      ( ( dvd_dvd_int @ C @ B2 )
     => ( ( times_times_int @ ( divide_divide_int @ B2 @ C ) @ A2 )
        = ( divide_divide_int @ ( times_times_int @ B2 @ A2 ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_6043_div__mult__swap,axiom,
    ! [C: code_integer,B2: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B2 )
     => ( ( times_3573771949741848930nteger @ A2 @ ( divide6298287555418463151nteger @ B2 @ C ) )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C ) ) ) ).

% div_mult_swap
thf(fact_6044_div__mult__swap,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ C @ B2 )
     => ( ( times_times_nat @ A2 @ ( divide_divide_nat @ B2 @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A2 @ B2 ) @ C ) ) ) ).

% div_mult_swap
thf(fact_6045_div__mult__swap,axiom,
    ! [C: int,B2: int,A2: int] :
      ( ( dvd_dvd_int @ C @ B2 )
     => ( ( times_times_int @ A2 @ ( divide_divide_int @ B2 @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A2 @ B2 ) @ C ) ) ) ).

% div_mult_swap
thf(fact_6046_div__div__eq__right,axiom,
    ! [C: code_integer,B2: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B2 )
     => ( ( dvd_dvd_Code_integer @ B2 @ A2 )
       => ( ( divide6298287555418463151nteger @ A2 @ ( divide6298287555418463151nteger @ B2 @ C ) )
          = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_6047_div__div__eq__right,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ C @ B2 )
     => ( ( dvd_dvd_nat @ B2 @ A2 )
       => ( ( divide_divide_nat @ A2 @ ( divide_divide_nat @ B2 @ C ) )
          = ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_6048_div__div__eq__right,axiom,
    ! [C: int,B2: int,A2: int] :
      ( ( dvd_dvd_int @ C @ B2 )
     => ( ( dvd_dvd_int @ B2 @ A2 )
       => ( ( divide_divide_int @ A2 @ ( divide_divide_int @ B2 @ C ) )
          = ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_6049_dvd__div__mult2__eq,axiom,
    ! [B2: code_integer,C: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B2 @ C ) @ A2 )
     => ( ( divide6298287555418463151nteger @ A2 @ ( times_3573771949741848930nteger @ B2 @ C ) )
        = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_6050_dvd__div__mult2__eq,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ C ) @ A2 )
     => ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B2 @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_6051_dvd__div__mult2__eq,axiom,
    ! [B2: int,C: int,A2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ B2 @ C ) @ A2 )
     => ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A2 @ B2 ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_6052_dvd__mult__imp__div,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ C ) @ B2 )
     => ( dvd_dvd_Code_integer @ A2 @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_6053_dvd__mult__imp__div,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ C ) @ B2 )
     => ( dvd_dvd_nat @ A2 @ ( divide_divide_nat @ B2 @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_6054_dvd__mult__imp__div,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A2 @ C ) @ B2 )
     => ( dvd_dvd_int @ A2 @ ( divide_divide_int @ B2 @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_6055_div__mult__div__if__dvd,axiom,
    ! [B2: code_integer,A2: code_integer,D: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ A2 )
     => ( ( dvd_dvd_Code_integer @ D @ C )
       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ ( divide6298287555418463151nteger @ C @ D ) )
          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ ( times_3573771949741848930nteger @ B2 @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_6056_div__mult__div__if__dvd,axiom,
    ! [B2: nat,A2: nat,D: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ A2 )
     => ( ( dvd_dvd_nat @ D @ C )
       => ( ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ ( divide_divide_nat @ C @ D ) )
          = ( divide_divide_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_6057_div__mult__div__if__dvd,axiom,
    ! [B2: int,A2: int,D: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ A2 )
     => ( ( dvd_dvd_int @ D @ C )
       => ( ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ ( divide_divide_int @ C @ D ) )
          = ( divide_divide_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_6058_real__minus__mult__self__le,axiom,
    ! [U2: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U2 @ U2 ) ) @ ( times_times_real @ X @ X ) ) ).

% real_minus_mult_self_le
thf(fact_6059_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_6060_minus__power__mult__self,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A2 ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A2 ) @ N ) )
      = ( power_8256067586552552935nteger @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_6061_minus__power__mult__self,axiom,
    ! [A2: complex,N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A2 ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A2 ) @ N ) )
      = ( power_power_complex @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_6062_minus__power__mult__self,axiom,
    ! [A2: uint32,N: nat] :
      ( ( times_times_uint32 @ ( power_power_uint32 @ ( uminus_uminus_uint32 @ A2 ) @ N ) @ ( power_power_uint32 @ ( uminus_uminus_uint32 @ A2 ) @ N ) )
      = ( power_power_uint32 @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_6063_minus__power__mult__self,axiom,
    ! [A2: real,N: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A2 ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ A2 ) @ N ) )
      = ( power_power_real @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_6064_minus__power__mult__self,axiom,
    ! [A2: rat,N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A2 ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ A2 ) @ N ) )
      = ( power_power_rat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_6065_minus__power__mult__self,axiom,
    ! [A2: int,N: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A2 ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A2 ) @ N ) )
      = ( power_power_int @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_6066_tanh__real__lt__1,axiom,
    ! [X: real] : ( ord_less_real @ ( tanh_real @ X ) @ one_one_real ) ).

% tanh_real_lt_1
thf(fact_6067_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_6068_mult__le__cancel__left,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A2 @ B2 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).

% mult_le_cancel_left
thf(fact_6069_mult__le__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A2 @ B2 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).

% mult_le_cancel_left
thf(fact_6070_mult__le__cancel__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A2 @ B2 ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).

% mult_le_cancel_left
thf(fact_6071_mult__le__cancel__right,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A2 @ B2 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).

% mult_le_cancel_right
thf(fact_6072_mult__le__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A2 @ B2 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).

% mult_le_cancel_right
thf(fact_6073_mult__le__cancel__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A2 @ B2 ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).

% mult_le_cancel_right
thf(fact_6074_mult__left__less__imp__less,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A2 @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_6075_mult__left__less__imp__less,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A2 @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_6076_mult__left__less__imp__less,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A2 @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_6077_mult__left__less__imp__less,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A2 @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_6078_mult__strict__mono,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ B2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_6079_mult__strict__mono,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ B2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_6080_mult__strict__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ B2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_6081_mult__strict__mono,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ B2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_6082_mult__less__cancel__left,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A2 @ B2 ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_left
thf(fact_6083_mult__less__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A2 @ B2 ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_left
thf(fact_6084_mult__less__cancel__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A2 @ B2 ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_left
thf(fact_6085_mult__right__less__imp__less,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ C ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A2 @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_6086_mult__right__less__imp__less,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A2 @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_6087_mult__right__less__imp__less,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A2 @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_6088_mult__right__less__imp__less,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A2 @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_6089_mult__strict__mono_H,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_6090_mult__strict__mono_H,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_6091_mult__strict__mono_H,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_6092_mult__strict__mono_H,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_6093_mult__less__cancel__right,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A2 @ B2 ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_right
thf(fact_6094_mult__less__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A2 @ B2 ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_right
thf(fact_6095_mult__less__cancel__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A2 @ B2 ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_right
thf(fact_6096_mult__le__cancel__left__neg,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) )
        = ( ord_less_eq_real @ B2 @ A2 ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_6097_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
        = ( ord_less_eq_rat @ B2 @ A2 ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_6098_mult__le__cancel__left__neg,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
        = ( ord_less_eq_int @ B2 @ A2 ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_6099_mult__le__cancel__left__pos,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) )
        = ( ord_less_eq_real @ A2 @ B2 ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_6100_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
        = ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_6101_mult__le__cancel__left__pos,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
        = ( ord_less_eq_int @ A2 @ B2 ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_6102_mult__left__le__imp__le,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B2 ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A2 @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_6103_mult__left__le__imp__le,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B2 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_6104_mult__left__le__imp__le,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_6105_mult__left__le__imp__le,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A2 @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_6106_mult__right__le__imp__le,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ C ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A2 @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_6107_mult__right__le__imp__le,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ C ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_6108_mult__right__le__imp__le,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_6109_mult__right__le__imp__le,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A2 @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_6110_mult__le__less__imp__less,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_6111_mult__le__less__imp__less,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_6112_mult__le__less__imp__less,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_6113_mult__le__less__imp__less,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_6114_mult__less__le__imp__less,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
         => ( ( ord_less_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_6115_mult__less__le__imp__less,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
         => ( ( ord_less_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_6116_mult__less__le__imp__less,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
         => ( ( ord_less_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_6117_mult__less__le__imp__less,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
         => ( ( ord_less_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_6118_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_6119_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_6120_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_6121_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_6122_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_6123_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_6124_mult__left__le,axiom,
    ! [C: real,A2: real] :
      ( ( ord_less_eq_real @ C @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
       => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ A2 ) ) ) ).

% mult_left_le
thf(fact_6125_mult__left__le,axiom,
    ! [C: rat,A2: rat] :
      ( ( ord_less_eq_rat @ C @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
       => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ A2 ) ) ) ).

% mult_left_le
thf(fact_6126_mult__left__le,axiom,
    ! [C: nat,A2: nat] :
      ( ( ord_less_eq_nat @ C @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ A2 ) ) ) ).

% mult_left_le
thf(fact_6127_mult__left__le,axiom,
    ! [C: int,A2: int] :
      ( ( ord_less_eq_int @ C @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ A2 ) ) ) ).

% mult_left_le
thf(fact_6128_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_6129_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_6130_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_6131_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_6132_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_6133_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_6134_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_6135_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_6136_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_6137_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_6138_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_6139_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_6140_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_6141_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_6142_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_6143_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_6144_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B2 @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_6145_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A2 @ B2 ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_6146_divide__less__eq,axiom,
    ! [B2: real,C: real,A2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ A2 )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B2 @ ( times_times_real @ A2 @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ B2 ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A2 ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_6147_divide__less__eq,axiom,
    ! [B2: rat,C: rat,A2: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ A2 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B2 @ ( times_times_rat @ A2 @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ B2 ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A2 ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_6148_less__divide__eq,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_real @ A2 @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ B2 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B2 @ ( times_times_real @ A2 @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ A2 @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_6149_less__divide__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ B2 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B2 @ ( times_times_rat @ A2 @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ A2 @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_6150_neg__divide__less__eq,axiom,
    ! [C: real,B2: real,A2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ A2 )
        = ( ord_less_real @ ( times_times_real @ A2 @ C ) @ B2 ) ) ) ).

% neg_divide_less_eq
thf(fact_6151_neg__divide__less__eq,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ A2 )
        = ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ B2 ) ) ) ).

% neg_divide_less_eq
thf(fact_6152_neg__less__divide__eq,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A2 @ ( divide_divide_real @ B2 @ C ) )
        = ( ord_less_real @ B2 @ ( times_times_real @ A2 @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_6153_neg__less__divide__eq,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B2 @ C ) )
        = ( ord_less_rat @ B2 @ ( times_times_rat @ A2 @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_6154_pos__divide__less__eq,axiom,
    ! [C: real,B2: real,A2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ A2 )
        = ( ord_less_real @ B2 @ ( times_times_real @ A2 @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_6155_pos__divide__less__eq,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ A2 )
        = ( ord_less_rat @ B2 @ ( times_times_rat @ A2 @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_6156_pos__less__divide__eq,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A2 @ ( divide_divide_real @ B2 @ C ) )
        = ( ord_less_real @ ( times_times_real @ A2 @ C ) @ B2 ) ) ) ).

% pos_less_divide_eq
thf(fact_6157_pos__less__divide__eq,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B2 @ C ) )
        = ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ B2 ) ) ) ).

% pos_less_divide_eq
thf(fact_6158_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_6159_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_6160_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_6161_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_6162_divide__strict__left__mono,axiom,
    ! [B2: real,A2: real,C: real] :
      ( ( ord_less_real @ B2 @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) )
         => ( ord_less_real @ ( divide_divide_real @ C @ A2 ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_6163_divide__strict__left__mono,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A2 ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_6164_divide__strict__left__mono__neg,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) )
         => ( ord_less_real @ ( divide_divide_real @ C @ A2 ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_6165_divide__strict__left__mono__neg,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A2 ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_6166_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: real,C: real,W: num] :
      ( ( ( divide_divide_real @ B2 @ C )
        = ( numeral_numeral_real @ W ) )
      = ( ( ( C != zero_zero_real )
         => ( B2
            = ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_6167_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: rat,C: rat,W: num] :
      ( ( ( divide_divide_rat @ B2 @ C )
        = ( numeral_numeral_rat @ W ) )
      = ( ( ( C != zero_zero_rat )
         => ( B2
            = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_6168_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: real,C: real] :
      ( ( ( numeral_numeral_real @ W )
        = ( divide_divide_real @ B2 @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
            = B2 ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_6169_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: rat,C: rat] :
      ( ( ( numeral_numeral_rat @ W )
        = ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
            = B2 ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_6170_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A2: real,E2: real,C: real,B2: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A2 @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E2 ) @ D ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A2 @ B2 ) @ E2 ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_6171_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A2: rat,E2: rat,C: rat,B2: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E2 ) @ D ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B2 ) @ E2 ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_6172_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A2: int,E2: int,C: int,B2: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E2 ) @ D ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ E2 ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_6173_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A2: real,E2: real,C: real,B2: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A2 @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E2 ) @ D ) )
      = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A2 ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_6174_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A2: rat,E2: rat,C: rat,B2: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E2 ) @ D ) )
      = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A2 ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_6175_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A2: int,E2: int,C: int,B2: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E2 ) @ D ) )
      = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A2 ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_6176_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_6177_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_6178_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_6179_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_6180_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_6181_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_6182_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_6183_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_6184_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_6185_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_6186_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_6187_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_6188_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_6189_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_6190_less__add__iff1,axiom,
    ! [A2: real,E2: real,C: real,B2: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A2 @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E2 ) @ D ) )
      = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A2 @ B2 ) @ E2 ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_6191_less__add__iff1,axiom,
    ! [A2: rat,E2: rat,C: rat,B2: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E2 ) @ D ) )
      = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B2 ) @ E2 ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_6192_less__add__iff1,axiom,
    ! [A2: int,E2: int,C: int,B2: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E2 ) @ D ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ E2 ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_6193_less__add__iff2,axiom,
    ! [A2: real,E2: real,C: real,B2: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A2 @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E2 ) @ D ) )
      = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A2 ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_6194_less__add__iff2,axiom,
    ! [A2: rat,E2: rat,C: rat,B2: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E2 ) @ D ) )
      = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A2 ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_6195_less__add__iff2,axiom,
    ! [A2: int,E2: int,C: int,B2: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E2 ) @ D ) )
      = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A2 ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_6196_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_6197_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_6198_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_6199_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_6200_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_6201_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_6202_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_6203_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_6204_square__diff__one__factored,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( minus_4019991460397169231l_num1 @ ( times_7065122842183080059l_num1 @ X @ X ) @ one_on7727431528512463931l_num1 )
      = ( times_7065122842183080059l_num1 @ ( plus_p361126936061061375l_num1 @ X @ one_on7727431528512463931l_num1 ) @ ( minus_4019991460397169231l_num1 @ X @ one_on7727431528512463931l_num1 ) ) ) ).

% square_diff_one_factored
thf(fact_6205_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_6206_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_6207_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_6208_mult__1s__ring__1_I2_J,axiom,
    ! [B2: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ B2 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ one ) ) )
      = ( uminus8244633308260627903l_num1 @ B2 ) ) ).

% mult_1s_ring_1(2)
thf(fact_6209_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_6210_mult__1s__ring__1_I2_J,axiom,
    ! [B2: uint32] :
      ( ( times_times_uint32 @ B2 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ one ) ) )
      = ( uminus_uminus_uint32 @ B2 ) ) ).

% mult_1s_ring_1(2)
thf(fact_6211_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_6212_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_6213_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_6214_mult__1s__ring__1_I1_J,axiom,
    ! [B2: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ one ) ) @ B2 )
      = ( uminus8244633308260627903l_num1 @ B2 ) ) ).

% mult_1s_ring_1(1)
thf(fact_6215_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_6216_mult__1s__ring__1_I1_J,axiom,
    ! [B2: uint32] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ one ) ) @ B2 )
      = ( uminus_uminus_uint32 @ B2 ) ) ).

% mult_1s_ring_1(1)
thf(fact_6217_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_6218_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_6219_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_6220_power__gt1__lemma,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
     => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_6221_power__gt1__lemma,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ord_less_real @ one_one_real @ ( times_times_real @ A2 @ ( power_power_real @ A2 @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_6222_power__gt1__lemma,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_6223_power__gt1__lemma,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_6224_power__gt1__lemma,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ord_less_int @ one_one_int @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_6225_power__less__power__Suc,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
     => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_6226_power__less__power__Suc,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ord_less_real @ ( power_power_real @ A2 @ N ) @ ( times_times_real @ A2 @ ( power_power_real @ A2 @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_6227_power__less__power__Suc,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ord_less_rat @ ( power_power_rat @ A2 @ N ) @ ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_6228_power__less__power__Suc,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_6229_power__less__power__Suc,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ord_less_int @ ( power_power_int @ A2 @ N ) @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_6230_ex__less__of__nat__mult,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ? [N2: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ X ) ) ) ).

% ex_less_of_nat_mult
thf(fact_6231_ex__less__of__nat__mult,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [N2: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).

% ex_less_of_nat_mult
thf(fact_6232_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: complex,C: complex,A2: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( C
          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ B2 ) ) )
        = ( ( times_times_complex @ C @ B2 )
          = ( uminus1482373934393186551omplex @ A2 ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_6233_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: real,C: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( ( C
          = ( uminus_uminus_real @ ( divide_divide_real @ A2 @ B2 ) ) )
        = ( ( times_times_real @ C @ B2 )
          = ( uminus_uminus_real @ A2 ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_6234_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: rat,C: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( C
          = ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ B2 ) ) )
        = ( ( times_times_rat @ C @ B2 )
          = ( uminus_uminus_rat @ A2 ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_6235_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: complex,A2: complex,C: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ B2 ) )
          = C )
        = ( ( uminus1482373934393186551omplex @ A2 )
          = ( times_times_complex @ C @ B2 ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_6236_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: real,A2: real,C: real] :
      ( ( B2 != zero_zero_real )
     => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A2 @ B2 ) )
          = C )
        = ( ( uminus_uminus_real @ A2 )
          = ( times_times_real @ C @ B2 ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_6237_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ B2 ) )
          = C )
        = ( ( uminus_uminus_rat @ A2 )
          = ( times_times_rat @ C @ B2 ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_6238_minus__divide__eq__eq,axiom,
    ! [B2: complex,C: complex,A2: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B2 @ C ) )
        = A2 )
      = ( ( ( C != zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ B2 )
            = ( times_times_complex @ A2 @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_6239_minus__divide__eq__eq,axiom,
    ! [B2: real,C: real,A2: real] :
      ( ( ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) )
        = A2 )
      = ( ( ( C != zero_zero_real )
         => ( ( uminus_uminus_real @ B2 )
            = ( times_times_real @ A2 @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_6240_minus__divide__eq__eq,axiom,
    ! [B2: rat,C: rat,A2: rat] :
      ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) )
        = A2 )
      = ( ( ( C != zero_zero_rat )
         => ( ( uminus_uminus_rat @ B2 )
            = ( times_times_rat @ A2 @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_6241_eq__minus__divide__eq,axiom,
    ! [A2: complex,B2: complex,C: complex] :
      ( ( A2
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B2 @ C ) ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ A2 @ C )
            = ( uminus1482373934393186551omplex @ B2 ) ) )
        & ( ( C = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_6242_eq__minus__divide__eq,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( A2
        = ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ A2 @ C )
            = ( uminus_uminus_real @ B2 ) ) )
        & ( ( C = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_6243_eq__minus__divide__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( A2
        = ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A2 @ C )
            = ( uminus_uminus_rat @ B2 ) ) )
        & ( ( C = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_6244_unity__coeff__ex,axiom,
    ! [P: uint32 > $o,L: uint32] :
      ( ( ? [X2: uint32] : ( P @ ( times_times_uint32 @ L @ X2 ) ) )
      = ( ? [X2: uint32] :
            ( ( dvd_dvd_uint32 @ L @ ( plus_plus_uint32 @ X2 @ zero_zero_uint32 ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_6245_unity__coeff__ex,axiom,
    ! [P: code_integer > $o,L: code_integer] :
      ( ( ? [X2: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X2 ) ) )
      = ( ? [X2: code_integer] :
            ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X2 @ zero_z3403309356797280102nteger ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_6246_unity__coeff__ex,axiom,
    ! [P: word_N3645301735248828278l_num1 > $o,L: word_N3645301735248828278l_num1] :
      ( ( ? [X2: word_N3645301735248828278l_num1] : ( P @ ( times_7065122842183080059l_num1 @ L @ X2 ) ) )
      = ( ? [X2: word_N3645301735248828278l_num1] :
            ( ( dvd_dv6812691276156420380l_num1 @ L @ ( plus_p361126936061061375l_num1 @ X2 @ zero_z3563351764282998399l_num1 ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_6247_unity__coeff__ex,axiom,
    ! [P: real > $o,L: real] :
      ( ( ? [X2: real] : ( P @ ( times_times_real @ L @ X2 ) ) )
      = ( ? [X2: real] :
            ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X2 @ zero_zero_real ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_6248_unity__coeff__ex,axiom,
    ! [P: rat > $o,L: rat] :
      ( ( ? [X2: rat] : ( P @ ( times_times_rat @ L @ X2 ) ) )
      = ( ? [X2: rat] :
            ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X2 @ zero_zero_rat ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_6249_unity__coeff__ex,axiom,
    ! [P: nat > $o,L: nat] :
      ( ( ? [X2: nat] : ( P @ ( times_times_nat @ L @ X2 ) ) )
      = ( ? [X2: nat] :
            ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X2 @ zero_zero_nat ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_6250_unity__coeff__ex,axiom,
    ! [P: int > $o,L: int] :
      ( ( ? [X2: int] : ( P @ ( times_times_int @ L @ X2 ) ) )
      = ( ? [X2: int] :
            ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X2 @ zero_zero_int ) )
            & ( P @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_6251_unit__dvdE,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
     => ~ ( ( A2 != zero_z3403309356797280102nteger )
         => ! [C4: code_integer] :
              ( B2
             != ( times_3573771949741848930nteger @ A2 @ C4 ) ) ) ) ).

% unit_dvdE
thf(fact_6252_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_6253_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_6254_power__minus,axiom,
    ! [A2: word_N3645301735248828278l_num1,N: nat] :
      ( ( power_2184487114949457152l_num1 @ ( uminus8244633308260627903l_num1 @ A2 ) @ N )
      = ( times_7065122842183080059l_num1 @ ( power_2184487114949457152l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ N ) @ ( power_2184487114949457152l_num1 @ A2 @ N ) ) ) ).

% power_minus
thf(fact_6255_power__minus,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A2 ) @ N )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ).

% power_minus
thf(fact_6256_power__minus,axiom,
    ! [A2: complex,N: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A2 ) @ N )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A2 @ N ) ) ) ).

% power_minus
thf(fact_6257_power__minus,axiom,
    ! [A2: uint32,N: nat] :
      ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ A2 ) @ N )
      = ( times_times_uint32 @ ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N ) @ ( power_power_uint32 @ A2 @ N ) ) ) ).

% power_minus
thf(fact_6258_power__minus,axiom,
    ! [A2: real,N: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ A2 ) @ N )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A2 @ N ) ) ) ).

% power_minus
thf(fact_6259_power__minus,axiom,
    ! [A2: rat,N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A2 ) @ N )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A2 @ N ) ) ) ).

% power_minus
thf(fact_6260_power__minus,axiom,
    ! [A2: int,N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A2 ) @ N )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A2 @ N ) ) ) ).

% power_minus
thf(fact_6261_dvd__div__div__eq__mult,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer,D: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( ( dvd_dvd_Code_integer @ A2 @ B2 )
         => ( ( dvd_dvd_Code_integer @ C @ D )
           => ( ( ( divide6298287555418463151nteger @ B2 @ A2 )
                = ( divide6298287555418463151nteger @ D @ C ) )
              = ( ( times_3573771949741848930nteger @ B2 @ C )
                = ( times_3573771949741848930nteger @ A2 @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_6262_dvd__div__div__eq__mult,axiom,
    ! [A2: nat,C: nat,B2: nat,D: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( ( dvd_dvd_nat @ A2 @ B2 )
         => ( ( dvd_dvd_nat @ C @ D )
           => ( ( ( divide_divide_nat @ B2 @ A2 )
                = ( divide_divide_nat @ D @ C ) )
              = ( ( times_times_nat @ B2 @ C )
                = ( times_times_nat @ A2 @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_6263_dvd__div__div__eq__mult,axiom,
    ! [A2: int,C: int,B2: int,D: int] :
      ( ( A2 != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( ( dvd_dvd_int @ A2 @ B2 )
         => ( ( dvd_dvd_int @ C @ D )
           => ( ( ( divide_divide_int @ B2 @ A2 )
                = ( divide_divide_int @ D @ C ) )
              = ( ( times_times_int @ B2 @ C )
                = ( times_times_int @ A2 @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_6264_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B2: code_integer,A2: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ C @ B2 )
       => ( ( dvd_dvd_Code_integer @ A2 @ ( divide6298287555418463151nteger @ B2 @ C ) )
          = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A2 @ C ) @ B2 ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_6265_dvd__div__iff__mult,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( ( C != zero_zero_nat )
     => ( ( dvd_dvd_nat @ C @ B2 )
       => ( ( dvd_dvd_nat @ A2 @ ( divide_divide_nat @ B2 @ C ) )
          = ( dvd_dvd_nat @ ( times_times_nat @ A2 @ C ) @ B2 ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_6266_dvd__div__iff__mult,axiom,
    ! [C: int,B2: int,A2: int] :
      ( ( C != zero_zero_int )
     => ( ( dvd_dvd_int @ C @ B2 )
       => ( ( dvd_dvd_int @ A2 @ ( divide_divide_int @ B2 @ C ) )
          = ( dvd_dvd_int @ ( times_times_int @ A2 @ C ) @ B2 ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_6267_div__dvd__iff__mult,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( ( B2 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B2 @ A2 )
       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ C )
          = ( dvd_dvd_Code_integer @ A2 @ ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_6268_div__dvd__iff__mult,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B2 @ A2 )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C )
          = ( dvd_dvd_nat @ A2 @ ( times_times_nat @ C @ B2 ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_6269_div__dvd__iff__mult,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( dvd_dvd_int @ B2 @ A2 )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ A2 @ B2 ) @ C )
          = ( dvd_dvd_int @ A2 @ ( times_times_int @ C @ B2 ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_6270_dvd__div__eq__mult,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ A2 @ B2 )
       => ( ( ( divide6298287555418463151nteger @ B2 @ A2 )
            = C )
          = ( B2
            = ( times_3573771949741848930nteger @ C @ A2 ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_6271_dvd__div__eq__mult,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ A2 @ B2 )
       => ( ( ( divide_divide_nat @ B2 @ A2 )
            = C )
          = ( B2
            = ( times_times_nat @ C @ A2 ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_6272_dvd__div__eq__mult,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( A2 != zero_zero_int )
     => ( ( dvd_dvd_int @ A2 @ B2 )
       => ( ( ( divide_divide_int @ B2 @ A2 )
            = C )
          = ( B2
            = ( times_times_int @ C @ A2 ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_6273_is__unit__div__mult2__eq,axiom,
    ! [B2: code_integer,C: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A2 @ ( times_3573771949741848930nteger @ B2 @ C ) )
          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_6274_is__unit__div__mult2__eq,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ C @ one_one_nat )
       => ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B2 @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_6275_is__unit__div__mult2__eq,axiom,
    ! [B2: int,C: int,A2: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ C @ one_one_int )
       => ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A2 @ B2 ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_6276_unit__div__mult__swap,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ A2 @ ( divide6298287555418463151nteger @ B2 @ C ) )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_6277_unit__div__mult__swap,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( times_times_nat @ A2 @ ( divide_divide_nat @ B2 @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A2 @ B2 ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_6278_unit__div__mult__swap,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( times_times_int @ A2 @ ( divide_divide_int @ B2 @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A2 @ B2 ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_6279_unit__div__commute,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ C )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A2 @ C ) @ B2 ) ) ) ).

% unit_div_commute
thf(fact_6280_unit__div__commute,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C )
        = ( divide_divide_nat @ ( times_times_nat @ A2 @ C ) @ B2 ) ) ) ).

% unit_div_commute
thf(fact_6281_unit__div__commute,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ C )
        = ( divide_divide_int @ ( times_times_int @ A2 @ C ) @ B2 ) ) ) ).

% unit_div_commute
thf(fact_6282_div__mult__unit2,axiom,
    ! [C: code_integer,B2: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B2 @ A2 )
       => ( ( divide6298287555418463151nteger @ A2 @ ( times_3573771949741848930nteger @ B2 @ C ) )
          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_6283_div__mult__unit2,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( dvd_dvd_nat @ B2 @ A2 )
       => ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B2 @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_6284_div__mult__unit2,axiom,
    ! [C: int,B2: int,A2: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( dvd_dvd_int @ B2 @ A2 )
       => ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A2 @ B2 ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_6285_unit__eq__div2,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( A2
          = ( divide6298287555418463151nteger @ C @ B2 ) )
        = ( ( times_3573771949741848930nteger @ A2 @ B2 )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_6286_unit__eq__div2,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( A2
          = ( divide_divide_nat @ C @ B2 ) )
        = ( ( times_times_nat @ A2 @ B2 )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_6287_unit__eq__div2,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( A2
          = ( divide_divide_int @ C @ B2 ) )
        = ( ( times_times_int @ A2 @ B2 )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_6288_unit__eq__div1,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ A2 @ B2 )
          = C )
        = ( A2
          = ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ).

% unit_eq_div1
thf(fact_6289_unit__eq__div1,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( ( divide_divide_nat @ A2 @ B2 )
          = C )
        = ( A2
          = ( times_times_nat @ C @ B2 ) ) ) ) ).

% unit_eq_div1
thf(fact_6290_unit__eq__div1,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( ( divide_divide_int @ A2 @ B2 )
          = C )
        = ( A2
          = ( times_times_int @ C @ B2 ) ) ) ) ).

% unit_eq_div1
thf(fact_6291_floor__le__round,axiom,
    ! [X: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X ) @ ( archim8280529875227126926d_real @ X ) ) ).

% floor_le_round
thf(fact_6292_floor__le__round,axiom,
    ! [X: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim7778729529865785530nd_rat @ X ) ) ).

% floor_le_round
thf(fact_6293_ceiling__ge__round,axiom,
    ! [X: real] : ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X ) @ ( archim7802044766580827645g_real @ X ) ) ).

% ceiling_ge_round
thf(fact_6294_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_6295_reals__Archimedean3,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ! [Y5: real] :
        ? [N2: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).

% reals_Archimedean3
thf(fact_6296_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_6297_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_6298_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_6299_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_6300_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_6301_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_6302_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_6303_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_6304_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_6305_mult__le__cancel__left1,axiom,
    ! [C: real,B2: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B2 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B2 @ one_one_real ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_6306_mult__le__cancel__left1,axiom,
    ! [C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B2 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B2 @ one_one_rat ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_6307_mult__le__cancel__left1,axiom,
    ! [C: int,B2: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B2 ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_6308_mult__le__cancel__left2,axiom,
    ! [C: real,A2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ C )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A2 @ one_one_real ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A2 ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_6309_mult__le__cancel__left2,axiom,
    ! [C: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A2 @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A2 ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_6310_mult__le__cancel__left2,axiom,
    ! [C: int,A2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A2 @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_6311_mult__le__cancel__right1,axiom,
    ! [C: real,B2: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B2 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B2 @ one_one_real ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_6312_mult__le__cancel__right1,axiom,
    ! [C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B2 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B2 @ one_one_rat ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_6313_mult__le__cancel__right1,axiom,
    ! [C: int,B2: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ B2 @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B2 ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_6314_mult__le__cancel__right2,axiom,
    ! [A2: real,C: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ C )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A2 @ one_one_real ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A2 ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_6315_mult__le__cancel__right2,axiom,
    ! [A2: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A2 @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A2 ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_6316_mult__le__cancel__right2,axiom,
    ! [A2: int,C: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A2 @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_6317_mult__less__cancel__left1,axiom,
    ! [C: real,B2: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ C @ B2 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B2 ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B2 @ one_one_real ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_6318_mult__less__cancel__left1,axiom,
    ! [C: rat,B2: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B2 ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B2 @ one_one_rat ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_6319_mult__less__cancel__left1,axiom,
    ! [C: int,B2: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ C @ B2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B2 ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_6320_mult__less__cancel__left2,axiom,
    ! [C: real,A2: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ C )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A2 @ one_one_real ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A2 ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_6321_mult__less__cancel__left2,axiom,
    ! [C: rat,A2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A2 @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A2 ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_6322_mult__less__cancel__left2,axiom,
    ! [C: int,A2: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A2 @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_6323_mult__less__cancel__right1,axiom,
    ! [C: real,B2: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ B2 @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B2 ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B2 @ one_one_real ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_6324_mult__less__cancel__right1,axiom,
    ! [C: rat,B2: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ B2 @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B2 ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B2 @ one_one_rat ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_6325_mult__less__cancel__right1,axiom,
    ! [C: int,B2: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ B2 @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B2 ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_6326_mult__less__cancel__right2,axiom,
    ! [A2: real,C: real] :
      ( ( ord_less_real @ ( times_times_real @ A2 @ C ) @ C )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A2 @ one_one_real ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A2 ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_6327_mult__less__cancel__right2,axiom,
    ! [A2: rat,C: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A2 @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A2 ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_6328_mult__less__cancel__right2,axiom,
    ! [A2: int,C: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A2 @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_6329_field__le__mult__one__interval,axiom,
    ! [X: real,Y: real] :
      ( ! [Z3: real] :
          ( ( ord_less_real @ zero_zero_real @ Z3 )
         => ( ( ord_less_real @ Z3 @ one_one_real )
           => ( ord_less_eq_real @ ( times_times_real @ Z3 @ X ) @ Y ) ) )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_6330_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z3: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ Z3 )
         => ( ( ord_less_rat @ Z3 @ one_one_rat )
           => ( ord_less_eq_rat @ ( times_times_rat @ Z3 @ X ) @ Y ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_6331_convex__bound__le,axiom,
    ! [X: real,A2: real,Y: real,U2: real,V: real] :
      ( ( ord_less_eq_real @ X @ A2 )
     => ( ( ord_less_eq_real @ Y @ A2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ U2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U2 @ V )
                = one_one_real )
             => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U2 @ X ) @ ( times_times_real @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_6332_convex__bound__le,axiom,
    ! [X: rat,A2: rat,Y: rat,U2: rat,V: rat] :
      ( ( ord_less_eq_rat @ X @ A2 )
     => ( ( ord_less_eq_rat @ Y @ A2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U2 @ V )
                = one_one_rat )
             => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U2 @ X ) @ ( times_times_rat @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_6333_convex__bound__le,axiom,
    ! [X: int,A2: int,Y: int,U2: int,V: int] :
      ( ( ord_less_eq_int @ X @ A2 )
     => ( ( ord_less_eq_int @ Y @ A2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ U2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U2 @ V )
                = one_one_int )
             => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U2 @ X ) @ ( times_times_int @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_6334_divide__left__mono__neg,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A2 ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_6335_divide__left__mono__neg,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A2 ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_6336_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_6337_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_6338_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_6339_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_6340_pos__le__divide__eq,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A2 @ ( divide_divide_real @ B2 @ C ) )
        = ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ B2 ) ) ) ).

% pos_le_divide_eq
thf(fact_6341_pos__le__divide__eq,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B2 @ C ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ B2 ) ) ) ).

% pos_le_divide_eq
thf(fact_6342_pos__divide__le__eq,axiom,
    ! [C: real,B2: real,A2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ A2 )
        = ( ord_less_eq_real @ B2 @ ( times_times_real @ A2 @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_6343_pos__divide__le__eq,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ A2 )
        = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A2 @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_6344_neg__le__divide__eq,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A2 @ ( divide_divide_real @ B2 @ C ) )
        = ( ord_less_eq_real @ B2 @ ( times_times_real @ A2 @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_6345_neg__le__divide__eq,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B2 @ C ) )
        = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A2 @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_6346_neg__divide__le__eq,axiom,
    ! [C: real,B2: real,A2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ A2 )
        = ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ B2 ) ) ) ).

% neg_divide_le_eq
thf(fact_6347_neg__divide__le__eq,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ A2 )
        = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ B2 ) ) ) ).

% neg_divide_le_eq
thf(fact_6348_divide__left__mono,axiom,
    ! [B2: real,A2: real,C: real] :
      ( ( ord_less_eq_real @ B2 @ A2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A2 ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).

% divide_left_mono
thf(fact_6349_divide__left__mono,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A2 ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).

% divide_left_mono
thf(fact_6350_le__divide__eq,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ B2 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B2 @ ( times_times_real @ A2 @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_6351_le__divide__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ B2 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A2 @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_6352_divide__le__eq,axiom,
    ! [B2: real,C: real,A2: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ A2 )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B2 @ ( times_times_real @ A2 @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ B2 ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_6353_divide__le__eq,axiom,
    ! [B2: rat,C: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ A2 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A2 @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ B2 ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_6354_divide__less__eq__numeral_I1_J,axiom,
    ! [B2: real,C: real,W: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ ( numeral_numeral_real @ W ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B2 ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_6355_divide__less__eq__numeral_I1_J,axiom,
    ! [B2: rat,C: rat,W: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ ( numeral_numeral_rat @ W ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B2 ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_6356_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: real,C: real] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B2 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_6357_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: rat,C: rat] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B2 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_6358_mult__2,axiom,
    ! [Z: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ Z )
      = ( plus_p361126936061061375l_num1 @ Z @ Z ) ) ).

% mult_2
thf(fact_6359_mult__2,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_real @ Z @ Z ) ) ).

% mult_2
thf(fact_6360_mult__2,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2
thf(fact_6361_mult__2,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2
thf(fact_6362_mult__2,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2
thf(fact_6363_mult__2__right,axiom,
    ! [Z: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ Z @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
      = ( plus_p361126936061061375l_num1 @ Z @ Z ) ) ).

% mult_2_right
thf(fact_6364_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_6365_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_6366_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_6367_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_6368_left__add__twice,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ A2 @ ( plus_p361126936061061375l_num1 @ A2 @ B2 ) )
      = ( plus_p361126936061061375l_num1 @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A2 ) @ B2 ) ) ).

% left_add_twice
thf(fact_6369_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_6370_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_6371_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_6372_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_6373_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_6374_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_6375_power__Suc__less,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N ) ) @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ) ).

% power_Suc_less
thf(fact_6376_power__Suc__less,axiom,
    ! [A2: real,N: 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 @ N ) ) @ ( power_power_real @ A2 @ N ) ) ) ) ).

% power_Suc_less
thf(fact_6377_power__Suc__less,axiom,
    ! [A2: rat,N: 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 @ N ) ) @ ( power_power_rat @ A2 @ N ) ) ) ) ).

% power_Suc_less
thf(fact_6378_power__Suc__less,axiom,
    ! [A2: nat,N: 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 @ N ) ) @ ( power_power_nat @ A2 @ N ) ) ) ) ).

% power_Suc_less
thf(fact_6379_power__Suc__less,axiom,
    ! [A2: int,N: 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 @ N ) ) @ ( power_power_int @ A2 @ N ) ) ) ) ).

% power_Suc_less
thf(fact_6380_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_6381_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_6382_less__minus__divide__eq,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B2 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A2 @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ A2 @ zero_zero_real ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_6383_less__minus__divide__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A2 @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ A2 @ zero_zero_rat ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_6384_minus__divide__less__eq,axiom,
    ! [B2: real,C: real,A2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A2 )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A2 @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B2 ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A2 ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_6385_minus__divide__less__eq,axiom,
    ! [B2: rat,C: rat,A2: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A2 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A2 @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B2 ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A2 ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_6386_neg__less__minus__divide__eq,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
        = ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A2 @ C ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_6387_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A2 @ C ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_6388_neg__minus__divide__less__eq,axiom,
    ! [C: real,B2: real,A2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A2 )
        = ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_6389_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A2 )
        = ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_6390_pos__less__minus__divide__eq,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
        = ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_6391_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
        = ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_6392_pos__minus__divide__less__eq,axiom,
    ! [C: real,B2: real,A2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A2 )
        = ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A2 @ C ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_6393_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A2 )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A2 @ C ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_6394_divide__eq__eq__numeral_I2_J,axiom,
    ! [B2: complex,C: complex,W: num] :
      ( ( ( divide1717551699836669952omplex @ B2 @ C )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( ( C != zero_zero_complex )
         => ( B2
            = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_6395_divide__eq__eq__numeral_I2_J,axiom,
    ! [B2: real,C: real,W: num] :
      ( ( ( divide_divide_real @ B2 @ C )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( C != zero_zero_real )
         => ( B2
            = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_6396_divide__eq__eq__numeral_I2_J,axiom,
    ! [B2: rat,C: rat,W: num] :
      ( ( ( divide_divide_rat @ B2 @ C )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( B2
            = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_6397_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: complex,C: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
        = ( divide1717551699836669952omplex @ B2 @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C )
            = B2 ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_6398_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: real,C: real] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
        = ( divide_divide_real @ B2 @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
            = B2 ) )
        & ( ( C = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_6399_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: rat,C: rat] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
        = ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
            = B2 ) )
        & ( ( C = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_6400_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_6401_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_6402_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_6403_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_6404_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_6405_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_6406_evenE,axiom,
    ! [A2: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A2 )
     => ~ ! [B6: uint32] :
            ( A2
           != ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B6 ) ) ) ).

% evenE
thf(fact_6407_evenE,axiom,
    ! [A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
     => ~ ! [B6: code_integer] :
            ( A2
           != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B6 ) ) ) ).

% evenE
thf(fact_6408_evenE,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A2 )
     => ~ ! [B6: word_N3645301735248828278l_num1] :
            ( A2
           != ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ B6 ) ) ) ).

% evenE
thf(fact_6409_evenE,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
     => ~ ! [B6: nat] :
            ( A2
           != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B6 ) ) ) ).

% evenE
thf(fact_6410_evenE,axiom,
    ! [A2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
     => ~ ! [B6: int] :
            ( A2
           != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B6 ) ) ) ).

% evenE
thf(fact_6411_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_6412_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_6413_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_6414_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_6415_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_6416_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_6417_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_6418_power2__eq__square,axiom,
    ! [A2: code_integer] :
      ( ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_3573771949741848930nteger @ A2 @ A2 ) ) ).

% power2_eq_square
thf(fact_6419_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_6420_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_6421_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_6422_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_6423_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_6424_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_6425_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_6426_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_6427_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_6428_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_6429_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_6430_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_6431_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_6432_is__unitE,axiom,
    ! [A2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
     => ~ ( ( A2 != zero_z3403309356797280102nteger )
         => ! [B6: code_integer] :
              ( ( B6 != zero_z3403309356797280102nteger )
             => ( ( dvd_dvd_Code_integer @ B6 @ one_one_Code_integer )
               => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A2 )
                    = B6 )
                 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B6 )
                      = A2 )
                   => ( ( ( times_3573771949741848930nteger @ A2 @ B6 )
                        = one_one_Code_integer )
                     => ( ( divide6298287555418463151nteger @ C @ A2 )
                       != ( times_3573771949741848930nteger @ C @ B6 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_6433_is__unitE,axiom,
    ! [A2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ~ ( ( A2 != zero_zero_nat )
         => ! [B6: nat] :
              ( ( B6 != zero_zero_nat )
             => ( ( dvd_dvd_nat @ B6 @ one_one_nat )
               => ( ( ( divide_divide_nat @ one_one_nat @ A2 )
                    = B6 )
                 => ( ( ( divide_divide_nat @ one_one_nat @ B6 )
                      = A2 )
                   => ( ( ( times_times_nat @ A2 @ B6 )
                        = one_one_nat )
                     => ( ( divide_divide_nat @ C @ A2 )
                       != ( times_times_nat @ C @ B6 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_6434_is__unitE,axiom,
    ! [A2: int,C: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ~ ( ( A2 != zero_zero_int )
         => ! [B6: int] :
              ( ( B6 != zero_zero_int )
             => ( ( dvd_dvd_int @ B6 @ one_one_int )
               => ( ( ( divide_divide_int @ one_one_int @ A2 )
                    = B6 )
                 => ( ( ( divide_divide_int @ one_one_int @ B6 )
                      = A2 )
                   => ( ( ( times_times_int @ A2 @ B6 )
                        = one_one_int )
                     => ( ( divide_divide_int @ C @ A2 )
                       != ( times_times_int @ C @ B6 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_6435_is__unit__div__mult__cancel__left,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A2 @ ( times_3573771949741848930nteger @ A2 @ B2 ) )
          = ( divide6298287555418463151nteger @ one_one_Code_integer @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_6436_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_6437_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_6438_is__unit__div__mult__cancel__right,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A2 @ ( times_3573771949741848930nteger @ B2 @ A2 ) )
          = ( divide6298287555418463151nteger @ one_one_Code_integer @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_6439_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_6440_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_6441_ln__realpow,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ln_ln_real @ ( power_power_real @ X @ N ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X ) ) ) ) ).

% ln_realpow
thf(fact_6442_linear__plus__1__le__power,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N ) ) ) ).

% linear_plus_1_le_power
thf(fact_6443_convex__bound__lt,axiom,
    ! [X: real,A2: real,Y: real,U2: real,V: real] :
      ( ( ord_less_real @ X @ A2 )
     => ( ( ord_less_real @ Y @ A2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ U2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U2 @ V )
                = one_one_real )
             => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U2 @ X ) @ ( times_times_real @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_6444_convex__bound__lt,axiom,
    ! [X: rat,A2: rat,Y: rat,U2: rat,V: rat] :
      ( ( ord_less_rat @ X @ A2 )
     => ( ( ord_less_rat @ Y @ A2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U2 @ V )
                = one_one_rat )
             => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U2 @ X ) @ ( times_times_rat @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_6445_convex__bound__lt,axiom,
    ! [X: int,A2: int,Y: int,U2: int,V: int] :
      ( ( ord_less_int @ X @ A2 )
     => ( ( ord_less_int @ Y @ A2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ U2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U2 @ V )
                = one_one_int )
             => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U2 @ X ) @ ( times_times_int @ V @ Y ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_6446_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: real,C: real] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B2 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_6447_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B2 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_6448_divide__le__eq__numeral_I1_J,axiom,
    ! [B2: real,C: real,W: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ ( numeral_numeral_real @ W ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B2 ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_6449_divide__le__eq__numeral_I1_J,axiom,
    ! [B2: rat,C: rat,W: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ ( numeral_numeral_rat @ W ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B2 ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_6450_scaling__mono,axiom,
    ! [U2: real,V: real,R3: real,S2: real] :
      ( ( ord_less_eq_real @ U2 @ V )
     => ( ( ord_less_eq_real @ zero_zero_real @ R3 )
       => ( ( ord_less_eq_real @ R3 @ S2 )
         => ( ord_less_eq_real @ ( plus_plus_real @ U2 @ ( divide_divide_real @ ( times_times_real @ R3 @ ( minus_minus_real @ V @ U2 ) ) @ S2 ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_6451_scaling__mono,axiom,
    ! [U2: rat,V: rat,R3: rat,S2: rat] :
      ( ( ord_less_eq_rat @ U2 @ V )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ R3 )
       => ( ( ord_less_eq_rat @ R3 @ S2 )
         => ( ord_less_eq_rat @ ( plus_plus_rat @ U2 @ ( divide_divide_rat @ ( times_times_rat @ R3 @ ( minus_minus_rat @ V @ U2 ) ) @ S2 ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_6452_le__minus__divide__eq,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B2 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A2 @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_6453_le__minus__divide__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A2 @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_6454_minus__divide__le__eq,axiom,
    ! [B2: real,C: real,A2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A2 )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A2 @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B2 ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_6455_minus__divide__le__eq,axiom,
    ! [B2: rat,C: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A2 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A2 @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B2 ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_6456_neg__le__minus__divide__eq,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A2 @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_6457_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A2 @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_6458_neg__minus__divide__le__eq,axiom,
    ! [C: real,B2: real,A2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A2 )
        = ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_6459_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A2 )
        = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_6460_pos__le__minus__divide__eq,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
        = ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_6461_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_6462_pos__minus__divide__le__eq,axiom,
    ! [C: real,B2: real,A2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A2 )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A2 @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_6463_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A2 )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A2 @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_6464_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: real,C: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B2 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_6465_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: rat,C: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B2 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_6466_divide__less__eq__numeral_I2_J,axiom,
    ! [B2: real,C: real,W: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B2 ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_6467_divide__less__eq__numeral_I2_J,axiom,
    ! [B2: rat,C: rat,W: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B2 ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_6468_even__two__times__div__two,axiom,
    ! [A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
     => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
        = A2 ) ) ).

% even_two_times_div_two
thf(fact_6469_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_6470_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_6471_power__eq__if,axiom,
    ( power_2184487114949457152l_num1
    = ( ^ [P6: word_N3645301735248828278l_num1,M3: nat] : ( if_wor5778924947035936048l_num1 @ ( M3 = zero_zero_nat ) @ one_on7727431528512463931l_num1 @ ( times_7065122842183080059l_num1 @ P6 @ ( power_2184487114949457152l_num1 @ P6 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_6472_power__eq__if,axiom,
    ( power_power_complex
    = ( ^ [P6: complex,M3: nat] : ( if_complex @ ( M3 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P6 @ ( power_power_complex @ P6 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_6473_power__eq__if,axiom,
    ( power_8256067586552552935nteger
    = ( ^ [P6: code_integer,M3: nat] : ( if_Code_integer @ ( M3 = zero_zero_nat ) @ one_one_Code_integer @ ( times_3573771949741848930nteger @ P6 @ ( power_8256067586552552935nteger @ P6 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_6474_power__eq__if,axiom,
    ( power_power_real
    = ( ^ [P6: real,M3: nat] : ( if_real @ ( M3 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P6 @ ( power_power_real @ P6 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_6475_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P6: rat,M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P6 @ ( power_power_rat @ P6 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_6476_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P6: nat,M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P6 @ ( power_power_nat @ P6 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_6477_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P6: int,M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P6 @ ( power_power_int @ P6 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_6478_power__minus__mult,axiom,
    ! [N: nat,A2: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_complex @ ( power_power_complex @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
        = ( power_power_complex @ A2 @ N ) ) ) ).

% power_minus_mult
thf(fact_6479_power__minus__mult,axiom,
    ! [N: nat,A2: code_integer] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
        = ( power_8256067586552552935nteger @ A2 @ N ) ) ) ).

% power_minus_mult
thf(fact_6480_power__minus__mult,axiom,
    ! [N: nat,A2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_real @ ( power_power_real @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
        = ( power_power_real @ A2 @ N ) ) ) ).

% power_minus_mult
thf(fact_6481_power__minus__mult,axiom,
    ! [N: nat,A2: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_rat @ ( power_power_rat @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
        = ( power_power_rat @ A2 @ N ) ) ) ).

% power_minus_mult
thf(fact_6482_power__minus__mult,axiom,
    ! [N: nat,A2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_nat @ ( power_power_nat @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
        = ( power_power_nat @ A2 @ N ) ) ) ).

% power_minus_mult
thf(fact_6483_power__minus__mult,axiom,
    ! [N: nat,A2: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_int @ ( power_power_int @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
        = ( power_power_int @ A2 @ N ) ) ) ).

% power_minus_mult
thf(fact_6484_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_6485_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ! [M4: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ M4 )
             => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X ) @ C ) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
thf(fact_6486_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_6487_shiftl__eq__mult,axiom,
    ( bit_Sh7051673377389942294nteger
    = ( ^ [X2: code_integer,N4: nat] : ( times_3573771949741848930nteger @ X2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% shiftl_eq_mult
thf(fact_6488_shiftl__eq__mult,axiom,
    ( bit_Sh9074413540854191407l_num1
    = ( ^ [X2: word_N3645301735248828278l_num1,N4: nat] : ( times_7065122842183080059l_num1 @ X2 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% shiftl_eq_mult
thf(fact_6489_shiftl__eq__mult,axiom,
    ( bit_Sh3963086678839698405tl_int
    = ( ^ [X2: int,N4: nat] : ( times_times_int @ X2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% shiftl_eq_mult
thf(fact_6490_shiftl__eq__mult,axiom,
    ( bit_Sh3965577149348748681tl_nat
    = ( ^ [X2: nat,N4: nat] : ( times_times_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% shiftl_eq_mult
thf(fact_6491_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_6492_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_6493_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_6494_log__nat__power,axiom,
    ! [X: real,B2: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( log @ B2 @ ( power_power_real @ X @ N ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ X ) ) ) ) ).

% log_nat_power
thf(fact_6495_divide__le__eq__numeral_I2_J,axiom,
    ! [B2: real,C: real,W: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B2 ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_6496_divide__le__eq__numeral_I2_J,axiom,
    ! [B2: rat,C: rat,W: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B2 ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_6497_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: real,C: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B2 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_6498_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B2 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_6499_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_6500_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_6501_power2__sum,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( power_2184487114949457152l_num1 @ ( plus_p361126936061061375l_num1 @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_p361126936061061375l_num1 @ ( plus_p361126936061061375l_num1 @ ( power_2184487114949457152l_num1 @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_2184487114949457152l_num1 @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_7065122842183080059l_num1 @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_6502_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_6503_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_6504_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_6505_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_6506_oddE,axiom,
    ! [A2: uint32] :
      ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A2 )
     => ~ ! [B6: uint32] :
            ( A2
           != ( plus_plus_uint32 @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B6 ) @ one_one_uint32 ) ) ) ).

% oddE
thf(fact_6507_oddE,axiom,
    ! [A2: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
     => ~ ! [B6: code_integer] :
            ( A2
           != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B6 ) @ one_one_Code_integer ) ) ) ).

% oddE
thf(fact_6508_oddE,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A2 )
     => ~ ! [B6: word_N3645301735248828278l_num1] :
            ( A2
           != ( plus_p361126936061061375l_num1 @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ B6 ) @ one_on7727431528512463931l_num1 ) ) ) ).

% oddE
thf(fact_6509_oddE,axiom,
    ! [A2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
     => ~ ! [B6: nat] :
            ( A2
           != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B6 ) @ one_one_nat ) ) ) ).

% oddE
thf(fact_6510_oddE,axiom,
    ! [A2: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
     => ~ ! [B6: int] :
            ( A2
           != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B6 ) @ one_one_int ) ) ) ).

% oddE
thf(fact_6511_floor__divide__lower,axiom,
    ! [Q3: real,P4: real] :
      ( ( ord_less_real @ zero_zero_real @ Q3 )
     => ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P4 @ Q3 ) ) ) @ Q3 ) @ P4 ) ) ).

% floor_divide_lower
thf(fact_6512_floor__divide__lower,axiom,
    ! [Q3: rat,P4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q3 )
     => ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P4 @ Q3 ) ) ) @ Q3 ) @ P4 ) ) ).

% floor_divide_lower
thf(fact_6513_ceiling__divide__upper,axiom,
    ! [Q3: real,P4: real] :
      ( ( ord_less_real @ zero_zero_real @ Q3 )
     => ( ord_less_eq_real @ P4 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P4 @ Q3 ) ) ) @ Q3 ) ) ) ).

% ceiling_divide_upper
thf(fact_6514_ceiling__divide__upper,axiom,
    ! [Q3: rat,P4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q3 )
     => ( ord_less_eq_rat @ P4 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P4 @ Q3 ) ) ) @ Q3 ) ) ) ).

% ceiling_divide_upper
thf(fact_6515_Bernoulli__inequality,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).

% Bernoulli_inequality
thf(fact_6516_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_6517_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_6518_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_6519_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_6520_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_6521_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_6522_power2__diff,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( power_2184487114949457152l_num1 @ ( minus_4019991460397169231l_num1 @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_4019991460397169231l_num1 @ ( plus_p361126936061061375l_num1 @ ( power_2184487114949457152l_num1 @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_2184487114949457152l_num1 @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_7065122842183080059l_num1 @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_6523_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_6524_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_6525_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_6526_floor__divide__upper,axiom,
    ! [Q3: real,P4: real] :
      ( ( ord_less_real @ zero_zero_real @ Q3 )
     => ( ord_less_real @ P4 @ ( times_times_real @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P4 @ Q3 ) ) ) @ one_one_real ) @ Q3 ) ) ) ).

% floor_divide_upper
thf(fact_6527_floor__divide__upper,axiom,
    ! [Q3: rat,P4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q3 )
     => ( ord_less_rat @ P4 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P4 @ Q3 ) ) ) @ one_one_rat ) @ Q3 ) ) ) ).

% floor_divide_upper
thf(fact_6528_ceiling__divide__lower,axiom,
    ! [Q3: real,P4: 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 @ P4 @ Q3 ) ) ) @ one_one_real ) @ Q3 ) @ P4 ) ) ).

% ceiling_divide_lower
thf(fact_6529_ceiling__divide__lower,axiom,
    ! [Q3: rat,P4: 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 @ P4 @ Q3 ) ) ) @ one_one_rat ) @ Q3 ) @ P4 ) ) ).

% ceiling_divide_lower
thf(fact_6530_L2__set__mult__ineq__lemma,axiom,
    ! [A2: real,C: real,B2: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A2 @ C ) ) @ ( times_times_real @ B2 @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% L2_set_mult_ineq_lemma
thf(fact_6531_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_6532_arith__geo__mean,axiom,
    ! [U2: real,X: real,Y: real] :
      ( ( ( power_power_real @ U2 @ ( 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 @ U2 @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_6533_arith__geo__mean,axiom,
    ! [U2: rat,X: rat,Y: rat] :
      ( ( ( power_power_rat @ U2 @ ( 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 @ U2 @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_6534_Bernoulli__inequality__even,axiom,
    ! [N: nat,X: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).

% Bernoulli_inequality_even
thf(fact_6535_round__def,axiom,
    ( archim8280529875227126926d_real
    = ( ^ [X2: real] : ( archim6058952711729229775r_real @ ( plus_plus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% round_def
thf(fact_6536_round__def,axiom,
    ( archim7778729529865785530nd_rat
    = ( ^ [X2: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% round_def
thf(fact_6537_even__mult__exp__div__exp__iff,axiom,
    ! [A2: code_integer,M: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M )
        | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
          = zero_z3403309356797280102nteger )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_6538_even__mult__exp__div__exp__iff,axiom,
    ! [A2: word_N3645301735248828278l_num1,M: nat,N: nat] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( divide1791077408188789448l_num1 @ ( times_7065122842183080059l_num1 @ A2 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ M ) ) @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M )
        | ( ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N )
          = zero_z3563351764282998399l_num1 )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( divide1791077408188789448l_num1 @ A2 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_6539_even__mult__exp__div__exp__iff,axiom,
    ! [A2: uint32,M: nat,N: nat] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( divide_divide_uint32 @ ( times_times_uint32 @ A2 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M )
        | ( ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N )
          = zero_zero_uint32 )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( divide_divide_uint32 @ A2 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_6540_even__mult__exp__div__exp__iff,axiom,
    ! [A2: nat,M: nat,N: 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 ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M )
        | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          = zero_zero_nat )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_6541_even__mult__exp__div__exp__iff,axiom,
    ! [A2: int,M: nat,N: 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 ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M )
        | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
          = zero_zero_int )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_6542_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_6543_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_6544_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_6545_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_6546_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_6547_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_6548_arsinh__def,axiom,
    ( arsinh_real
    = ( ^ [X2: real] : ( ln_ln_real @ ( plus_plus_real @ X2 @ ( powr_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% arsinh_def
thf(fact_6549_of__real__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V1803761363581548252l_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ).

% of_real_neg_numeral
thf(fact_6550_of__real__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V4546457046886955230omplex @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).

% of_real_neg_numeral
thf(fact_6551_of__real__of__nat__eq,axiom,
    ! [N: nat] :
      ( ( real_V1803761363581548252l_real @ ( semiri5074537144036343181t_real @ N ) )
      = ( semiri5074537144036343181t_real @ N ) ) ).

% of_real_of_nat_eq
thf(fact_6552_of__real__of__nat__eq,axiom,
    ! [N: nat] :
      ( ( real_V4546457046886955230omplex @ ( semiri5074537144036343181t_real @ N ) )
      = ( semiri8010041392384452111omplex @ N ) ) ).

% of_real_of_nat_eq
thf(fact_6553_of__real__power,axiom,
    ! [X: real,N: nat] :
      ( ( real_V1803761363581548252l_real @ ( power_power_real @ X @ N ) )
      = ( power_power_real @ ( real_V1803761363581548252l_real @ X ) @ N ) ) ).

% of_real_power
thf(fact_6554_of__real__power,axiom,
    ! [X: real,N: nat] :
      ( ( real_V4546457046886955230omplex @ ( power_power_real @ X @ N ) )
      = ( power_power_complex @ ( real_V4546457046886955230omplex @ X ) @ N ) ) ).

% of_real_power
thf(fact_6555_of__real__numeral,axiom,
    ! [W: num] :
      ( ( real_V1803761363581548252l_real @ ( numeral_numeral_real @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% of_real_numeral
thf(fact_6556_of__real__numeral,axiom,
    ! [W: num] :
      ( ( real_V4546457046886955230omplex @ ( numeral_numeral_real @ W ) )
      = ( numera6690914467698888265omplex @ W ) ) ).

% of_real_numeral
thf(fact_6557_of__real__eq__1__iff,axiom,
    ! [X: real] :
      ( ( ( real_V1803761363581548252l_real @ X )
        = one_one_real )
      = ( X = one_one_real ) ) ).

% of_real_eq_1_iff
thf(fact_6558_of__real__eq__1__iff,axiom,
    ! [X: real] :
      ( ( ( real_V4546457046886955230omplex @ X )
        = one_one_complex )
      = ( X = one_one_real ) ) ).

% of_real_eq_1_iff
thf(fact_6559_of__real__1,axiom,
    ( ( real_V1803761363581548252l_real @ one_one_real )
    = one_one_real ) ).

% of_real_1
thf(fact_6560_of__real__1,axiom,
    ( ( real_V4546457046886955230omplex @ one_one_real )
    = one_one_complex ) ).

% of_real_1
thf(fact_6561_of__real__eq__0__iff,axiom,
    ! [X: real] :
      ( ( ( real_V1803761363581548252l_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% of_real_eq_0_iff
thf(fact_6562_of__real__eq__0__iff,axiom,
    ! [X: real] :
      ( ( ( real_V4546457046886955230omplex @ X )
        = zero_zero_complex )
      = ( X = zero_zero_real ) ) ).

% of_real_eq_0_iff
thf(fact_6563_star__false__right,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ P @ bot_bot_assn )
      = bot_bot_assn ) ).

% star_false_right
thf(fact_6564_star__false__left,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ bot_bot_assn @ P )
      = bot_bot_assn ) ).

% star_false_left
thf(fact_6565_mult__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ( times_times_nat @ M @ K )
        = ( times_times_nat @ N @ K ) )
      = ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_6566_mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N ) )
      = ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_6567_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_6568_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_6569_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = one_one_nat )
      = ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_6570_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M @ N ) )
      = ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_6571_semiring__norm_I13_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).

% semiring_norm(13)
thf(fact_6572_semiring__norm_I11_J,axiom,
    ! [M: num] :
      ( ( times_times_num @ M @ one )
      = M ) ).

% semiring_norm(11)
thf(fact_6573_semiring__norm_I12_J,axiom,
    ! [N: num] :
      ( ( times_times_num @ one @ N )
      = N ) ).

% semiring_norm(12)
thf(fact_6574_norm__pre__pure__iff,axiom,
    ! [P: assn,B2: $o,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ P @ ( pure_assn @ B2 ) ) @ F @ Q )
      = ( B2
       => ( hoare_1429296392585015714_VEBTi @ P @ F @ Q ) ) ) ).

% norm_pre_pure_iff
thf(fact_6575_norm__pre__pure__iff,axiom,
    ! [P: assn,B2: $o,F: heap_Time_Heap_o,Q: $o > assn] :
      ( ( hoare_hoare_triple_o @ ( times_times_assn @ P @ ( pure_assn @ B2 ) ) @ F @ Q )
      = ( B2
       => ( hoare_hoare_triple_o @ P @ F @ Q ) ) ) ).

% norm_pre_pure_iff
thf(fact_6576_nat__0__less__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% nat_0_less_mult_iff
thf(fact_6577_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N ) ) ) ).

% mult_less_cancel2
thf(fact_6578_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_6579_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( K = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
          = zero_zero_nat ) )
      & ( ( K != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
          = ( divide_divide_nat @ M @ N ) ) ) ) ).

% nat_mult_div_cancel_disj
thf(fact_6580_of__real__0,axiom,
    ( ( real_V1803761363581548252l_real @ zero_zero_real )
    = zero_zero_real ) ).

% of_real_0
thf(fact_6581_of__real__0,axiom,
    ( ( real_V4546457046886955230omplex @ zero_zero_real )
    = zero_zero_complex ) ).

% of_real_0
thf(fact_6582_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( K = zero_zero_nat )
        | ( dvd_dvd_nat @ M @ N ) ) ) ).

% nat_mult_dvd_cancel_disj
thf(fact_6583_num__double,axiom,
    ! [N: num] :
      ( ( times_times_num @ ( bit0 @ one ) @ N )
      = ( bit0 @ N ) ) ).

% num_double
thf(fact_6584_power__mult__numeral,axiom,
    ! [A2: nat,M: num,N: num] :
      ( ( power_power_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_6585_power__mult__numeral,axiom,
    ! [A2: real,M: num,N: num] :
      ( ( power_power_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_6586_power__mult__numeral,axiom,
    ! [A2: int,M: num,N: num] :
      ( ( power_power_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_6587_power__mult__numeral,axiom,
    ! [A2: complex,M: num,N: num] :
      ( ( power_power_complex @ ( power_power_complex @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_complex @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_6588_power__mult__numeral,axiom,
    ! [A2: code_integer,M: num,N: num] :
      ( ( power_8256067586552552935nteger @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_6589_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% mult_le_cancel2
thf(fact_6590_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_6591_div__mult__self__is__m,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
        = M ) ) ).

% div_mult_self_is_m
thf(fact_6592_div__mult__self1__is__m,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
        = M ) ) ).

% div_mult_self1_is_m
thf(fact_6593_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_8256067586552552935nteger @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_6594_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A2: complex,N: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_complex @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_6595_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A2: uint32,N: nat] :
      ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ A2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_uint32 @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_6596_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A2: real,N: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ A2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_real @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_6597_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A2: rat,N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_rat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_6598_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A2: int,N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_int @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_6599_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_2184487114949457152l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_on7727431528512463931l_num1 ) ).

% power_minus1_even
thf(fact_6600_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_Code_integer ) ).

% power_minus1_even
thf(fact_6601_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_complex ) ).

% power_minus1_even
thf(fact_6602_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_uint32 ) ).

% power_minus1_even
thf(fact_6603_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_real ) ).

% power_minus1_even
thf(fact_6604_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_rat ) ).

% power_minus1_even
thf(fact_6605_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_int ) ).

% power_minus1_even
thf(fact_6606_odd__two__times__div__two__nat,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( minus_minus_nat @ N @ one_one_nat ) ) ) ).

% odd_two_times_div_two_nat
thf(fact_6607_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_6608_wi__hom__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( times_7065122842183080059l_num1 @ ( ring_17408606157368542149l_num1 @ A2 ) @ ( ring_17408606157368542149l_num1 @ B2 ) )
      = ( ring_17408606157368542149l_num1 @ ( times_times_int @ A2 @ B2 ) ) ) ).

% wi_hom_mult
thf(fact_6609_word__mult__def,axiom,
    ( times_7065122842183080059l_num1
    = ( ^ [A4: word_N3645301735248828278l_num1,B4: word_N3645301735248828278l_num1] : ( ring_17408606157368542149l_num1 @ ( times_times_int @ ( semiri7338730514057886004m1_int @ A4 ) @ ( semiri7338730514057886004m1_int @ B4 ) ) ) ) ) ).

% word_mult_def
thf(fact_6610_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N ) )
      = ( ( K = zero_zero_nat )
        | ( M = N ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_6611_mult__0,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% mult_0
thf(fact_6612_power__mult,axiom,
    ! [A2: nat,M: nat,N: nat] :
      ( ( power_power_nat @ A2 @ ( times_times_nat @ M @ N ) )
      = ( power_power_nat @ ( power_power_nat @ A2 @ M ) @ N ) ) ).

% power_mult
thf(fact_6613_power__mult,axiom,
    ! [A2: real,M: nat,N: nat] :
      ( ( power_power_real @ A2 @ ( times_times_nat @ M @ N ) )
      = ( power_power_real @ ( power_power_real @ A2 @ M ) @ N ) ) ).

% power_mult
thf(fact_6614_power__mult,axiom,
    ! [A2: int,M: nat,N: nat] :
      ( ( power_power_int @ A2 @ ( times_times_nat @ M @ N ) )
      = ( power_power_int @ ( power_power_int @ A2 @ M ) @ N ) ) ).

% power_mult
thf(fact_6615_power__mult,axiom,
    ! [A2: complex,M: nat,N: nat] :
      ( ( power_power_complex @ A2 @ ( times_times_nat @ M @ N ) )
      = ( power_power_complex @ ( power_power_complex @ A2 @ M ) @ N ) ) ).

% power_mult
thf(fact_6616_power__mult,axiom,
    ! [A2: code_integer,M: nat,N: nat] :
      ( ( power_8256067586552552935nteger @ A2 @ ( times_times_nat @ M @ N ) )
      = ( power_8256067586552552935nteger @ ( power_8256067586552552935nteger @ A2 @ M ) @ N ) ) ).

% power_mult
thf(fact_6617_add__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
      = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).

% add_mult_distrib
thf(fact_6618_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).

% add_mult_distrib2
thf(fact_6619_left__add__mult__distrib,axiom,
    ! [I: nat,U2: nat,J: nat,K: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ K ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U2 ) @ K ) ) ).

% left_add_mult_distrib
thf(fact_6620_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_6621_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_6622_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).

% mult_le_mono
thf(fact_6623_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).

% mult_le_mono1
thf(fact_6624_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).

% mult_le_mono2
thf(fact_6625_nat__mult__1__right,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ N @ one_one_nat )
      = N ) ).

% nat_mult_1_right
thf(fact_6626_nat__mult__1,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ one_one_nat @ N )
      = N ) ).

% nat_mult_1
thf(fact_6627_times__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( times_times_int @ zero_zero_int @ L )
      = zero_zero_int ) ).

% times_int_code(2)
thf(fact_6628_times__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( times_times_int @ K @ zero_zero_int )
      = zero_zero_int ) ).

% times_int_code(1)
thf(fact_6629_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
      = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).

% diff_mult_distrib
thf(fact_6630_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
      = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).

% diff_mult_distrib2
thf(fact_6631_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] :
      ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
      = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).

% int_distrib(2)
thf(fact_6632_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] :
      ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
      = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).

% int_distrib(1)
thf(fact_6633_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_6634_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_6635_assn__one__left,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ one_one_assn @ P )
      = P ) ).

% assn_one_left
thf(fact_6636_imult__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( ( times_7803423173614009249d_enat @ M @ N )
        = zero_z5237406670263579293d_enat )
      = ( ( M = zero_z5237406670263579293d_enat )
        | ( N = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
thf(fact_6637_frame__rule,axiom,
    ! [P: assn,C: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,R: assn] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ C @ Q )
     => ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ P @ R ) @ C
        @ ^ [X2: vEBT_VEBTi] : ( times_times_assn @ ( Q @ X2 ) @ R ) ) ) ).

% frame_rule
thf(fact_6638_frame__rule,axiom,
    ! [P: assn,C: heap_Time_Heap_o,Q: $o > assn,R: assn] :
      ( ( hoare_hoare_triple_o @ P @ C @ Q )
     => ( hoare_hoare_triple_o @ ( times_times_assn @ P @ R ) @ C
        @ ^ [X2: $o] : ( times_times_assn @ ( Q @ X2 ) @ R ) ) ) ).

% frame_rule
thf(fact_6639_div__mult2__numeral__eq,axiom,
    ! [A2: nat,K: num,L: num] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
      = ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_6640_div__mult2__numeral__eq,axiom,
    ! [A2: int,K: num,L: num] :
      ( ( divide_divide_int @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L ) )
      = ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( times_times_num @ K @ L ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_6641_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).

% mult_less_mono2
thf(fact_6642_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).

% mult_less_mono1
thf(fact_6643_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ( times_times_nat @ K @ M )
          = ( times_times_nat @ K @ N ) )
        = ( M = N ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_6644_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( ord_less_nat @ M @ N ) ) ) ).

% nat_mult_less_cancel1
thf(fact_6645_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M
        = ( times_times_nat @ M @ N ) )
     => ( ( N = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_6646_mlex__snd__decrI,axiom,
    ! [A2: nat,A6: nat,B2: nat,B7: nat,N3: nat] :
      ( ( A2 = A6 )
     => ( ( ord_less_nat @ B2 @ B7 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N3 ) @ B2 ) @ ( plus_plus_nat @ ( times_times_nat @ A6 @ N3 ) @ B7 ) ) ) ) ).

% mlex_snd_decrI
thf(fact_6647_mlex__fst__decrI,axiom,
    ! [A2: nat,A6: nat,B2: nat,N3: nat,B7: nat] :
      ( ( ord_less_nat @ A2 @ A6 )
     => ( ( ord_less_nat @ B2 @ N3 )
       => ( ( ord_less_nat @ B7 @ N3 )
         => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N3 ) @ B2 ) @ ( plus_plus_nat @ ( times_times_nat @ A6 @ N3 ) @ B7 ) ) ) ) ) ).

% mlex_fst_decrI
thf(fact_6648_mlex__bound,axiom,
    ! [A2: nat,A: nat,B2: nat,N3: nat] :
      ( ( ord_less_nat @ A2 @ A )
     => ( ( ord_less_nat @ B2 @ N3 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N3 ) @ B2 ) @ ( times_times_nat @ A @ N3 ) ) ) ) ).

% mlex_bound
thf(fact_6649_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
     => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).

% less_mult_imp_div_less
thf(fact_6650_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_int @ I @ J )
     => ( ( ord_less_int @ zero_zero_int @ K )
       => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).

% zmult_zless_mono2
thf(fact_6651_zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( ( times_times_int @ M @ N )
        = one_one_int )
      = ( ( ( M = one_one_int )
          & ( N = one_one_int ) )
        | ( ( M
            = ( uminus_uminus_int @ one_one_int ) )
          & ( N
            = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zmult_eq_1_iff
thf(fact_6652_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N: int] :
      ( ( ( times_times_int @ M @ N )
        = one_one_int )
     => ( ( M = one_one_int )
        | ( M
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
thf(fact_6653_zdvd__mono,axiom,
    ! [K: int,M: int,T: int] :
      ( ( K != zero_zero_int )
     => ( ( dvd_dvd_int @ M @ T )
        = ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).

% zdvd_mono
thf(fact_6654_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,N: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
     => ( ( K != zero_zero_int )
       => ( dvd_dvd_int @ M @ N ) ) ) ).

% zdvd_mult_cancel
thf(fact_6655_zdvd__period,axiom,
    ! [A2: int,D: int,X: int,T: int,C: int] :
      ( ( dvd_dvd_int @ A2 @ D )
     => ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ X @ T ) )
        = ( dvd_dvd_int @ A2 @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).

% zdvd_period
thf(fact_6656_zdvd__reduce,axiom,
    ! [K: int,N: int,M: int] :
      ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
      = ( dvd_dvd_int @ K @ N ) ) ).

% zdvd_reduce
thf(fact_6657_norm__pre__pure__rule1,axiom,
    ! [B2: $o,P: assn,F: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( B2
       => ( hoare_1429296392585015714_VEBTi @ P @ F @ Q ) )
     => ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ P @ ( pure_assn @ B2 ) ) @ F @ Q ) ) ).

% norm_pre_pure_rule1
thf(fact_6658_norm__pre__pure__rule1,axiom,
    ! [B2: $o,P: assn,F: heap_Time_Heap_o,Q: $o > assn] :
      ( ( B2
       => ( hoare_hoare_triple_o @ P @ F @ Q ) )
     => ( hoare_hoare_triple_o @ ( times_times_assn @ P @ ( pure_assn @ B2 ) ) @ F @ Q ) ) ).

% norm_pre_pure_rule1
thf(fact_6659_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
      = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
        & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).

% enat_0_less_mult_iff
thf(fact_6660_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( ord_less_eq_nat @ M @ N ) ) ) ).

% nat_mult_le_cancel1
thf(fact_6661_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( divide_divide_nat @ M @ N ) ) ) ).

% nat_mult_div_cancel1
thf(fact_6662_td__gal__lt,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( ( ord_less_nat @ A2 @ ( times_times_nat @ B2 @ C ) )
        = ( ord_less_nat @ ( divide_divide_nat @ A2 @ C ) @ B2 ) ) ) ).

% td_gal_lt
thf(fact_6663_div__less__iff__less__mult,axiom,
    ! [Q3: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q3 )
     => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q3 ) @ N )
        = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q3 ) ) ) ) ).

% div_less_iff_less_mult
thf(fact_6664_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U2: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U2 ) @ M )
          = N ) ) ) ).

% nat_eq_add_iff1
thf(fact_6665_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U2: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
        = ( M
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U2 ) @ N ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_6666_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U2: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U2 ) @ M ) @ N ) ) ) ).

% nat_le_add_iff1
thf(fact_6667_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U2: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
        = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U2 ) @ N ) ) ) ) ).

% nat_le_add_iff2
thf(fact_6668_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U2: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U2 ) @ M ) @ N ) ) ) ).

% nat_diff_add_eq1
thf(fact_6669_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U2: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
        = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U2 ) @ N ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_6670_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( dvd_dvd_nat @ M @ N ) ) ) ).

% dvd_mult_cancel
thf(fact_6671_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( dvd_dvd_nat @ M @ N ) ) ) ).

% nat_mult_dvd_cancel1
thf(fact_6672_bezout__add__strong__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ? [D6: nat,X3: nat,Y4: nat] :
          ( ( dvd_dvd_nat @ D6 @ A2 )
          & ( dvd_dvd_nat @ D6 @ B2 )
          & ( ( times_times_nat @ A2 @ X3 )
            = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y4 ) @ D6 ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_6673_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ( times_times_int @ M @ N )
          = one_one_int )
        = ( ( M = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
thf(fact_6674_uint__mult__ge0,axiom,
    ! [Xa: word_N3645301735248828278l_num1,X: word_N3645301735248828278l_num1] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ ( semiri7338730514057886004m1_int @ Xa ) @ ( semiri7338730514057886004m1_int @ X ) ) ) ).

% uint_mult_ge0
thf(fact_6675_minusinfinity,axiom,
    ! [D: int,P1: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K2: int] :
            ( ( P1 @ X3 )
            = ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ? [Z5: int] :
            ! [X3: int] :
              ( ( ord_less_int @ X3 @ Z5 )
             => ( ( P @ X3 )
                = ( P1 @ X3 ) ) )
         => ( ? [X_12: int] : ( P1 @ X_12 )
           => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).

% minusinfinity
thf(fact_6676_plusinfinity,axiom,
    ! [D: int,P5: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K2: int] :
            ( ( P5 @ X3 )
            = ( P5 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ? [Z5: int] :
            ! [X3: int] :
              ( ( ord_less_int @ Z5 @ X3 )
             => ( ( P @ X3 )
                = ( P5 @ X3 ) ) )
         => ( ? [X_12: int] : ( P5 @ X_12 )
           => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).

% plusinfinity
thf(fact_6677_zdiv__zmult2__eq,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A2 @ B2 ) @ C ) ) ) ).

% zdiv_zmult2_eq
thf(fact_6678_zdiv__mult__self,axiom,
    ! [M: int,A2: int,N: int] :
      ( ( M != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ ( times_times_int @ M @ N ) ) @ M )
        = ( plus_plus_int @ ( divide_divide_int @ A2 @ M ) @ N ) ) ) ).

% zdiv_mult_self
thf(fact_6679_div__lt__mult,axiom,
    ! [I: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1,X: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ I @ ( divide1791077408188789448l_num1 @ K @ X ) )
     => ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ X )
       => ( ord_le750835935415966154l_num1 @ ( times_7065122842183080059l_num1 @ I @ X ) @ K ) ) ) ).

% div_lt_mult
thf(fact_6680_More__Word_Oword__div__mult,axiom,
    ! [C: word_N3645301735248828278l_num1,A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ C )
     => ( ( ord_le750835935415966154l_num1 @ A2 @ ( times_7065122842183080059l_num1 @ B2 @ C ) )
       => ( ord_le750835935415966154l_num1 @ ( divide1791077408188789448l_num1 @ A2 @ C ) @ B2 ) ) ) ).

% More_Word.word_div_mult
thf(fact_6681_power__even__eq,axiom,
    ! [A2: nat,N: nat] :
      ( ( power_power_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_nat @ ( power_power_nat @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_6682_power__even__eq,axiom,
    ! [A2: real,N: nat] :
      ( ( power_power_real @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_real @ ( power_power_real @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_6683_power__even__eq,axiom,
    ! [A2: int,N: nat] :
      ( ( power_power_int @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_int @ ( power_power_int @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_6684_power__even__eq,axiom,
    ! [A2: complex,N: nat] :
      ( ( power_power_complex @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_complex @ ( power_power_complex @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_6685_power__even__eq,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_8256067586552552935nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_6686_msrevs_I1_J,axiom,
    ! [N: nat,K: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) @ N )
        = ( plus_plus_nat @ ( divide_divide_nat @ M @ N ) @ K ) ) ) ).

% msrevs(1)
thf(fact_6687_split__div,axiom,
    ! [P: nat > $o,M: nat,N: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ zero_zero_nat ) )
        & ( ( N != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
               => ( P @ I4 ) ) ) ) ) ) ).

% split_div
thf(fact_6688_dividend__less__div__times,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).

% dividend_less_div_times
thf(fact_6689_dividend__less__times__div,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).

% dividend_less_times_div
thf(fact_6690_td__gal,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ B2 @ C ) @ A2 )
        = ( ord_less_eq_nat @ B2 @ ( divide_divide_nat @ A2 @ C ) ) ) ) ).

% td_gal
thf(fact_6691_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q3 )
     => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q3 ) )
        = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q3 ) @ N ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_6692_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M3: nat,N4: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N4 ) ) ) ) ) ).

% mult_eq_if
thf(fact_6693_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U2: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
        = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U2 ) @ N ) ) ) ) ).

% nat_less_add_iff2
thf(fact_6694_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U2: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U2 ) @ M ) @ N ) ) ) ).

% nat_less_add_iff1
thf(fact_6695_nat__mult__power__less__eq,axiom,
    ! [B2: nat,A2: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ( ord_less_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ B2 @ N ) ) @ ( power_power_nat @ B2 @ M ) )
        = ( ord_less_nat @ A2 @ ( power_power_nat @ B2 @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).

% nat_mult_power_less_eq
thf(fact_6696_dvd__mult__cancel1,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
        = ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel1
thf(fact_6697_dvd__mult__cancel2,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
        = ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel2
thf(fact_6698_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( ( ord_less_int @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).

% zmult_zless_mono2_lemma
thf(fact_6699_dvd__minus__add,axiom,
    ! [Q3: nat,N: nat,R3: nat,M: nat] :
      ( ( ord_less_eq_nat @ Q3 @ N )
     => ( ( ord_less_eq_nat @ Q3 @ ( times_times_nat @ R3 @ M ) )
       => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q3 ) )
          = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R3 @ M ) @ Q3 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_6700_pos__mult__pos__ge,axiom,
    ! [X: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ N )
       => ( ord_less_eq_int @ ( times_times_int @ N @ one_one_int ) @ ( times_times_int @ N @ X ) ) ) ) ).

% pos_mult_pos_ge
thf(fact_6701_incr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int] :
            ( ( P @ X3 )
           => ( P @ ( plus_plus_int @ X3 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X6: int] :
              ( ( P @ X6 )
             => ( P @ ( plus_plus_int @ X6 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% incr_mult_lemma
thf(fact_6702_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q4: int,R4: int,Q3: int,R3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q4 ) @ 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 @ Q4 ) ) ) ) ) ).

% unique_quotient_lemma_neg
thf(fact_6703_unique__quotient__lemma,axiom,
    ! [B2: int,Q4: int,R4: int,Q3: int,R3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q4 ) @ 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 @ Q4 @ Q3 ) ) ) ) ) ).

% unique_quotient_lemma
thf(fact_6704_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q3: int,R3: int,B7: int,Q4: int,R4: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B2 @ Q3 ) @ R3 )
        = ( plus_plus_int @ ( times_times_int @ B7 @ Q4 ) @ R4 ) )
     => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q4 ) @ R4 ) @ zero_zero_int )
       => ( ( ord_less_int @ R3 @ B2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
           => ( ( ord_less_int @ zero_zero_int @ B7 )
             => ( ( ord_less_eq_int @ B7 @ B2 )
               => ( ord_less_eq_int @ Q4 @ Q3 ) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
thf(fact_6705_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q3: int,R3: int,B7: int,Q4: int,R4: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B2 @ Q3 ) @ R3 )
        = ( plus_plus_int @ ( times_times_int @ B7 @ Q4 ) @ R4 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q4 ) @ R4 ) )
       => ( ( ord_less_int @ R4 @ B7 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
           => ( ( ord_less_int @ zero_zero_int @ B7 )
             => ( ( ord_less_eq_int @ B7 @ B2 )
               => ( ord_less_eq_int @ Q3 @ Q4 ) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
thf(fact_6706_q__pos__lemma,axiom,
    ! [B7: int,Q4: int,R4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q4 ) @ R4 ) )
     => ( ( ord_less_int @ R4 @ B7 )
       => ( ( ord_less_int @ zero_zero_int @ B7 )
         => ( ord_less_eq_int @ zero_zero_int @ Q4 ) ) ) ) ).

% q_pos_lemma
thf(fact_6707_decr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int] :
            ( ( P @ X3 )
           => ( P @ ( minus_minus_int @ X3 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X6: int] :
              ( ( P @ X6 )
             => ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% decr_mult_lemma
thf(fact_6708_udvd__incr__lem0,axiom,
    ! [Up: int,Uq: int,N: int,K4: word_N3645301735248828278l_num1,N7: int] :
      ( ( ord_less_int @ Up @ Uq )
     => ( ( Up
          = ( times_times_int @ N @ ( semiri7338730514057886004m1_int @ K4 ) ) )
       => ( ( Uq
            = ( times_times_int @ N7 @ ( semiri7338730514057886004m1_int @ K4 ) ) )
         => ( ord_less_eq_int @ ( plus_plus_int @ Up @ ( semiri7338730514057886004m1_int @ K4 ) ) @ Uq ) ) ) ) ).

% udvd_incr_lem0
thf(fact_6709_udvd__incr__lem,axiom,
    ! [Up: int,Uq: int,Ua: int,N: int,K4: word_N3645301735248828278l_num1,N7: int] :
      ( ( ord_less_int @ Up @ Uq )
     => ( ( Up
          = ( plus_plus_int @ Ua @ ( times_times_int @ N @ ( semiri7338730514057886004m1_int @ K4 ) ) ) )
       => ( ( Uq
            = ( plus_plus_int @ Ua @ ( times_times_int @ N7 @ ( semiri7338730514057886004m1_int @ K4 ) ) ) )
         => ( ord_less_eq_int @ ( plus_plus_int @ Up @ ( semiri7338730514057886004m1_int @ K4 ) ) @ Uq ) ) ) ) ).

% udvd_incr_lem
thf(fact_6710_udvd__incr0,axiom,
    ! [P4: word_N3645301735248828278l_num1,Q3: word_N3645301735248828278l_num1,N: int,K4: word_N3645301735248828278l_num1,N7: int] :
      ( ( ord_le750835935415966154l_num1 @ P4 @ Q3 )
     => ( ( ( semiri7338730514057886004m1_int @ P4 )
          = ( times_times_int @ N @ ( semiri7338730514057886004m1_int @ K4 ) ) )
       => ( ( ( semiri7338730514057886004m1_int @ Q3 )
            = ( times_times_int @ N7 @ ( semiri7338730514057886004m1_int @ K4 ) ) )
         => ( ord_le3335648743751981014l_num1 @ ( plus_p361126936061061375l_num1 @ P4 @ K4 ) @ Q3 ) ) ) ) ).

% udvd_incr0
thf(fact_6711_udvd__decr0,axiom,
    ! [P4: word_N3645301735248828278l_num1,Q3: word_N3645301735248828278l_num1,N: int,K4: word_N3645301735248828278l_num1,N7: int] :
      ( ( ord_le750835935415966154l_num1 @ P4 @ Q3 )
     => ( ( ( semiri7338730514057886004m1_int @ P4 )
          = ( times_times_int @ N @ ( semiri7338730514057886004m1_int @ K4 ) ) )
       => ( ( ( semiri7338730514057886004m1_int @ Q3 )
            = ( times_times_int @ N7 @ ( semiri7338730514057886004m1_int @ K4 ) ) )
         => ( ( ( semiri7338730514057886004m1_int @ Q3 )
              = ( times_times_int @ N7 @ ( semiri7338730514057886004m1_int @ K4 ) ) )
           => ( ord_le3335648743751981014l_num1 @ P4 @ ( minus_4019991460397169231l_num1 @ Q3 @ K4 ) ) ) ) ) ) ).

% udvd_decr0
thf(fact_6712_div__le__mult,axiom,
    ! [I: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1,X: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ I @ ( divide1791077408188789448l_num1 @ K @ X ) )
     => ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ X )
       => ( ord_le3335648743751981014l_num1 @ ( times_7065122842183080059l_num1 @ I @ X ) @ K ) ) ) ).

% div_le_mult
thf(fact_6713_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_6714_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_6715_split__zdiv,axiom,
    ! [P: int > $o,N: int,K: int] :
      ( ( P @ ( divide_divide_int @ N @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ zero_zero_int ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I4: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ I4 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I4: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ I4 ) ) ) ) ) ).

% split_zdiv
thf(fact_6716_udvd__incr_H,axiom,
    ! [P4: word_N3645301735248828278l_num1,Q3: word_N3645301735248828278l_num1,Ua: int,N: int,K4: word_N3645301735248828278l_num1,N7: int] :
      ( ( ord_le750835935415966154l_num1 @ P4 @ Q3 )
     => ( ( ( semiri7338730514057886004m1_int @ P4 )
          = ( plus_plus_int @ Ua @ ( times_times_int @ N @ ( semiri7338730514057886004m1_int @ K4 ) ) ) )
       => ( ( ( semiri7338730514057886004m1_int @ Q3 )
            = ( plus_plus_int @ Ua @ ( times_times_int @ N7 @ ( semiri7338730514057886004m1_int @ K4 ) ) ) )
         => ( ord_le3335648743751981014l_num1 @ ( plus_p361126936061061375l_num1 @ P4 @ K4 ) @ Q3 ) ) ) ) ).

% udvd_incr'
thf(fact_6717_udvd__decr_H,axiom,
    ! [P4: word_N3645301735248828278l_num1,Q3: word_N3645301735248828278l_num1,Ua: int,N: int,K4: word_N3645301735248828278l_num1,N7: int] :
      ( ( ord_le750835935415966154l_num1 @ P4 @ Q3 )
     => ( ( ( semiri7338730514057886004m1_int @ P4 )
          = ( plus_plus_int @ Ua @ ( times_times_int @ N @ ( semiri7338730514057886004m1_int @ K4 ) ) ) )
       => ( ( ( semiri7338730514057886004m1_int @ Q3 )
            = ( plus_plus_int @ Ua @ ( times_times_int @ N7 @ ( semiri7338730514057886004m1_int @ K4 ) ) ) )
         => ( ( ( semiri7338730514057886004m1_int @ Q3 )
              = ( plus_plus_int @ Ua @ ( times_times_int @ N7 @ ( semiri7338730514057886004m1_int @ K4 ) ) ) )
           => ( ord_le3335648743751981014l_num1 @ P4 @ ( minus_4019991460397169231l_num1 @ Q3 @ K4 ) ) ) ) ) ) ).

% udvd_decr'
thf(fact_6718_zero__le__even__power_H,axiom,
    ! [A2: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% zero_le_even_power'
thf(fact_6719_zero__le__even__power_H,axiom,
    ! [A2: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% zero_le_even_power'
thf(fact_6720_zero__le__even__power_H,axiom,
    ! [A2: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% zero_le_even_power'
thf(fact_6721_zero__le__even__power_H,axiom,
    ! [A2: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% zero_le_even_power'
thf(fact_6722_nat__add__offset__less,axiom,
    ! [Y: nat,N: nat,X: nat,M: nat,Sz: nat] :
      ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
       => ( ( Sz
            = ( plus_plus_nat @ M @ N ) )
         => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ Y ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Sz ) ) ) ) ) ).

% nat_add_offset_less
thf(fact_6723_nat__power__less__diff,axiom,
    ! [N: nat,Q3: nat,M: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ 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 @ N ) ) ) ) ).

% nat_power_less_diff
thf(fact_6724_nat__le__power__trans,axiom,
    ! [N: nat,M: nat,K: nat] :
      ( ( ord_less_eq_nat @ N @ ( 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 ) @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% nat_le_power_trans
thf(fact_6725_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_6726_nat__less__power__trans,axiom,
    ! [N: nat,M: nat,K: nat] :
      ( ( ord_less_nat @ N @ ( 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 ) @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% nat_less_power_trans
thf(fact_6727_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_6728_power__2__mult__step__le,axiom,
    ! [N7: nat,N: nat,K5: nat,K: nat] :
      ( ( ord_less_eq_nat @ N7 @ N )
     => ( ( ord_less_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) @ K5 ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) )
       => ( ord_less_eq_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) @ ( plus_plus_nat @ K5 @ one_one_nat ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).

% power_2_mult_step_le
thf(fact_6729_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_6730_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_6731_word__div__eq__1__iff,axiom,
    ! [N: word_N3645301735248828278l_num1,M: word_N3645301735248828278l_num1] :
      ( ( ( divide1791077408188789448l_num1 @ N @ M )
        = one_on7727431528512463931l_num1 )
      = ( ( ord_le3335648743751981014l_num1 @ M @ N )
        & ( ord_less_nat @ ( semiri7341220984566936280m1_nat @ N ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri7341220984566936280m1_nat @ M ) ) ) ) ) ).

% word_div_eq_1_iff
thf(fact_6732_nonzero__of__real__divide,axiom,
    ! [Y: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X @ Y ) )
        = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ) ).

% nonzero_of_real_divide
thf(fact_6733_nonzero__of__real__divide,axiom,
    ! [Y: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X @ Y ) )
        = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ) ).

% nonzero_of_real_divide
thf(fact_6734_VEBT__internal_Omulcomm,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 ) ) ).

% VEBT_internal.mulcomm
thf(fact_6735_set__bit__0,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( bit_se4894374433684937756l_num1 @ zero_zero_nat @ A2 )
      = ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( divide1791077408188789448l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_6736_set__bit__0,axiom,
    ! [A2: uint32] :
      ( ( bit_se6647067497041451410uint32 @ zero_zero_nat @ A2 )
      = ( plus_plus_uint32 @ one_one_uint32 @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( divide_divide_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_6737_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_6738_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_6739_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_6740_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_6741_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_6742_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_6743_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_6744_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_6745_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_6746_unset__bit__0,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( bit_se5331074070815623765l_num1 @ zero_zero_nat @ A2 )
      = ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( divide1791077408188789448l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_6747_unset__bit__0,axiom,
    ! [A2: uint32] :
      ( ( bit_se4315839071623982667uint32 @ zero_zero_nat @ A2 )
      = ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( divide_divide_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_6748_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_6749_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_6750_abs__idempotent,axiom,
    ! [A2: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A2 ) )
      = ( abs_abs_real @ A2 ) ) ).

% abs_idempotent
thf(fact_6751_abs__idempotent,axiom,
    ! [A2: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A2 ) )
      = ( abs_abs_int @ A2 ) ) ).

% abs_idempotent
thf(fact_6752_abs__idempotent,axiom,
    ! [A2: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A2 ) )
      = ( abs_abs_rat @ A2 ) ) ).

% abs_idempotent
thf(fact_6753_abs__idempotent,axiom,
    ! [A2: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A2 ) )
      = ( abs_abs_Code_integer @ A2 ) ) ).

% abs_idempotent
thf(fact_6754_abs__abs,axiom,
    ! [A2: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A2 ) )
      = ( abs_abs_real @ A2 ) ) ).

% abs_abs
thf(fact_6755_abs__abs,axiom,
    ! [A2: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A2 ) )
      = ( abs_abs_int @ A2 ) ) ).

% abs_abs
thf(fact_6756_abs__abs,axiom,
    ! [A2: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A2 ) )
      = ( abs_abs_rat @ A2 ) ) ).

% abs_abs
thf(fact_6757_abs__abs,axiom,
    ! [A2: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A2 ) )
      = ( abs_abs_Code_integer @ A2 ) ) ).

% abs_abs
thf(fact_6758_abs__0,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_0
thf(fact_6759_abs__0,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_0
thf(fact_6760_abs__0,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_0
thf(fact_6761_abs__0,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_0
thf(fact_6762_abs__zero,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_zero
thf(fact_6763_abs__zero,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_zero
thf(fact_6764_abs__zero,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_zero
thf(fact_6765_abs__zero,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_zero
thf(fact_6766_abs__eq__0,axiom,
    ! [A2: code_integer] :
      ( ( ( abs_abs_Code_integer @ A2 )
        = zero_z3403309356797280102nteger )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
thf(fact_6767_abs__eq__0,axiom,
    ! [A2: real] :
      ( ( ( abs_abs_real @ A2 )
        = zero_zero_real )
      = ( A2 = zero_zero_real ) ) ).

% abs_eq_0
thf(fact_6768_abs__eq__0,axiom,
    ! [A2: rat] :
      ( ( ( abs_abs_rat @ A2 )
        = zero_zero_rat )
      = ( A2 = zero_zero_rat ) ) ).

% abs_eq_0
thf(fact_6769_abs__eq__0,axiom,
    ! [A2: int] :
      ( ( ( abs_abs_int @ A2 )
        = zero_zero_int )
      = ( A2 = zero_zero_int ) ) ).

% abs_eq_0
thf(fact_6770_abs__0__eq,axiom,
    ! [A2: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( abs_abs_Code_integer @ A2 ) )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
thf(fact_6771_abs__0__eq,axiom,
    ! [A2: real] :
      ( ( zero_zero_real
        = ( abs_abs_real @ A2 ) )
      = ( A2 = zero_zero_real ) ) ).

% abs_0_eq
thf(fact_6772_abs__0__eq,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat
        = ( abs_abs_rat @ A2 ) )
      = ( A2 = zero_zero_rat ) ) ).

% abs_0_eq
thf(fact_6773_abs__0__eq,axiom,
    ! [A2: int] :
      ( ( zero_zero_int
        = ( abs_abs_int @ A2 ) )
      = ( A2 = zero_zero_int ) ) ).

% abs_0_eq
thf(fact_6774_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% abs_numeral
thf(fact_6775_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ N ) ) ).

% abs_numeral
thf(fact_6776_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ N ) ) ).

% abs_numeral
thf(fact_6777_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% abs_numeral
thf(fact_6778_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_6779_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_6780_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_6781_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_6782_abs__1,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_1
thf(fact_6783_abs__1,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_1
thf(fact_6784_abs__1,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_1
thf(fact_6785_abs__1,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_1
thf(fact_6786_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_6787_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_6788_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_6789_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_6790_abs__minus__cancel,axiom,
    ! [A2: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A2 ) )
      = ( abs_abs_Code_integer @ A2 ) ) ).

% abs_minus_cancel
thf(fact_6791_abs__minus__cancel,axiom,
    ! [A2: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A2 ) )
      = ( abs_abs_real @ A2 ) ) ).

% abs_minus_cancel
thf(fact_6792_abs__minus__cancel,axiom,
    ! [A2: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A2 ) )
      = ( abs_abs_rat @ A2 ) ) ).

% abs_minus_cancel
thf(fact_6793_abs__minus__cancel,axiom,
    ! [A2: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A2 ) )
      = ( abs_abs_int @ A2 ) ) ).

% abs_minus_cancel
thf(fact_6794_abs__minus,axiom,
    ! [A2: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A2 ) )
      = ( abs_abs_Code_integer @ A2 ) ) ).

% abs_minus
thf(fact_6795_abs__minus,axiom,
    ! [A2: complex] :
      ( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A2 ) )
      = ( abs_abs_complex @ A2 ) ) ).

% abs_minus
thf(fact_6796_abs__minus,axiom,
    ! [A2: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A2 ) )
      = ( abs_abs_real @ A2 ) ) ).

% abs_minus
thf(fact_6797_abs__minus,axiom,
    ! [A2: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A2 ) )
      = ( abs_abs_rat @ A2 ) ) ).

% abs_minus
thf(fact_6798_abs__minus,axiom,
    ! [A2: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A2 ) )
      = ( abs_abs_int @ A2 ) ) ).

% abs_minus
thf(fact_6799_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
      = ( semiri681578069525770553at_rat @ N ) ) ).

% abs_of_nat
thf(fact_6800_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% abs_of_nat
thf(fact_6801_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
      = ( semiri5074537144036343181t_real @ N ) ) ).

% abs_of_nat
thf(fact_6802_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
      = ( semiri4939895301339042750nteger @ N ) ) ).

% abs_of_nat
thf(fact_6803_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_6804_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_6805_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_6806_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_6807_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_6808_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_6809_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_6810_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_6811_unset__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% unset_bit_nonnegative_int_iff
thf(fact_6812_set__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% set_bit_nonnegative_int_iff
thf(fact_6813_unset__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% unset_bit_negative_int_iff
thf(fact_6814_set__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% set_bit_negative_int_iff
thf(fact_6815_of__int__abs,axiom,
    ! [X: int] :
      ( ( ring_1_of_int_int @ ( abs_abs_int @ X ) )
      = ( abs_abs_int @ ( ring_1_of_int_int @ X ) ) ) ).

% of_int_abs
thf(fact_6816_of__int__abs,axiom,
    ! [X: int] :
      ( ( ring_18347121197199848620nteger @ ( abs_abs_int @ X ) )
      = ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ X ) ) ) ).

% of_int_abs
thf(fact_6817_of__int__abs,axiom,
    ! [X: int] :
      ( ( ring_1_of_int_real @ ( abs_abs_int @ X ) )
      = ( abs_abs_real @ ( ring_1_of_int_real @ X ) ) ) ).

% of_int_abs
thf(fact_6818_of__int__abs,axiom,
    ! [X: int] :
      ( ( ring_1_of_int_rat @ ( abs_abs_int @ X ) )
      = ( abs_abs_rat @ ( ring_1_of_int_rat @ X ) ) ) ).

% of_int_abs
thf(fact_6819_abs__of__nonneg,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( abs_abs_Code_integer @ A2 )
        = A2 ) ) ).

% abs_of_nonneg
thf(fact_6820_abs__of__nonneg,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( abs_abs_real @ A2 )
        = A2 ) ) ).

% abs_of_nonneg
thf(fact_6821_abs__of__nonneg,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( abs_abs_rat @ A2 )
        = A2 ) ) ).

% abs_of_nonneg
thf(fact_6822_abs__of__nonneg,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( abs_abs_int @ A2 )
        = A2 ) ) ).

% abs_of_nonneg
thf(fact_6823_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_6824_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_6825_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_6826_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_6827_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_6828_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_6829_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_6830_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_6831_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_6832_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_6833_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_6834_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_6835_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% abs_neg_numeral
thf(fact_6836_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ N ) ) ).

% abs_neg_numeral
thf(fact_6837_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ N ) ) ).

% abs_neg_numeral
thf(fact_6838_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ N ) ) ).

% abs_neg_numeral
thf(fact_6839_abs__neg__one,axiom,
    ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = one_one_Code_integer ) ).

% abs_neg_one
thf(fact_6840_abs__neg__one,axiom,
    ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
    = one_one_real ) ).

% abs_neg_one
thf(fact_6841_abs__neg__one,axiom,
    ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = one_one_rat ) ).

% abs_neg_one
thf(fact_6842_abs__neg__one,axiom,
    ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
    = one_one_int ) ).

% abs_neg_one
thf(fact_6843_abs__power__minus,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A2 ) @ N ) )
      = ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ).

% abs_power_minus
thf(fact_6844_abs__power__minus,axiom,
    ! [A2: real,N: nat] :
      ( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A2 ) @ N ) )
      = ( abs_abs_real @ ( power_power_real @ A2 @ N ) ) ) ).

% abs_power_minus
thf(fact_6845_abs__power__minus,axiom,
    ! [A2: rat,N: nat] :
      ( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A2 ) @ N ) )
      = ( abs_abs_rat @ ( power_power_rat @ A2 @ N ) ) ) ).

% abs_power_minus
thf(fact_6846_abs__power__minus,axiom,
    ! [A2: int,N: nat] :
      ( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A2 ) @ N ) )
      = ( abs_abs_int @ ( power_power_int @ A2 @ N ) ) ) ).

% abs_power_minus
thf(fact_6847_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_6848_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_6849_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_6850_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_6851_abs__of__nonpos,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A2 )
        = ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% abs_of_nonpos
thf(fact_6852_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_6853_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_6854_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_6855_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_6856_zero__less__power__abs__iff,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A2 ) @ N ) )
      = ( ( A2 != zero_z3403309356797280102nteger )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_6857_zero__less__power__abs__iff,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A2 ) @ N ) )
      = ( ( A2 != zero_zero_real )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_6858_zero__less__power__abs__iff,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A2 ) @ N ) )
      = ( ( A2 != zero_zero_rat )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_6859_zero__less__power__abs__iff,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A2 ) @ N ) )
      = ( ( A2 != zero_zero_int )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_6860_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_6861_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_6862_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_6863_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_6864_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_6865_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_6866_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_6867_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_6868_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_6869_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_6870_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_6871_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_6872_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_6873_abs__eq__0__iff,axiom,
    ! [A2: code_integer] :
      ( ( ( abs_abs_Code_integer @ A2 )
        = zero_z3403309356797280102nteger )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
thf(fact_6874_abs__eq__0__iff,axiom,
    ! [A2: real] :
      ( ( ( abs_abs_real @ A2 )
        = zero_zero_real )
      = ( A2 = zero_zero_real ) ) ).

% abs_eq_0_iff
thf(fact_6875_abs__eq__0__iff,axiom,
    ! [A2: rat] :
      ( ( ( abs_abs_rat @ A2 )
        = zero_zero_rat )
      = ( A2 = zero_zero_rat ) ) ).

% abs_eq_0_iff
thf(fact_6876_abs__eq__0__iff,axiom,
    ! [A2: int] :
      ( ( ( abs_abs_int @ A2 )
        = zero_zero_int )
      = ( A2 = zero_zero_int ) ) ).

% abs_eq_0_iff
thf(fact_6877_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_6878_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_6879_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_6880_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_6881_abs__ge__self,axiom,
    ! [A2: real] : ( ord_less_eq_real @ A2 @ ( abs_abs_real @ A2 ) ) ).

% abs_ge_self
thf(fact_6882_abs__ge__self,axiom,
    ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ A2 @ ( abs_abs_Code_integer @ A2 ) ) ).

% abs_ge_self
thf(fact_6883_abs__ge__self,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ A2 @ ( abs_abs_rat @ A2 ) ) ).

% abs_ge_self
thf(fact_6884_abs__ge__self,axiom,
    ! [A2: int] : ( ord_less_eq_int @ A2 @ ( abs_abs_int @ A2 ) ) ).

% abs_ge_self
thf(fact_6885_abs__one,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_one
thf(fact_6886_abs__one,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_one
thf(fact_6887_abs__one,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_one
thf(fact_6888_abs__one,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_one
thf(fact_6889_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_6890_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_6891_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_6892_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_6893_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_6894_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_6895_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_6896_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_6897_power__abs,axiom,
    ! [A2: rat,N: nat] :
      ( ( abs_abs_rat @ ( power_power_rat @ A2 @ N ) )
      = ( power_power_rat @ ( abs_abs_rat @ A2 ) @ N ) ) ).

% power_abs
thf(fact_6898_power__abs,axiom,
    ! [A2: real,N: nat] :
      ( ( abs_abs_real @ ( power_power_real @ A2 @ N ) )
      = ( power_power_real @ ( abs_abs_real @ A2 ) @ N ) ) ).

% power_abs
thf(fact_6899_power__abs,axiom,
    ! [A2: int,N: nat] :
      ( ( abs_abs_int @ ( power_power_int @ A2 @ N ) )
      = ( power_power_int @ ( abs_abs_int @ A2 ) @ N ) ) ).

% power_abs
thf(fact_6900_power__abs,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A2 @ N ) )
      = ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A2 ) @ N ) ) ).

% power_abs
thf(fact_6901_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_6902_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_6903_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_6904_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_6905_dvd__if__abs__eq,axiom,
    ! [L: real,K: real] :
      ( ( ( abs_abs_real @ L )
        = ( abs_abs_real @ K ) )
     => ( dvd_dvd_real @ L @ K ) ) ).

% dvd_if_abs_eq
thf(fact_6906_dvd__if__abs__eq,axiom,
    ! [L: int,K: int] :
      ( ( ( abs_abs_int @ L )
        = ( abs_abs_int @ K ) )
     => ( dvd_dvd_int @ L @ K ) ) ).

% dvd_if_abs_eq
thf(fact_6907_dvd__if__abs__eq,axiom,
    ! [L: rat,K: rat] :
      ( ( ( abs_abs_rat @ L )
        = ( abs_abs_rat @ K ) )
     => ( dvd_dvd_rat @ L @ K ) ) ).

% dvd_if_abs_eq
thf(fact_6908_dvd__if__abs__eq,axiom,
    ! [L: code_integer,K: code_integer] :
      ( ( ( abs_abs_Code_integer @ L )
        = ( abs_abs_Code_integer @ K ) )
     => ( dvd_dvd_Code_integer @ L @ K ) ) ).

% dvd_if_abs_eq
thf(fact_6909_abs__ge__zero,axiom,
    ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A2 ) ) ).

% abs_ge_zero
thf(fact_6910_abs__ge__zero,axiom,
    ! [A2: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A2 ) ) ).

% abs_ge_zero
thf(fact_6911_abs__ge__zero,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A2 ) ) ).

% abs_ge_zero
thf(fact_6912_abs__ge__zero,axiom,
    ! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A2 ) ) ).

% abs_ge_zero
thf(fact_6913_abs__of__pos,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( abs_abs_Code_integer @ A2 )
        = A2 ) ) ).

% abs_of_pos
thf(fact_6914_abs__of__pos,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( abs_abs_real @ A2 )
        = A2 ) ) ).

% abs_of_pos
thf(fact_6915_abs__of__pos,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( abs_abs_rat @ A2 )
        = A2 ) ) ).

% abs_of_pos
thf(fact_6916_abs__of__pos,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( abs_abs_int @ A2 )
        = A2 ) ) ).

% abs_of_pos
thf(fact_6917_abs__not__less__zero,axiom,
    ! [A2: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A2 ) @ zero_z3403309356797280102nteger ) ).

% abs_not_less_zero
thf(fact_6918_abs__not__less__zero,axiom,
    ! [A2: real] :
      ~ ( ord_less_real @ ( abs_abs_real @ A2 ) @ zero_zero_real ) ).

% abs_not_less_zero
thf(fact_6919_abs__not__less__zero,axiom,
    ! [A2: rat] :
      ~ ( ord_less_rat @ ( abs_abs_rat @ A2 ) @ zero_zero_rat ) ).

% abs_not_less_zero
thf(fact_6920_abs__not__less__zero,axiom,
    ! [A2: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A2 ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_6921_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_6922_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_6923_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_6924_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_6925_abs__mult__less,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer,D: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A2 ) @ C )
     => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B2 ) @ D )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_6926_abs__mult__less,axiom,
    ! [A2: real,C: real,B2: real,D: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A2 ) @ C )
     => ( ( ord_less_real @ ( abs_abs_real @ B2 ) @ D )
       => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B2 ) ) @ ( times_times_real @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_6927_abs__mult__less,axiom,
    ! [A2: rat,C: rat,B2: rat,D: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A2 ) @ C )
     => ( ( ord_less_rat @ ( abs_abs_rat @ B2 ) @ D )
       => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B2 ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_6928_abs__mult__less,axiom,
    ! [A2: int,C: int,B2: int,D: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A2 ) @ C )
     => ( ( ord_less_int @ ( abs_abs_int @ B2 ) @ D )
       => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) @ ( times_times_int @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_6929_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_6930_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_6931_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_6932_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_6933_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_6934_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_6935_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_6936_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_6937_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_6938_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_6939_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_6940_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_6941_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_6942_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_6943_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_6944_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_6945_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_6946_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_6947_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_6948_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_6949_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_6950_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_6951_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_6952_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_6953_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_6954_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_6955_abs__ge__minus__self,axiom,
    ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( abs_abs_Code_integer @ A2 ) ) ).

% abs_ge_minus_self
thf(fact_6956_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_6957_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_6958_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_6959_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_6960_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_6961_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_6962_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_6963_dense__eq0__I,axiom,
    ! [X: real] :
      ( ! [E: real] :
          ( ( ord_less_real @ zero_zero_real @ E )
         => ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E ) )
     => ( X = zero_zero_real ) ) ).

% dense_eq0_I
thf(fact_6964_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E )
         => ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E ) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
thf(fact_6965_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_6966_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_6967_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_6968_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_6969_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_6970_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_6971_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_6972_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_6973_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_6974_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_6975_zero__le__power__abs,axiom,
    ! [A2: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A2 ) @ N ) ) ).

% zero_le_power_abs
thf(fact_6976_zero__le__power__abs,axiom,
    ! [A2: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A2 ) @ N ) ) ).

% zero_le_power_abs
thf(fact_6977_zero__le__power__abs,axiom,
    ! [A2: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A2 ) @ N ) ) ).

% zero_le_power_abs
thf(fact_6978_zero__le__power__abs,axiom,
    ! [A2: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A2 ) @ N ) ) ).

% zero_le_power_abs
thf(fact_6979_abs__minus__le__zero,axiom,
    ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A2 ) ) @ zero_z3403309356797280102nteger ) ).

% abs_minus_le_zero
thf(fact_6980_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_6981_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_6982_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_6983_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_6984_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_6985_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_6986_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_6987_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_6988_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_6989_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_6990_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_6991_abs__if,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A4 ) @ A4 ) ) ) ).

% abs_if
thf(fact_6992_abs__if,axiom,
    ( abs_abs_real
    = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).

% abs_if
thf(fact_6993_abs__if,axiom,
    ( abs_abs_rat
    = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).

% abs_if
thf(fact_6994_abs__if,axiom,
    ( abs_abs_int
    = ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).

% abs_if
thf(fact_6995_abs__if__raw,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A4 ) @ A4 ) ) ) ).

% abs_if_raw
thf(fact_6996_abs__if__raw,axiom,
    ( abs_abs_real
    = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).

% abs_if_raw
thf(fact_6997_abs__if__raw,axiom,
    ( abs_abs_rat
    = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).

% abs_if_raw
thf(fact_6998_abs__if__raw,axiom,
    ( abs_abs_int
    = ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).

% abs_if_raw
thf(fact_6999_abs__of__neg,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A2 )
        = ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% abs_of_neg
thf(fact_7000_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_7001_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_7002_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_7003_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_7004_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_7005_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_7006_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_7007_abs__diff__triangle__ineq,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A2 @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B2 @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_7008_abs__diff__triangle__ineq,axiom,
    ! [A2: real,B2: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A2 @ B2 ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A2 @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B2 @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_7009_abs__diff__triangle__ineq,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B2 ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A2 @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B2 @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_7010_abs__diff__triangle__ineq,axiom,
    ! [A2: int,B2: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A2 @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B2 @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_7011_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_7012_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_7013_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_7014_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_7015_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_7016_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_7017_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_7018_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_7019_abs__real__def,axiom,
    ( abs_abs_real
    = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).

% abs_real_def
thf(fact_7020_lemma__interval__lt,axiom,
    ! [A2: real,X: real,B2: real] :
      ( ( ord_less_real @ A2 @ X )
     => ( ( ord_less_real @ X @ B2 )
       => ? [D6: real] :
            ( ( ord_less_real @ zero_zero_real @ D6 )
            & ! [Y5: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D6 )
               => ( ( ord_less_real @ A2 @ Y5 )
                  & ( ord_less_real @ Y5 @ B2 ) ) ) ) ) ) ).

% lemma_interval_lt
thf(fact_7021_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_7022_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_7023_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_7024_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_7025_of__int__leD,axiom,
    ! [N: int,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).

% of_int_leD
thf(fact_7026_of__int__leD,axiom,
    ! [N: int,X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).

% of_int_leD
thf(fact_7027_of__int__leD,axiom,
    ! [N: int,X: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).

% of_int_leD
thf(fact_7028_of__int__leD,axiom,
    ! [N: int,X: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).

% of_int_leD
thf(fact_7029_of__int__lessD,axiom,
    ! [N: int,X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).

% of_int_lessD
thf(fact_7030_of__int__lessD,axiom,
    ! [N: int,X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_real @ one_one_real @ X ) ) ) ).

% of_int_lessD
thf(fact_7031_of__int__lessD,axiom,
    ! [N: int,X: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_rat @ one_one_rat @ X ) ) ) ).

% of_int_lessD
thf(fact_7032_of__int__lessD,axiom,
    ! [N: int,X: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_int @ one_one_int @ X ) ) ) ).

% of_int_lessD
thf(fact_7033_lemma__interval,axiom,
    ! [A2: real,X: real,B2: real] :
      ( ( ord_less_real @ A2 @ X )
     => ( ( ord_less_real @ X @ B2 )
       => ? [D6: real] :
            ( ( ord_less_real @ zero_zero_real @ D6 )
            & ! [Y5: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D6 )
               => ( ( ord_less_eq_real @ A2 @ Y5 )
                  & ( ord_less_eq_real @ Y5 @ B2 ) ) ) ) ) ) ).

% lemma_interval
thf(fact_7034_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_7035_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_7036_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_7037_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_7038_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_7039_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_7040_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_7041_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_7042_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_7043_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_7044_power__even__abs,axiom,
    ! [N: nat,A2: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( abs_abs_rat @ A2 ) @ N )
        = ( power_power_rat @ A2 @ N ) ) ) ).

% power_even_abs
thf(fact_7045_power__even__abs,axiom,
    ! [N: nat,A2: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( abs_abs_real @ A2 ) @ N )
        = ( power_power_real @ A2 @ N ) ) ) ).

% power_even_abs
thf(fact_7046_power__even__abs,axiom,
    ! [N: nat,A2: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( abs_abs_int @ A2 ) @ N )
        = ( power_power_int @ A2 @ N ) ) ) ).

% power_even_abs
thf(fact_7047_power__even__abs,axiom,
    ! [N: nat,A2: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A2 ) @ N )
        = ( power_8256067586552552935nteger @ A2 @ N ) ) ) ).

% power_even_abs
thf(fact_7048_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_7049_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_7050_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_7051_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_7052_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_7053_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_7054_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_7055_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_7056_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_7057_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_7058_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_7059_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_7060_power__mono__even,axiom,
    ! [N: nat,A2: real,B2: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B2 ) )
       => ( ord_less_eq_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ).

% power_mono_even
thf(fact_7061_power__mono__even,axiom,
    ! [N: nat,A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B2 ) )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ ( power_8256067586552552935nteger @ B2 @ N ) ) ) ) ).

% power_mono_even
thf(fact_7062_power__mono__even,axiom,
    ! [N: nat,A2: rat,B2: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B2 ) )
       => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ).

% power_mono_even
thf(fact_7063_power__mono__even,axiom,
    ! [N: nat,A2: int,B2: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) )
       => ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).

% power_mono_even
thf(fact_7064_even__set__bit__iff,axiom,
    ! [M: nat,A2: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_se6647067497041451410uint32 @ M @ A2 ) )
      = ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A2 )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_7065_even__set__bit__iff,axiom,
    ! [M: nat,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A2 ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_7066_even__set__bit__iff,axiom,
    ! [M: nat,A2: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( bit_se4894374433684937756l_num1 @ M @ A2 ) )
      = ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A2 )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_7067_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_7068_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_7069_even__unset__bit__iff,axiom,
    ! [M: nat,A2: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_se4315839071623982667uint32 @ M @ A2 ) )
      = ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A2 )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_7070_even__unset__bit__iff,axiom,
    ! [M: nat,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A2 ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_7071_even__unset__bit__iff,axiom,
    ! [M: nat,A2: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( bit_se5331074070815623765l_num1 @ M @ A2 ) )
      = ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A2 )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_7072_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_7073_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_7074_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_7075_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_7076_round__unique_H,axiom,
    ! [X: real,N: int] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ N ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( archim8280529875227126926d_real @ X )
        = N ) ) ).

% round_unique'
thf(fact_7077_round__unique_H,axiom,
    ! [X: rat,N: int] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
     => ( ( archim7778729529865785530nd_rat @ X )
        = N ) ) ).

% round_unique'
thf(fact_7078_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_7079_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_7080_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_7081_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_7082_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_7083_abs__sqrt__wlog,axiom,
    ! [P: real > real > $o,X: real] :
      ( ! [X3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X3 )
         => ( P @ X3 @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_7084_abs__sqrt__wlog,axiom,
    ! [P: code_integer > code_integer > $o,X: code_integer] :
      ( ! [X3: code_integer] :
          ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
         => ( P @ X3 @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_Code_integer @ X ) @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_7085_abs__sqrt__wlog,axiom,
    ! [P: rat > rat > $o,X: rat] :
      ( ! [X3: rat] :
          ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
         => ( P @ X3 @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_rat @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_7086_abs__sqrt__wlog,axiom,
    ! [P: int > int > $o,X: int] :
      ( ! [X3: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X3 )
         => ( P @ X3 @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_7087_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( topolo6980174941875973593q_real
        @ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% monoseq_arctan_series
thf(fact_7088_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_7089_round__altdef,axiom,
    ( archim8280529875227126926d_real
    = ( ^ [X2: real] : ( if_int @ ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( archim2898591450579166408c_real @ X2 ) ) @ ( archim7802044766580827645g_real @ X2 ) @ ( archim6058952711729229775r_real @ X2 ) ) ) ) ).

% round_altdef
thf(fact_7090_round__altdef,axiom,
    ( archim7778729529865785530nd_rat
    = ( ^ [X2: rat] : ( if_int @ ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( archimedean_frac_rat @ X2 ) ) @ ( archim2889992004027027881ng_rat @ X2 ) @ ( archim3151403230148437115or_rat @ X2 ) ) ) ) ).

% round_altdef
thf(fact_7091_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_7092_pochhammer__double,axiom,
    ! [Z: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).

% pochhammer_double
thf(fact_7093_pochhammer__double,axiom,
    ! [Z: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).

% pochhammer_double
thf(fact_7094_pochhammer__double,axiom,
    ! [Z: complex,N: nat] :
      ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s2602460028002588243omplex @ Z @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).

% pochhammer_double
thf(fact_7095_arctan__zero__zero,axiom,
    ( ( arctan @ zero_zero_real )
    = zero_zero_real ) ).

% arctan_zero_zero
thf(fact_7096_arctan__eq__zero__iff,axiom,
    ! [X: real] :
      ( ( ( arctan @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% arctan_eq_zero_iff
thf(fact_7097_pochhammer__0,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( comm_s6431939913906641691l_num1 @ A2 @ zero_zero_nat )
      = one_on7727431528512463931l_num1 ) ).

% pochhammer_0
thf(fact_7098_pochhammer__0,axiom,
    ! [A2: real] :
      ( ( comm_s7457072308508201937r_real @ A2 @ zero_zero_nat )
      = one_one_real ) ).

% pochhammer_0
thf(fact_7099_pochhammer__0,axiom,
    ! [A2: rat] :
      ( ( comm_s4028243227959126397er_rat @ A2 @ zero_zero_nat )
      = one_one_rat ) ).

% pochhammer_0
thf(fact_7100_pochhammer__0,axiom,
    ! [A2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A2 @ zero_zero_nat )
      = one_one_nat ) ).

% pochhammer_0
thf(fact_7101_pochhammer__0,axiom,
    ! [A2: int] :
      ( ( comm_s4660882817536571857er_int @ A2 @ zero_zero_nat )
      = one_one_int ) ).

% pochhammer_0
thf(fact_7102_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_7103_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_7104_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_7105_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_7106_zdvd1__eq,axiom,
    ! [X: int] :
      ( ( dvd_dvd_int @ X @ one_one_int )
      = ( ( abs_abs_int @ X )
        = one_one_int ) ) ).

% zdvd1_eq
thf(fact_7107_frac__of__int,axiom,
    ! [Z: int] :
      ( ( archim2898591450579166408c_real @ ( ring_1_of_int_real @ Z ) )
      = zero_zero_real ) ).

% frac_of_int
thf(fact_7108_frac__of__int,axiom,
    ! [Z: int] :
      ( ( archimedean_frac_rat @ ( ring_1_of_int_rat @ Z ) )
      = zero_zero_rat ) ).

% frac_of_int
thf(fact_7109_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_7110_pochhammer__of__nat,axiom,
    ! [X: nat,N: nat] :
      ( ( comm_s4660882817536571857er_int @ ( semiri1314217659103216013at_int @ X ) @ N )
      = ( semiri1314217659103216013at_int @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).

% pochhammer_of_nat
thf(fact_7111_pochhammer__of__nat,axiom,
    ! [X: nat,N: nat] :
      ( ( comm_s7457072308508201937r_real @ ( semiri5074537144036343181t_real @ X ) @ N )
      = ( semiri5074537144036343181t_real @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).

% pochhammer_of_nat
thf(fact_7112_pochhammer__of__nat,axiom,
    ! [X: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ ( semiri1316708129612266289at_nat @ X ) @ N )
      = ( semiri1316708129612266289at_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).

% pochhammer_of_nat
thf(fact_7113_pochhammer__of__nat,axiom,
    ! [X: nat,N: nat] :
      ( ( comm_s8582702949713902594nteger @ ( semiri4939895301339042750nteger @ X ) @ N )
      = ( semiri4939895301339042750nteger @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).

% pochhammer_of_nat
thf(fact_7114_pochhammer__of__nat,axiom,
    ! [X: nat,N: nat] :
      ( ( comm_s2602460028002588243omplex @ ( semiri8010041392384452111omplex @ X ) @ N )
      = ( semiri8010041392384452111omplex @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).

% pochhammer_of_nat
thf(fact_7115_zdvd__antisym__abs,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ B2 )
     => ( ( dvd_dvd_int @ B2 @ A2 )
       => ( ( abs_abs_int @ A2 )
          = ( abs_abs_int @ B2 ) ) ) ) ).

% zdvd_antisym_abs
thf(fact_7116_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_7117_arctan__monotone,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).

% arctan_monotone
thf(fact_7118_abs__zmult__eq__1,axiom,
    ! [M: int,N: int] :
      ( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
        = one_one_int )
     => ( ( abs_abs_int @ M )
        = one_one_int ) ) ).

% abs_zmult_eq_1
thf(fact_7119_pochhammer__pos,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).

% pochhammer_pos
thf(fact_7120_pochhammer__pos,axiom,
    ! [X: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).

% pochhammer_pos
thf(fact_7121_pochhammer__pos,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X )
     => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).

% pochhammer_pos
thf(fact_7122_pochhammer__pos,axiom,
    ! [X: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).

% pochhammer_pos
thf(fact_7123_pochhammer__eq__0__mono,axiom,
    ! [A2: real,N: nat,M: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A2 @ N )
        = zero_zero_real )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( comm_s7457072308508201937r_real @ A2 @ M )
          = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_7124_pochhammer__eq__0__mono,axiom,
    ! [A2: rat,N: nat,M: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A2 @ N )
        = zero_zero_rat )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( comm_s4028243227959126397er_rat @ A2 @ M )
          = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_7125_pochhammer__neq__0__mono,axiom,
    ! [A2: real,M: nat,N: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A2 @ M )
       != zero_zero_real )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( comm_s7457072308508201937r_real @ A2 @ N )
         != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_7126_pochhammer__neq__0__mono,axiom,
    ! [A2: rat,M: nat,N: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A2 @ M )
       != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( comm_s4028243227959126397er_rat @ A2 @ N )
         != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_7127_frac__ge__0,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X ) ) ).

% frac_ge_0
thf(fact_7128_frac__ge__0,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X ) ) ).

% frac_ge_0
thf(fact_7129_frac__lt__1,axiom,
    ! [X: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X ) @ one_one_real ) ).

% frac_lt_1
thf(fact_7130_frac__lt__1,axiom,
    ! [X: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X ) @ one_one_rat ) ).

% frac_lt_1
thf(fact_7131_frac__1__eq,axiom,
    ! [X: real] :
      ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ one_one_real ) )
      = ( archim2898591450579166408c_real @ X ) ) ).

% frac_1_eq
thf(fact_7132_frac__1__eq,axiom,
    ! [X: rat] :
      ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
      = ( archimedean_frac_rat @ X ) ) ).

% frac_1_eq
thf(fact_7133_zabs__def,axiom,
    ( abs_abs_int
    = ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).

% zabs_def
thf(fact_7134_pochhammer__nonneg,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_7135_pochhammer__nonneg,axiom,
    ! [X: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_7136_pochhammer__nonneg,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_7137_pochhammer__nonneg,axiom,
    ! [X: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_7138_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero_int )
     => ( ( dvd_dvd_int @ D @ I )
       => ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I ) ) ) ) ).

% dvd_imp_le_int
thf(fact_7139_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s6431939913906641691l_num1 @ zero_z3563351764282998399l_num1 @ N )
          = one_on7727431528512463931l_num1 ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s6431939913906641691l_num1 @ zero_z3563351764282998399l_num1 @ N )
          = zero_z3563351764282998399l_num1 ) ) ) ).

% pochhammer_0_left
thf(fact_7140_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
          = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
          = zero_zero_real ) ) ) ).

% pochhammer_0_left
thf(fact_7141_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
          = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
          = zero_zero_rat ) ) ) ).

% pochhammer_0_left
thf(fact_7142_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
          = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
          = zero_zero_nat ) ) ) ).

% pochhammer_0_left
thf(fact_7143_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
          = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
          = zero_zero_int ) ) ) ).

% pochhammer_0_left
thf(fact_7144_zdvd__mult__cancel1,axiom,
    ! [M: int,N: int] :
      ( ( M != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ M )
        = ( ( abs_abs_int @ N )
          = one_one_int ) ) ) ).

% zdvd_mult_cancel1
thf(fact_7145_frac__def,axiom,
    ( archim2898591450579166408c_real
    = ( ^ [X2: real] : ( minus_minus_real @ X2 @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X2 ) ) ) ) ) ).

% frac_def
thf(fact_7146_frac__def,axiom,
    ( archimedean_frac_rat
    = ( ^ [X2: rat] : ( minus_minus_rat @ X2 @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X2 ) ) ) ) ) ).

% frac_def
thf(fact_7147_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
        = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_7148_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
        = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_7149_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
        = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_7150_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
        = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_7151_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
        = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_7152_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
        = zero_zero_rat )
      = ( ord_less_nat @ N @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_7153_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
        = zero_zero_int )
      = ( ord_less_nat @ N @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_7154_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
        = zero_zero_real )
      = ( ord_less_nat @ N @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_7155_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
        = zero_z3403309356797280102nteger )
      = ( ord_less_nat @ N @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_7156_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
        = zero_zero_complex )
      = ( ord_less_nat @ N @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_7157_pochhammer__eq__0__iff,axiom,
    ! [A2: rat,N: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A2 @ N )
        = zero_zero_rat )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N )
            & ( A2
              = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_7158_pochhammer__eq__0__iff,axiom,
    ! [A2: real,N: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A2 @ N )
        = zero_zero_real )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N )
            & ( A2
              = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_7159_pochhammer__eq__0__iff,axiom,
    ! [A2: complex,N: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A2 @ N )
        = zero_zero_complex )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N )
            & ( A2
              = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_7160_pochhammer__product_H,axiom,
    ! [Z: rat,N: nat,M: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N @ M ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_7161_pochhammer__product_H,axiom,
    ! [Z: int,N: nat,M: nat] :
      ( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N @ M ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_7162_pochhammer__product_H,axiom,
    ! [Z: real,N: nat,M: nat] :
      ( ( comm_s7457072308508201937r_real @ Z @ ( plus_plus_nat @ N @ M ) )
      = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ N ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_7163_pochhammer__product_H,axiom,
    ! [Z: nat,N: nat,M: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N @ M ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_7164_pochhammer__product_H,axiom,
    ! [Z: code_integer,N: nat,M: nat] :
      ( ( comm_s8582702949713902594nteger @ Z @ ( plus_plus_nat @ N @ M ) )
      = ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ Z @ N ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ Z @ ( semiri4939895301339042750nteger @ N ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_7165_pochhammer__product_H,axiom,
    ! [Z: complex,N: nat,M: nat] :
      ( ( comm_s2602460028002588243omplex @ Z @ ( plus_plus_nat @ N @ M ) )
      = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ N ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_7166_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
       != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_7167_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
       != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_7168_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
       != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_7169_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
       != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_7170_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
       != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_7171_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L ) ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).

% even_add_abs_iff
thf(fact_7172_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).

% even_abs_add_iff
thf(fact_7173_frac__eq,axiom,
    ! [X: real] :
      ( ( ( archim2898591450579166408c_real @ X )
        = X )
      = ( ( ord_less_eq_real @ zero_zero_real @ X )
        & ( ord_less_real @ X @ one_one_real ) ) ) ).

% frac_eq
thf(fact_7174_frac__eq,axiom,
    ! [X: rat] :
      ( ( ( archimedean_frac_rat @ X )
        = X )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ X )
        & ( ord_less_rat @ X @ one_one_rat ) ) ) ).

% frac_eq
thf(fact_7175_frac__add,axiom,
    ! [X: real,Y: real] :
      ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
       => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
          = ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) ) )
      & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
       => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X @ Y ) )
          = ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real ) ) ) ) ).

% frac_add
thf(fact_7176_frac__add,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
       => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
          = ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) ) )
      & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
       => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat ) ) ) ) ).

% frac_add
thf(fact_7177_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( comm_s4028243227959126397er_rat @ Z @ N )
        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_7178_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( comm_s4660882817536571857er_int @ Z @ N )
        = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_7179_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( comm_s7457072308508201937r_real @ Z @ N )
        = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_7180_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( comm_s4663373288045622133er_nat @ Z @ N )
        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_7181_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: code_integer] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( comm_s8582702949713902594nteger @ Z @ N )
        = ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ Z @ M ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ Z @ ( semiri4939895301339042750nteger @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_7182_pochhammer__product,axiom,
    ! [M: nat,N: nat,Z: complex] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( comm_s2602460028002588243omplex @ Z @ N )
        = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ M ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_7183_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_7184_incr__lemma,axiom,
    ! [D: int,Z: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( 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 ) @ D ) ) ) ) ).

% incr_lemma
thf(fact_7185_decr__lemma,axiom,
    ! [D: int,X: int,Z: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).

% decr_lemma
thf(fact_7186_pochhammer__absorb__comp,axiom,
    ! [R3: word_N3645301735248828278l_num1,K: nat] :
      ( ( times_7065122842183080059l_num1 @ ( minus_4019991460397169231l_num1 @ R3 @ ( semiri8819519690708144855l_num1 @ K ) ) @ ( comm_s6431939913906641691l_num1 @ ( uminus8244633308260627903l_num1 @ R3 ) @ K ) )
      = ( times_7065122842183080059l_num1 @ R3 @ ( comm_s6431939913906641691l_num1 @ ( plus_p361126936061061375l_num1 @ ( uminus8244633308260627903l_num1 @ R3 ) @ one_on7727431528512463931l_num1 ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7187_pochhammer__absorb__comp,axiom,
    ! [R3: uint32,K: nat] :
      ( ( times_times_uint32 @ ( minus_minus_uint32 @ R3 @ ( semiri2565882477558803405uint32 @ K ) ) @ ( comm_s6516030829397196305uint32 @ ( uminus_uminus_uint32 @ R3 ) @ K ) )
      = ( times_times_uint32 @ R3 @ ( comm_s6516030829397196305uint32 @ ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ R3 ) @ one_one_uint32 ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7188_pochhammer__absorb__comp,axiom,
    ! [R3: rat,K: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ R3 @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R3 ) @ K ) )
      = ( times_times_rat @ R3 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R3 ) @ one_one_rat ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7189_pochhammer__absorb__comp,axiom,
    ! [R3: int,K: nat] :
      ( ( times_times_int @ ( minus_minus_int @ R3 @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R3 ) @ K ) )
      = ( times_times_int @ R3 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R3 ) @ one_one_int ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7190_pochhammer__absorb__comp,axiom,
    ! [R3: real,K: nat] :
      ( ( times_times_real @ ( minus_minus_real @ R3 @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R3 ) @ K ) )
      = ( times_times_real @ R3 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R3 ) @ one_one_real ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7191_pochhammer__absorb__comp,axiom,
    ! [R3: code_integer,K: nat] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R3 @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R3 ) @ K ) )
      = ( times_3573771949741848930nteger @ R3 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R3 ) @ one_one_Code_integer ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7192_pochhammer__absorb__comp,axiom,
    ! [R3: complex,K: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ R3 @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R3 ) @ K ) )
      = ( times_times_complex @ R3 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R3 ) @ one_one_complex ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7193_pochhammer__minus,axiom,
    ! [B2: word_N3645301735248828278l_num1,K: nat] :
      ( ( comm_s6431939913906641691l_num1 @ ( uminus8244633308260627903l_num1 @ B2 ) @ K )
      = ( times_7065122842183080059l_num1 @ ( power_2184487114949457152l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ K ) @ ( comm_s6431939913906641691l_num1 @ ( plus_p361126936061061375l_num1 @ ( minus_4019991460397169231l_num1 @ B2 @ ( semiri8819519690708144855l_num1 @ K ) ) @ one_on7727431528512463931l_num1 ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_7194_pochhammer__minus,axiom,
    ! [B2: uint32,K: nat] :
      ( ( comm_s6516030829397196305uint32 @ ( uminus_uminus_uint32 @ B2 ) @ K )
      = ( times_times_uint32 @ ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ K ) @ ( comm_s6516030829397196305uint32 @ ( plus_plus_uint32 @ ( minus_minus_uint32 @ B2 @ ( semiri2565882477558803405uint32 @ K ) ) @ one_one_uint32 ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_7195_pochhammer__minus,axiom,
    ! [B2: rat,K: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B2 ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B2 @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_7196_pochhammer__minus,axiom,
    ! [B2: int,K: nat] :
      ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B2 ) @ K )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B2 @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_7197_pochhammer__minus,axiom,
    ! [B2: real,K: nat] :
      ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B2 ) @ K )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B2 @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_7198_pochhammer__minus,axiom,
    ! [B2: code_integer,K: nat] :
      ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B2 ) @ K )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B2 @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_7199_pochhammer__minus,axiom,
    ! [B2: complex,K: nat] :
      ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B2 ) @ K )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B2 @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_7200_pochhammer__minus_H,axiom,
    ! [B2: word_N3645301735248828278l_num1,K: nat] :
      ( ( comm_s6431939913906641691l_num1 @ ( plus_p361126936061061375l_num1 @ ( minus_4019991460397169231l_num1 @ B2 @ ( semiri8819519690708144855l_num1 @ K ) ) @ one_on7727431528512463931l_num1 ) @ K )
      = ( times_7065122842183080059l_num1 @ ( power_2184487114949457152l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ K ) @ ( comm_s6431939913906641691l_num1 @ ( uminus8244633308260627903l_num1 @ B2 ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_7201_pochhammer__minus_H,axiom,
    ! [B2: uint32,K: nat] :
      ( ( comm_s6516030829397196305uint32 @ ( plus_plus_uint32 @ ( minus_minus_uint32 @ B2 @ ( semiri2565882477558803405uint32 @ K ) ) @ one_one_uint32 ) @ K )
      = ( times_times_uint32 @ ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ K ) @ ( comm_s6516030829397196305uint32 @ ( uminus_uminus_uint32 @ B2 ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_7202_pochhammer__minus_H,axiom,
    ! [B2: rat,K: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B2 @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B2 ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_7203_pochhammer__minus_H,axiom,
    ! [B2: int,K: nat] :
      ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B2 @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B2 ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_7204_pochhammer__minus_H,axiom,
    ! [B2: real,K: nat] :
      ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B2 @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B2 ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_7205_pochhammer__minus_H,axiom,
    ! [B2: code_integer,K: nat] :
      ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B2 @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B2 ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_7206_pochhammer__minus_H,axiom,
    ! [B2: complex,K: nat] :
      ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B2 @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B2 ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_7207_floor__add,axiom,
    ! [X: real,Y: real] :
      ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
       => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
          = ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) ) )
      & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
       => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ Y ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X ) @ ( archim6058952711729229775r_real @ Y ) ) @ one_one_int ) ) ) ) ).

% floor_add
thf(fact_7208_floor__add,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
       => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
          = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) )
      & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
       => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ one_one_int ) ) ) ) ).

% floor_add
thf(fact_7209_nat0__intermed__int__val,axiom,
    ! [N: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N ) )
         => ? [I3: nat] :
              ( ( ord_less_eq_nat @ I3 @ N )
              & ( ( F @ I3 )
                = K ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_7210_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_7211_summable__divide__iff,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( divide1717551699836669952omplex @ ( F @ N4 ) @ C ) )
      = ( ( C = zero_zero_complex )
        | ( summable_complex @ F ) ) ) ).

% summable_divide_iff
thf(fact_7212_summable__divide__iff,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( divide_divide_real @ ( F @ N4 ) @ C ) )
      = ( ( C = zero_zero_real )
        | ( summable_real @ F ) ) ) ).

% summable_divide_iff
thf(fact_7213_summable__cmult__iff,axiom,
    ! [C: complex,F: nat > complex] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ C @ ( F @ N4 ) ) )
      = ( ( C = zero_zero_complex )
        | ( summable_complex @ F ) ) ) ).

% summable_cmult_iff
thf(fact_7214_summable__cmult__iff,axiom,
    ! [C: real,F: nat > real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ C @ ( F @ N4 ) ) )
      = ( ( C = zero_zero_real )
        | ( summable_real @ F ) ) ) ).

% summable_cmult_iff
thf(fact_7215_summable__power__series,axiom,
    ! [F: nat > real,Z: real] :
      ( ! [I3: nat] : ( ord_less_eq_real @ ( F @ I3 ) @ one_one_real )
     => ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ Z )
         => ( ( ord_less_real @ Z @ one_one_real )
           => ( summable_real
              @ ^ [I4: nat] : ( times_times_real @ ( F @ I4 ) @ ( power_power_real @ Z @ I4 ) ) ) ) ) ) ) ).

% summable_power_series
thf(fact_7216_summable__iff__shift,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real
        @ ^ [N4: nat] : ( F @ ( plus_plus_nat @ N4 @ K ) ) )
      = ( summable_real @ F ) ) ).

% summable_iff_shift
thf(fact_7217_summable__iff__shift,axiom,
    ! [F: nat > complex,K: nat] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( F @ ( plus_plus_nat @ N4 @ K ) ) )
      = ( summable_complex @ F ) ) ).

% summable_iff_shift
thf(fact_7218_summable__single,axiom,
    ! [I: nat,F: nat > complex] :
      ( summable_complex
      @ ^ [R5: nat] : ( if_complex @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_complex ) ) ).

% summable_single
thf(fact_7219_summable__single,axiom,
    ! [I: nat,F: nat > real] :
      ( summable_real
      @ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real ) ) ).

% summable_single
thf(fact_7220_summable__single,axiom,
    ! [I: nat,F: nat > nat] :
      ( summable_nat
      @ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat ) ) ).

% summable_single
thf(fact_7221_summable__single,axiom,
    ! [I: nat,F: nat > int] :
      ( summable_int
      @ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int ) ) ).

% summable_single
thf(fact_7222_summable__zero,axiom,
    ( summable_complex
    @ ^ [N4: nat] : zero_zero_complex ) ).

% summable_zero
thf(fact_7223_summable__zero,axiom,
    ( summable_real
    @ ^ [N4: nat] : zero_zero_real ) ).

% summable_zero
thf(fact_7224_summable__zero,axiom,
    ( summable_nat
    @ ^ [N4: nat] : zero_zero_nat ) ).

% summable_zero
thf(fact_7225_summable__zero,axiom,
    ( summable_int
    @ ^ [N4: nat] : zero_zero_int ) ).

% summable_zero
thf(fact_7226_summable__complex__of__real,axiom,
    ! [F: nat > real] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( real_V4546457046886955230omplex @ ( F @ N4 ) ) )
      = ( summable_real @ F ) ) ).

% summable_complex_of_real
thf(fact_7227_summable__const__iff,axiom,
    ! [C: complex] :
      ( ( summable_complex
        @ ^ [Uu: nat] : C )
      = ( C = zero_zero_complex ) ) ).

% summable_const_iff
thf(fact_7228_summable__const__iff,axiom,
    ! [C: real] :
      ( ( summable_real
        @ ^ [Uu: nat] : C )
      = ( C = zero_zero_real ) ) ).

% summable_const_iff
thf(fact_7229_summable__add,axiom,
    ! [F: nat > complex,G: nat > complex] :
      ( ( summable_complex @ F )
     => ( ( summable_complex @ G )
       => ( summable_complex
          @ ^ [N4: nat] : ( plus_plus_complex @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ).

% summable_add
thf(fact_7230_summable__add,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N4: nat] : ( plus_plus_real @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ).

% summable_add
thf(fact_7231_summable__add,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ( summable_nat @ F )
     => ( ( summable_nat @ G )
       => ( summable_nat
          @ ^ [N4: nat] : ( plus_plus_nat @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ).

% summable_add
thf(fact_7232_summable__add,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ( summable_int @ F )
     => ( ( summable_int @ G )
       => ( summable_int
          @ ^ [N4: nat] : ( plus_plus_int @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ).

% summable_add
thf(fact_7233_summable__mult,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ C @ ( F @ N4 ) ) ) ) ).

% summable_mult
thf(fact_7234_summable__mult,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ C @ ( F @ N4 ) ) ) ) ).

% summable_mult
thf(fact_7235_summable__mult2,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ C ) ) ) ).

% summable_mult2
thf(fact_7236_summable__mult2,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ C ) ) ) ).

% summable_mult2
thf(fact_7237_summable__diff,axiom,
    ! [F: nat > complex,G: nat > complex] :
      ( ( summable_complex @ F )
     => ( ( summable_complex @ G )
       => ( summable_complex
          @ ^ [N4: nat] : ( minus_minus_complex @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ).

% summable_diff
thf(fact_7238_summable__diff,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N4: nat] : ( minus_minus_real @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ).

% summable_diff
thf(fact_7239_summable__divide,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( summable_complex
        @ ^ [N4: nat] : ( divide1717551699836669952omplex @ ( F @ N4 ) @ C ) ) ) ).

% summable_divide
thf(fact_7240_summable__divide,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N4: nat] : ( divide_divide_real @ ( F @ N4 ) @ C ) ) ) ).

% summable_divide
thf(fact_7241_summable__minus,axiom,
    ! [F: nat > complex] :
      ( ( summable_complex @ F )
     => ( summable_complex
        @ ^ [N4: nat] : ( uminus1482373934393186551omplex @ ( F @ N4 ) ) ) ) ).

% summable_minus
thf(fact_7242_summable__minus,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N4: nat] : ( uminus_uminus_real @ ( F @ N4 ) ) ) ) ).

% summable_minus
thf(fact_7243_summable__minus__iff,axiom,
    ! [F: nat > complex] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( uminus1482373934393186551omplex @ ( F @ N4 ) ) )
      = ( summable_complex @ F ) ) ).

% summable_minus_iff
thf(fact_7244_summable__minus__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( uminus_uminus_real @ ( F @ N4 ) ) )
      = ( summable_real @ F ) ) ).

% summable_minus_iff
thf(fact_7245_summable__ignore__initial__segment,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N4: nat] : ( F @ ( plus_plus_nat @ N4 @ K ) ) ) ) ).

% summable_ignore_initial_segment
thf(fact_7246_summable__ignore__initial__segment,axiom,
    ! [F: nat > complex,K: nat] :
      ( ( summable_complex @ F )
     => ( summable_complex
        @ ^ [N4: nat] : ( F @ ( plus_plus_nat @ N4 @ K ) ) ) ) ).

% summable_ignore_initial_segment
thf(fact_7247_summable__rabs__cancel,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( abs_abs_real @ ( F @ N4 ) ) )
     => ( summable_real @ F ) ) ).

% summable_rabs_cancel
thf(fact_7248_summable__mult__D,axiom,
    ! [C: complex,F: nat > complex] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ C @ ( F @ N4 ) ) )
     => ( ( C != zero_zero_complex )
       => ( summable_complex @ F ) ) ) ).

% summable_mult_D
thf(fact_7249_summable__mult__D,axiom,
    ! [C: real,F: nat > real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ C @ ( F @ N4 ) ) )
     => ( ( C != zero_zero_real )
       => ( summable_real @ F ) ) ) ).

% summable_mult_D
thf(fact_7250_summable__zero__power,axiom,
    summable_real @ ( power_power_real @ zero_zero_real ) ).

% summable_zero_power
thf(fact_7251_summable__zero__power,axiom,
    summable_int @ ( power_power_int @ zero_zero_int ) ).

% summable_zero_power
thf(fact_7252_summable__zero__power,axiom,
    summable_complex @ ( power_power_complex @ zero_zero_complex ) ).

% summable_zero_power
thf(fact_7253_summable__of__real,axiom,
    ! [X5: nat > real] :
      ( ( summable_real @ X5 )
     => ( summable_real
        @ ^ [N4: nat] : ( real_V1803761363581548252l_real @ ( X5 @ N4 ) ) ) ) ).

% summable_of_real
thf(fact_7254_summable__of__real,axiom,
    ! [X5: nat > real] :
      ( ( summable_real @ X5 )
     => ( summable_complex
        @ ^ [N4: nat] : ( real_V4546457046886955230omplex @ ( X5 @ N4 ) ) ) ) ).

% summable_of_real
thf(fact_7255_summable__0__powser,axiom,
    ! [F: nat > complex] :
      ( summable_complex
      @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ zero_zero_complex @ N4 ) ) ) ).

% summable_0_powser
thf(fact_7256_summable__0__powser,axiom,
    ! [F: nat > real] :
      ( summable_real
      @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ zero_zero_real @ N4 ) ) ) ).

% summable_0_powser
thf(fact_7257_summable__zero__power_H,axiom,
    ! [F: nat > complex] :
      ( summable_complex
      @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ zero_zero_complex @ N4 ) ) ) ).

% summable_zero_power'
thf(fact_7258_summable__zero__power_H,axiom,
    ! [F: nat > real] :
      ( summable_real
      @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ zero_zero_real @ N4 ) ) ) ).

% summable_zero_power'
thf(fact_7259_summable__zero__power_H,axiom,
    ! [F: nat > int] :
      ( summable_int
      @ ^ [N4: nat] : ( times_times_int @ ( F @ N4 ) @ ( power_power_int @ zero_zero_int @ N4 ) ) ) ).

% summable_zero_power'
thf(fact_7260_summable__powser__ignore__initial__segment,axiom,
    ! [F: nat > complex,M: nat,Z: complex] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N4 @ M ) ) @ ( power_power_complex @ Z @ N4 ) ) )
      = ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_7261_summable__powser__ignore__initial__segment,axiom,
    ! [F: nat > real,M: nat,Z: real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N4 @ M ) ) @ ( power_power_real @ Z @ N4 ) ) )
      = ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_7262_summable__rabs__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N8: nat] :
        ! [N2: nat] :
          ( ( ord_less_eq_nat @ N8 @ N2 )
         => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N4: nat] : ( abs_abs_real @ ( F @ N4 ) ) ) ) ) ).

% summable_rabs_comparison_test
thf(fact_7263_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_7264_pochhammer__code,axiom,
    ( comm_s6431939913906641691l_num1
    = ( ^ [A4: word_N3645301735248828278l_num1,N4: nat] :
          ( if_wor5778924947035936048l_num1 @ ( N4 = zero_zero_nat ) @ one_on7727431528512463931l_num1
          @ ( set_fo4709898541180519304l_num1
            @ ^ [O: nat] : ( times_7065122842183080059l_num1 @ ( plus_p361126936061061375l_num1 @ A4 @ ( semiri8819519690708144855l_num1 @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_on7727431528512463931l_num1 ) ) ) ) ).

% pochhammer_code
thf(fact_7265_pochhammer__code,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A4: rat,N4: nat] :
          ( if_rat @ ( N4 = zero_zero_nat ) @ one_one_rat
          @ ( set_fo1949268297981939178at_rat
            @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A4 @ ( semiri681578069525770553at_rat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_rat ) ) ) ) ).

% pochhammer_code
thf(fact_7266_pochhammer__code,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A4: int,N4: nat] :
          ( if_int @ ( N4 = zero_zero_nat ) @ one_one_int
          @ ( set_fo2581907887559384638at_int
            @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A4 @ ( semiri1314217659103216013at_int @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_int ) ) ) ) ).

% pochhammer_code
thf(fact_7267_pochhammer__code,axiom,
    ( comm_s7457072308508201937r_real
    = ( ^ [A4: real,N4: nat] :
          ( if_real @ ( N4 = zero_zero_nat ) @ one_one_real
          @ ( set_fo3111899725591712190t_real
            @ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A4 @ ( semiri5074537144036343181t_real @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_real ) ) ) ) ).

% pochhammer_code
thf(fact_7268_pochhammer__code,axiom,
    ( comm_s8582702949713902594nteger
    = ( ^ [A4: code_integer,N4: nat] :
          ( if_Code_integer @ ( N4 = zero_zero_nat ) @ one_one_Code_integer
          @ ( set_fo1084959871951514735nteger
            @ ^ [O: nat] : ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A4 @ ( semiri4939895301339042750nteger @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_Code_integer ) ) ) ) ).

% pochhammer_code
thf(fact_7269_pochhammer__code,axiom,
    ( comm_s2602460028002588243omplex
    = ( ^ [A4: complex,N4: nat] :
          ( if_complex @ ( N4 = zero_zero_nat ) @ one_one_complex
          @ ( set_fo1517530859248394432omplex
            @ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A4 @ ( semiri8010041392384452111omplex @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_complex ) ) ) ) ).

% pochhammer_code
thf(fact_7270_pochhammer__code,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A4: nat,N4: nat] :
          ( if_nat @ ( N4 = zero_zero_nat ) @ one_one_nat
          @ ( set_fo2584398358068434914at_nat
            @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A4 @ ( semiri1316708129612266289at_nat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_nat ) ) ) ) ).

% pochhammer_code
thf(fact_7271_central__binomial__lower__bound,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ) ) ).

% central_binomial_lower_bound
thf(fact_7272_of__int__code__if,axiom,
    ( ring_17408606157368542149l_num1
    = ( ^ [K3: int] :
          ( if_wor5778924947035936048l_num1 @ ( K3 = zero_zero_int ) @ zero_z3563351764282998399l_num1
          @ ( if_wor5778924947035936048l_num1 @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus8244633308260627903l_num1 @ ( ring_17408606157368542149l_num1 @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_wor5778924947035936048l_num1
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( ring_17408606157368542149l_num1 @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p361126936061061375l_num1 @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( ring_17408606157368542149l_num1 @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_on7727431528512463931l_num1 ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_7273_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_7274_of__int__code__if,axiom,
    ( ring_1_of_int_uint32
    = ( ^ [K3: int] :
          ( if_uint32 @ ( K3 = zero_zero_int ) @ zero_zero_uint32
          @ ( if_uint32 @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_uint32 @ ( ring_1_of_int_uint32 @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_uint32
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( ring_1_of_int_uint32 @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_uint32 @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( ring_1_of_int_uint32 @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_uint32 ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_7275_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_7276_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_7277_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_7278_VEBT__internal_Obit__concat__def,axiom,
    ( vEBT_VEBT_bit_concat
    = ( ^ [H: nat,L2: nat,D4: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D4 ) ) @ L2 ) ) ) ).

% VEBT_internal.bit_concat_def
thf(fact_7279_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_7280_bits__mod__0,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( modulo1504961113040953224l_num1 @ zero_z3563351764282998399l_num1 @ A2 )
      = zero_z3563351764282998399l_num1 ) ).

% bits_mod_0
thf(fact_7281_bits__mod__0,axiom,
    ! [A2: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A2 )
      = zero_zero_int ) ).

% bits_mod_0
thf(fact_7282_bits__mod__0,axiom,
    ! [A2: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% bits_mod_0
thf(fact_7283_bits__mod__0,axiom,
    ! [A2: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A2 )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_0
thf(fact_7284_bits__mod__0,axiom,
    ! [A2: uint32] :
      ( ( modulo_modulo_uint32 @ zero_zero_uint32 @ A2 )
      = zero_zero_uint32 ) ).

% bits_mod_0
thf(fact_7285_mod__0,axiom,
    ! [A2: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A2 )
      = zero_zero_int ) ).

% mod_0
thf(fact_7286_mod__0,axiom,
    ! [A2: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% mod_0
thf(fact_7287_mod__0,axiom,
    ! [A2: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A2 )
      = zero_z3403309356797280102nteger ) ).

% mod_0
thf(fact_7288_mod__by__0,axiom,
    ! [A2: int] :
      ( ( modulo_modulo_int @ A2 @ zero_zero_int )
      = A2 ) ).

% mod_by_0
thf(fact_7289_mod__by__0,axiom,
    ! [A2: nat] :
      ( ( modulo_modulo_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% mod_by_0
thf(fact_7290_mod__by__0,axiom,
    ! [A2: code_integer] :
      ( ( modulo364778990260209775nteger @ A2 @ zero_z3403309356797280102nteger )
      = A2 ) ).

% mod_by_0
thf(fact_7291_mod__self,axiom,
    ! [A2: int] :
      ( ( modulo_modulo_int @ A2 @ A2 )
      = zero_zero_int ) ).

% mod_self
thf(fact_7292_mod__self,axiom,
    ! [A2: nat] :
      ( ( modulo_modulo_nat @ A2 @ A2 )
      = zero_zero_nat ) ).

% mod_self
thf(fact_7293_mod__self,axiom,
    ! [A2: code_integer] :
      ( ( modulo364778990260209775nteger @ A2 @ A2 )
      = zero_z3403309356797280102nteger ) ).

% mod_self
thf(fact_7294_suminf__zero,axiom,
    ( ( suminf_real
      @ ^ [N4: nat] : zero_zero_real )
    = zero_zero_real ) ).

% suminf_zero
thf(fact_7295_suminf__zero,axiom,
    ( ( suminf_nat
      @ ^ [N4: nat] : zero_zero_nat )
    = zero_zero_nat ) ).

% suminf_zero
thf(fact_7296_suminf__zero,axiom,
    ( ( suminf_int
      @ ^ [N4: nat] : zero_zero_int )
    = zero_zero_int ) ).

% suminf_zero
thf(fact_7297_bits__mod__by__1,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( modulo1504961113040953224l_num1 @ A2 @ one_on7727431528512463931l_num1 )
      = zero_z3563351764282998399l_num1 ) ).

% bits_mod_by_1
thf(fact_7298_bits__mod__by__1,axiom,
    ! [A2: int] :
      ( ( modulo_modulo_int @ A2 @ one_one_int )
      = zero_zero_int ) ).

% bits_mod_by_1
thf(fact_7299_bits__mod__by__1,axiom,
    ! [A2: nat] :
      ( ( modulo_modulo_nat @ A2 @ one_one_nat )
      = zero_zero_nat ) ).

% bits_mod_by_1
thf(fact_7300_bits__mod__by__1,axiom,
    ! [A2: code_integer] :
      ( ( modulo364778990260209775nteger @ A2 @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_by_1
thf(fact_7301_bits__mod__by__1,axiom,
    ! [A2: uint32] :
      ( ( modulo_modulo_uint32 @ A2 @ one_one_uint32 )
      = zero_zero_uint32 ) ).

% bits_mod_by_1
thf(fact_7302_mod__by__1,axiom,
    ! [A2: int] :
      ( ( modulo_modulo_int @ A2 @ one_one_int )
      = zero_zero_int ) ).

% mod_by_1
thf(fact_7303_mod__by__1,axiom,
    ! [A2: nat] :
      ( ( modulo_modulo_nat @ A2 @ one_one_nat )
      = zero_zero_nat ) ).

% mod_by_1
thf(fact_7304_mod__by__1,axiom,
    ! [A2: code_integer] :
      ( ( modulo364778990260209775nteger @ A2 @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% mod_by_1
thf(fact_7305_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_7306_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_7307_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_7308_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_7309_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_7310_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_7311_bits__mod__div__trivial,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( divide1791077408188789448l_num1 @ ( modulo1504961113040953224l_num1 @ A2 @ B2 ) @ B2 )
      = zero_z3563351764282998399l_num1 ) ).

% bits_mod_div_trivial
thf(fact_7312_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_7313_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_7314_bits__mod__div__trivial,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A2 @ B2 ) @ B2 )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_div_trivial
thf(fact_7315_bits__mod__div__trivial,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( divide_divide_uint32 @ ( modulo_modulo_uint32 @ A2 @ B2 ) @ B2 )
      = zero_zero_uint32 ) ).

% bits_mod_div_trivial
thf(fact_7316_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_7317_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_7318_mod__div__trivial,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A2 @ B2 ) @ B2 )
      = zero_z3403309356797280102nteger ) ).

% mod_div_trivial
thf(fact_7319_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_7320_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_7321_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_7322_set__decode__zero,axiom,
    ( ( nat_set_decode @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% set_decode_zero
thf(fact_7323_mod__minus1__right,axiom,
    ! [A2: code_integer] :
      ( ( modulo364778990260209775nteger @ A2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = zero_z3403309356797280102nteger ) ).

% mod_minus1_right
thf(fact_7324_mod__minus1__right,axiom,
    ! [A2: int] :
      ( ( modulo_modulo_int @ A2 @ ( uminus_uminus_int @ one_one_int ) )
      = zero_zero_int ) ).

% mod_minus1_right
thf(fact_7325_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L )
       => ( ( modulo_modulo_int @ K @ L )
          = K ) ) ) ).

% mod_pos_pos_trivial
thf(fact_7326_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L @ K )
       => ( ( modulo_modulo_int @ K @ L )
          = K ) ) ) ).

% mod_neg_neg_trivial
thf(fact_7327_bits__one__mod__two__eq__one,axiom,
    ( ( modulo1504961113040953224l_num1 @ one_on7727431528512463931l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
    = one_on7727431528512463931l_num1 ) ).

% bits_one_mod_two_eq_one
thf(fact_7328_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_7329_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_7330_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_7331_bits__one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_uint32 @ one_one_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
    = one_one_uint32 ) ).

% bits_one_mod_two_eq_one
thf(fact_7332_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_7333_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_7334_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_7335_even__mod__2__iff,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( modulo1504961113040953224l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) )
      = ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A2 ) ) ).

% even_mod_2_iff
thf(fact_7336_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_7337_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_7338_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_7339_even__mod__2__iff,axiom,
    ! [A2: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( modulo_modulo_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A2 ) ) ).

% even_mod_2_iff
thf(fact_7340_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_7341_powser__zero,axiom,
    ! [F: nat > complex] :
      ( ( suminf_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ zero_zero_complex @ N4 ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_7342_powser__zero,axiom,
    ! [F: nat > real] :
      ( ( suminf_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ zero_zero_real @ N4 ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_7343_not__mod__2__eq__1__eq__0,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( ( modulo1504961113040953224l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
       != one_on7727431528512463931l_num1 )
      = ( ( modulo1504961113040953224l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
        = zero_z3563351764282998399l_num1 ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_7344_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_7345_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_7346_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_7347_not__mod__2__eq__1__eq__0,axiom,
    ! [A2: uint32] :
      ( ( ( modulo_modulo_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
       != one_one_uint32 )
      = ( ( modulo_modulo_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
        = zero_zero_uint32 ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_7348_not__mod__2__eq__0__eq__1,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( ( modulo1504961113040953224l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
       != zero_z3563351764282998399l_num1 )
      = ( ( modulo1504961113040953224l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
        = one_on7727431528512463931l_num1 ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_7349_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_7350_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_7351_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_7352_not__mod__2__eq__0__eq__1,axiom,
    ! [A2: uint32] :
      ( ( ( modulo_modulo_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
       != zero_zero_uint32 )
      = ( ( modulo_modulo_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
        = one_one_uint32 ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_7353_bits__minus__1__mod__2__eq,axiom,
    ( ( modulo1504961113040953224l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
    = one_on7727431528512463931l_num1 ) ).

% bits_minus_1_mod_2_eq
thf(fact_7354_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_7355_bits__minus__1__mod__2__eq,axiom,
    ( ( modulo_modulo_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
    = one_one_uint32 ) ).

% bits_minus_1_mod_2_eq
thf(fact_7356_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_7357_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_7358_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_7359_one__mod__exp__eq__one,axiom,
    ! [N: nat] :
      ( ( modulo_modulo_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
      = one_one_int ) ).

% one_mod_exp_eq_one
thf(fact_7360_even__succ__mod__exp,axiom,
    ! [A2: word_N3645301735248828278l_num1,N: nat] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo1504961113040953224l_num1 @ ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ A2 ) @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N ) )
          = ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ ( modulo1504961113040953224l_num1 @ A2 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_7361_even__succ__mod__exp,axiom,
    ! [A2: int,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
          = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_7362_even__succ__mod__exp,axiom,
    ! [A2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
          = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_7363_even__succ__mod__exp,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
          = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_7364_even__succ__mod__exp,axiom,
    ! [A2: uint32,N: nat] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo_modulo_uint32 @ ( plus_plus_uint32 @ one_one_uint32 @ A2 ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N ) )
          = ( plus_plus_uint32 @ one_one_uint32 @ ( modulo_modulo_uint32 @ A2 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_7365_of__nat__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
      = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_mod
thf(fact_7366_of__nat__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N ) )
      = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_mod
thf(fact_7367_of__nat__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N ) )
      = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% of_nat_mod
thf(fact_7368_power__mod,axiom,
    ! [A2: int,B2: int,N: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A2 @ B2 ) @ N ) @ B2 )
      = ( modulo_modulo_int @ ( power_power_int @ A2 @ N ) @ B2 ) ) ).

% power_mod
thf(fact_7369_power__mod,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A2 @ B2 ) @ N ) @ B2 )
      = ( modulo_modulo_nat @ ( power_power_nat @ A2 @ N ) @ B2 ) ) ).

% power_mod
thf(fact_7370_power__mod,axiom,
    ! [A2: code_integer,B2: code_integer,N: nat] :
      ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A2 @ B2 ) @ N ) @ B2 )
      = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ B2 ) ) ).

% power_mod
thf(fact_7371_dvd__mod__imp__dvd,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A2 @ B2 ) )
     => ( ( dvd_dvd_int @ C @ B2 )
       => ( dvd_dvd_int @ C @ A2 ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_7372_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A2 @ B2 ) )
     => ( ( dvd_dvd_nat @ C @ B2 )
       => ( dvd_dvd_nat @ C @ A2 ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_7373_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A2 @ B2 ) )
     => ( ( dvd_dvd_Code_integer @ C @ B2 )
       => ( dvd_dvd_Code_integer @ C @ A2 ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_7374_dvd__mod__iff,axiom,
    ! [C: int,B2: int,A2: int] :
      ( ( dvd_dvd_int @ C @ B2 )
     => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A2 @ B2 ) )
        = ( dvd_dvd_int @ C @ A2 ) ) ) ).

% dvd_mod_iff
thf(fact_7375_dvd__mod__iff,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ C @ B2 )
     => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A2 @ B2 ) )
        = ( dvd_dvd_nat @ C @ A2 ) ) ) ).

% dvd_mod_iff
thf(fact_7376_dvd__mod__iff,axiom,
    ! [C: code_integer,B2: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B2 )
     => ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A2 @ B2 ) )
        = ( dvd_dvd_Code_integer @ C @ A2 ) ) ) ).

% dvd_mod_iff
thf(fact_7377_suminf__of__real,axiom,
    ! [X5: nat > real] :
      ( ( summable_real @ X5 )
     => ( ( real_V1803761363581548252l_real @ ( suminf_real @ X5 ) )
        = ( suminf_real
          @ ^ [N4: nat] : ( real_V1803761363581548252l_real @ ( X5 @ N4 ) ) ) ) ) ).

% suminf_of_real
thf(fact_7378_suminf__of__real,axiom,
    ! [X5: nat > real] :
      ( ( summable_real @ X5 )
     => ( ( real_V4546457046886955230omplex @ ( suminf_real @ X5 ) )
        = ( suminf_complex
          @ ^ [N4: nat] : ( real_V4546457046886955230omplex @ ( X5 @ N4 ) ) ) ) ) ).

% suminf_of_real
thf(fact_7379_mod__plus__cong,axiom,
    ! [B2: int,B7: int,X: int,X7: int,Y: int,Y6: int,Z6: int] :
      ( ( B2 = B7 )
     => ( ( ( modulo_modulo_int @ X @ B7 )
          = ( modulo_modulo_int @ X7 @ B7 ) )
       => ( ( ( modulo_modulo_int @ Y @ B7 )
            = ( modulo_modulo_int @ Y6 @ B7 ) )
         => ( ( ( plus_plus_int @ X7 @ Y6 )
              = Z6 )
           => ( ( modulo_modulo_int @ ( plus_plus_int @ X @ Y ) @ B2 )
              = ( modulo_modulo_int @ Z6 @ B7 ) ) ) ) ) ) ).

% mod_plus_cong
thf(fact_7380_Word_Omod__minus__cong,axiom,
    ! [B2: int,B7: int,X: int,X7: int,Y: int,Y6: int,Z6: int] :
      ( ( B2 = B7 )
     => ( ( ( modulo_modulo_int @ X @ B7 )
          = ( modulo_modulo_int @ X7 @ B7 ) )
       => ( ( ( modulo_modulo_int @ Y @ B7 )
            = ( modulo_modulo_int @ Y6 @ B7 ) )
         => ( ( ( minus_minus_int @ X7 @ Y6 )
              = Z6 )
           => ( ( modulo_modulo_int @ ( minus_minus_int @ X @ Y ) @ B2 )
              = ( modulo_modulo_int @ Z6 @ B7 ) ) ) ) ) ) ).

% Word.mod_minus_cong
thf(fact_7381_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_7382_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_7383_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_7384_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_7385_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_7386_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_7387_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q3: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_7388_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q3: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_7389_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q3: num,N: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_7390_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
      = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_7391_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
      = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_7392_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
      = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_7393_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_7394_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_7395_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_7396_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_7397_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_7398_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_7399_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_int
    = ( ^ [A4: int,B4: int] :
          ( ( modulo_modulo_int @ B4 @ A4 )
          = zero_zero_int ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_7400_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_nat
    = ( ^ [A4: nat,B4: nat] :
          ( ( modulo_modulo_nat @ B4 @ A4 )
          = zero_zero_nat ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_7401_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [A4: code_integer,B4: code_integer] :
          ( ( modulo364778990260209775nteger @ B4 @ A4 )
          = zero_z3403309356797280102nteger ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_7402_mod__0__imp__dvd,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( ( modulo1504961113040953224l_num1 @ A2 @ B2 )
        = zero_z3563351764282998399l_num1 )
     => ( dvd_dv6812691276156420380l_num1 @ B2 @ A2 ) ) ).

% mod_0_imp_dvd
thf(fact_7403_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_7404_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_7405_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_7406_mod__0__imp__dvd,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( ( modulo_modulo_uint32 @ A2 @ B2 )
        = zero_zero_uint32 )
     => ( dvd_dvd_uint32 @ B2 @ A2 ) ) ).

% mod_0_imp_dvd
thf(fact_7407_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_7408_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_7409_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_7410_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).

% zmod_le_nonneg_dividend
thf(fact_7411_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).

% neg_mod_bound
thf(fact_7412_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).

% Euclidean_Division.pos_mod_bound
thf(fact_7413_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
       != zero_zero_int )
     => ( ( modulo_modulo_int @ K @ L )
       != zero_zero_int ) ) ).

% zmod_zminus1_not_zero
thf(fact_7414_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( ( modulo_modulo_int @ K @ ( uminus_uminus_int @ L ) )
       != zero_zero_int )
     => ( ( modulo_modulo_int @ K @ L )
       != zero_zero_int ) ) ).

% zmod_zminus2_not_zero
thf(fact_7415_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( ( modulo_modulo_int @ M @ D )
        = zero_zero_int )
     => ? [Q5: int] :
          ( M
          = ( times_times_int @ D @ Q5 ) ) ) ).

% zmod_eq_0D
thf(fact_7416_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( ( modulo_modulo_int @ M @ D )
        = zero_zero_int )
      = ( ? [Q6: int] :
            ( M
            = ( times_times_int @ D @ Q6 ) ) ) ) ).

% zmod_eq_0_iff
thf(fact_7417_suminf__add,axiom,
    ! [F: nat > complex,G: nat > complex] :
      ( ( summable_complex @ F )
     => ( ( summable_complex @ G )
       => ( ( plus_plus_complex @ ( suminf_complex @ F ) @ ( suminf_complex @ G ) )
          = ( suminf_complex
            @ ^ [N4: nat] : ( plus_plus_complex @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ) ).

% suminf_add
thf(fact_7418_suminf__add,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( ( plus_plus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
          = ( suminf_real
            @ ^ [N4: nat] : ( plus_plus_real @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ) ).

% suminf_add
thf(fact_7419_suminf__add,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ( summable_nat @ F )
     => ( ( summable_nat @ G )
       => ( ( plus_plus_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) )
          = ( suminf_nat
            @ ^ [N4: nat] : ( plus_plus_nat @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ) ).

% suminf_add
thf(fact_7420_suminf__add,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ( summable_int @ F )
     => ( ( summable_int @ G )
       => ( ( plus_plus_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) )
          = ( suminf_int
            @ ^ [N4: nat] : ( plus_plus_int @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ) ).

% suminf_add
thf(fact_7421_suminf__mult2,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( ( times_times_complex @ ( suminf_complex @ F ) @ C )
        = ( suminf_complex
          @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ C ) ) ) ) ).

% suminf_mult2
thf(fact_7422_suminf__mult2,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( ( times_times_real @ ( suminf_real @ F ) @ C )
        = ( suminf_real
          @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ C ) ) ) ) ).

% suminf_mult2
thf(fact_7423_suminf__mult,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( ( suminf_complex
          @ ^ [N4: nat] : ( times_times_complex @ C @ ( F @ N4 ) ) )
        = ( times_times_complex @ C @ ( suminf_complex @ F ) ) ) ) ).

% suminf_mult
thf(fact_7424_suminf__mult,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N4: nat] : ( times_times_real @ C @ ( F @ N4 ) ) )
        = ( times_times_real @ C @ ( suminf_real @ F ) ) ) ) ).

% suminf_mult
thf(fact_7425_suminf__diff,axiom,
    ! [F: nat > complex,G: nat > complex] :
      ( ( summable_complex @ F )
     => ( ( summable_complex @ G )
       => ( ( minus_minus_complex @ ( suminf_complex @ F ) @ ( suminf_complex @ G ) )
          = ( suminf_complex
            @ ^ [N4: nat] : ( minus_minus_complex @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ) ).

% suminf_diff
thf(fact_7426_suminf__diff,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( ( minus_minus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
          = ( suminf_real
            @ ^ [N4: nat] : ( minus_minus_real @ ( F @ N4 ) @ ( G @ N4 ) ) ) ) ) ) ).

% suminf_diff
thf(fact_7427_suminf__divide,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( ( suminf_complex
          @ ^ [N4: nat] : ( divide1717551699836669952omplex @ ( F @ N4 ) @ C ) )
        = ( divide1717551699836669952omplex @ ( suminf_complex @ F ) @ C ) ) ) ).

% suminf_divide
thf(fact_7428_suminf__divide,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N4: nat] : ( divide_divide_real @ ( F @ N4 ) @ C ) )
        = ( divide_divide_real @ ( suminf_real @ F ) @ C ) ) ) ).

% suminf_divide
thf(fact_7429_suminf__minus,axiom,
    ! [F: nat > complex] :
      ( ( summable_complex @ F )
     => ( ( suminf_complex
          @ ^ [N4: nat] : ( uminus1482373934393186551omplex @ ( F @ N4 ) ) )
        = ( uminus1482373934393186551omplex @ ( suminf_complex @ F ) ) ) ) ).

% suminf_minus
thf(fact_7430_suminf__minus,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N4: nat] : ( uminus_uminus_real @ ( F @ N4 ) ) )
        = ( uminus_uminus_real @ ( suminf_real @ F ) ) ) ) ).

% suminf_minus
thf(fact_7431_suminf__nonneg,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7432_suminf__nonneg,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7433_suminf__nonneg,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
       => ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7434_suminf__eq__zero__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
       => ( ( ( suminf_real @ F )
            = zero_zero_real )
          = ( ! [N4: nat] :
                ( ( F @ N4 )
                = zero_zero_real ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7435_suminf__eq__zero__iff,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
       => ( ( ( suminf_nat @ F )
            = zero_zero_nat )
          = ( ! [N4: nat] :
                ( ( F @ N4 )
                = zero_zero_nat ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7436_suminf__eq__zero__iff,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
       => ( ( ( suminf_int @ F )
            = zero_zero_int )
          = ( ! [N4: nat] :
                ( ( F @ N4 )
                = zero_zero_int ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7437_suminf__pos,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N2: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N2 ) )
       => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).

% suminf_pos
thf(fact_7438_suminf__pos,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N2 ) )
       => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).

% suminf_pos
thf(fact_7439_suminf__pos,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N2: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N2 ) )
       => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).

% suminf_pos
thf(fact_7440_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_7441_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_7442_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_7443_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_7444_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_7445_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_7446_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q3: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = zero_zero_int )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
thf(fact_7447_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q3: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = zero_zero_nat )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
thf(fact_7448_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q3: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
thf(fact_7449_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
      = zero_zero_int ) ).

% cong_exp_iff_simps(1)
thf(fact_7450_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
      = zero_zero_nat ) ).

% cong_exp_iff_simps(1)
thf(fact_7451_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
      = zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(1)
thf(fact_7452_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_7453_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_7454_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_7455_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_7456_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_7457_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_7458_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_7459_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_7460_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_7461_mult__div__mod__eq,axiom,
    ! [B2: uint32,A2: uint32] :
      ( ( plus_plus_uint32 @ ( times_times_uint32 @ B2 @ ( divide_divide_uint32 @ A2 @ B2 ) ) @ ( modulo_modulo_uint32 @ A2 @ B2 ) )
      = A2 ) ).

% mult_div_mod_eq
thf(fact_7462_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_7463_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_7464_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_7465_mod__mult__div__eq,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( plus_plus_uint32 @ ( modulo_modulo_uint32 @ A2 @ B2 ) @ ( times_times_uint32 @ B2 @ ( divide_divide_uint32 @ A2 @ B2 ) ) )
      = A2 ) ).

% mod_mult_div_eq
thf(fact_7466_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_7467_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_7468_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_7469_mod__div__mult__eq,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( plus_plus_uint32 @ ( modulo_modulo_uint32 @ A2 @ B2 ) @ ( times_times_uint32 @ ( divide_divide_uint32 @ A2 @ B2 ) @ B2 ) )
      = A2 ) ).

% mod_div_mult_eq
thf(fact_7470_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_7471_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_7472_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_7473_div__mult__mod__eq,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( plus_plus_uint32 @ ( times_times_uint32 @ ( divide_divide_uint32 @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo_uint32 @ A2 @ B2 ) )
      = A2 ) ).

% div_mult_mod_eq
thf(fact_7474_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_7475_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_7476_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_7477_mod__div__decomp,axiom,
    ! [A2: uint32,B2: uint32] :
      ( A2
      = ( plus_plus_uint32 @ ( times_times_uint32 @ ( divide_divide_uint32 @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo_uint32 @ A2 @ B2 ) ) ) ).

% mod_div_decomp
thf(fact_7478_cancel__div__mod__rules_I1_J,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A2 @ B2 ) ) @ C )
      = ( plus_plus_int @ A2 @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_7479_cancel__div__mod__rules_I1_J,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A2 @ B2 ) ) @ C )
      = ( plus_plus_nat @ A2 @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_7480_cancel__div__mod__rules_I1_J,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ B2 ) @ ( modulo364778990260209775nteger @ A2 @ B2 ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A2 @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_7481_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B2 @ ( divide_divide_int @ A2 @ B2 ) ) @ ( modulo_modulo_int @ A2 @ B2 ) ) @ C )
      = ( plus_plus_int @ A2 @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_7482_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A2 @ B2 ) ) @ ( modulo_modulo_nat @ A2 @ B2 ) ) @ C )
      = ( plus_plus_nat @ A2 @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_7483_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A2 @ B2 ) ) @ ( modulo364778990260209775nteger @ A2 @ B2 ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A2 @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_7484_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_7485_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_7486_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_7487_minus__mult__div__eq__mod,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( minus_minus_uint32 @ A2 @ ( times_times_uint32 @ B2 @ ( divide_divide_uint32 @ A2 @ B2 ) ) )
      = ( modulo_modulo_uint32 @ A2 @ B2 ) ) ).

% minus_mult_div_eq_mod
thf(fact_7488_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_7489_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_7490_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_7491_minus__mod__eq__mult__div,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( minus_minus_uint32 @ A2 @ ( modulo_modulo_uint32 @ A2 @ B2 ) )
      = ( times_times_uint32 @ B2 @ ( divide_divide_uint32 @ A2 @ B2 ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_7492_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_7493_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_7494_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_7495_minus__mod__eq__div__mult,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( minus_minus_uint32 @ A2 @ ( modulo_modulo_uint32 @ A2 @ B2 ) )
      = ( times_times_uint32 @ ( divide_divide_uint32 @ A2 @ B2 ) @ B2 ) ) ).

% minus_mod_eq_div_mult
thf(fact_7496_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_7497_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_7498_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_7499_minus__div__mult__eq__mod,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( minus_minus_uint32 @ A2 @ ( times_times_uint32 @ ( divide_divide_uint32 @ A2 @ B2 ) @ B2 ) )
      = ( modulo_modulo_uint32 @ A2 @ B2 ) ) ).

% minus_div_mult_eq_mod
thf(fact_7500_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_7501_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_7502_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_7503_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_7504_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_7505_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_7506_int__mod__ge,axiom,
    ! [A2: int,N: int] :
      ( ( ord_less_int @ A2 @ N )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int @ A2 @ ( modulo_modulo_int @ A2 @ N ) ) ) ) ).

% int_mod_ge
thf(fact_7507_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).

% Euclidean_Division.pos_mod_sign
thf(fact_7508_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).

% neg_mod_sign
thf(fact_7509_int__mod__lem,axiom,
    ! [N: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ N )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ B2 )
          & ( ord_less_int @ B2 @ N ) )
        = ( ( modulo_modulo_int @ B2 @ N )
          = B2 ) ) ) ).

% int_mod_lem
thf(fact_7510_int__mod__eq,axiom,
    ! [B2: int,N: int,A2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B2 )
     => ( ( ord_less_int @ B2 @ N )
       => ( ( ( modulo_modulo_int @ A2 @ N )
            = ( modulo_modulo_int @ B2 @ N ) )
         => ( ( modulo_modulo_int @ A2 @ N )
            = B2 ) ) ) ) ).

% int_mod_eq
thf(fact_7511_int__mod__le_H,axiom,
    ! [B2: int,N: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ B2 @ N ) )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ B2 @ N ) @ ( minus_minus_int @ B2 @ N ) ) ) ).

% int_mod_le'
thf(fact_7512_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_7513_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_7514_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_7515_zdiv__mono__strict,axiom,
    ! [A: int,B: int,N: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ( ( modulo_modulo_int @ A @ N )
            = zero_zero_int )
         => ( ( ( modulo_modulo_int @ B @ N )
              = zero_zero_int )
           => ( ord_less_int @ ( divide_divide_int @ A @ N ) @ ( divide_divide_int @ B @ N ) ) ) ) ) ) ).

% zdiv_mono_strict
thf(fact_7516_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).

% abs_mod_less
thf(fact_7517_div__mod__decomp__int,axiom,
    ! [A: int,N: int] :
      ( A
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ N ) @ N ) @ ( modulo_modulo_int @ A @ N ) ) ) ).

% div_mod_decomp_int
thf(fact_7518_summable__rabs,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( abs_abs_real @ ( F @ N4 ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
        @ ( suminf_real
          @ ^ [N4: nat] : ( abs_abs_real @ ( F @ N4 ) ) ) ) ) ).

% summable_rabs
thf(fact_7519_suminf__pos2,axiom,
    ! [F: nat > real,I: nat] :
      ( ( summable_real @ F )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_7520_suminf__pos2,axiom,
    ! [F: nat > nat,I: nat] :
      ( ( summable_nat @ F )
     => ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_7521_suminf__pos2,axiom,
    ! [F: nat > int,I: nat] :
      ( ( summable_int @ F )
     => ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
       => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
         => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_7522_suminf__pos__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
       => ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
          = ( ? [I4: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7523_suminf__pos__iff,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
          = ( ? [I4: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7524_suminf__pos__iff,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
       => ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
          = ( ? [I4: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I4 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7525_mod__mult2__eq_H,axiom,
    ! [A2: int,M: nat,N: nat] :
      ( ( modulo_modulo_int @ A2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A2 @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) @ ( modulo_modulo_int @ A2 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_7526_mod__mult2__eq_H,axiom,
    ! [A2: nat,M: nat,N: nat] :
      ( ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A2 @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) @ ( modulo_modulo_nat @ A2 @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_7527_mod__mult2__eq_H,axiom,
    ! [A2: code_integer,M: nat,N: nat] :
      ( ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A2 @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A2 @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_7528_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_7529_int__mod__ge_H,axiom,
    ! [B2: int,N: int] :
      ( ( ord_less_int @ B2 @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int @ ( plus_plus_int @ B2 @ N ) @ ( modulo_modulo_int @ B2 @ N ) ) ) ) ).

% int_mod_ge'
thf(fact_7530_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
       => ( ( modulo_modulo_int @ K @ L )
          = ( plus_plus_int @ K @ L ) ) ) ) ).

% mod_pos_neg_trivial
thf(fact_7531_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ( ord_less_eq_int @ L @ K )
       => ( ( modulo_modulo_int @ K @ L )
          = ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).

% mod_pos_geq
thf(fact_7532_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( ( L = zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ K ) )
        | ( ord_less_int @ zero_zero_int @ L ) ) ) ).

% mod_int_pos_iff
thf(fact_7533_real__of__int__div__aux,axiom,
    ! [X: int,D: int] :
      ( ( divide_divide_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ D ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).

% real_of_int_div_aux
thf(fact_7534_binomial__antimono,axiom,
    ! [K: nat,K5: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ K5 )
     => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
       => ( ( ord_less_eq_nat @ K5 @ N )
         => ( ord_less_eq_nat @ ( binomial @ N @ K5 ) @ ( binomial @ N @ K ) ) ) ) ) ).

% binomial_antimono
thf(fact_7535_binomial__maximum,axiom,
    ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% binomial_maximum
thf(fact_7536_binomial__maximum_H,axiom,
    ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).

% binomial_maximum'
thf(fact_7537_binomial__mono,axiom,
    ! [K: nat,K5: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ K5 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K5 ) @ N )
       => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K5 ) ) ) ) ).

% binomial_mono
thf(fact_7538_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ B2 @ C ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ C ) ) @ ( modulo364778990260209775nteger @ A2 @ B2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_7539_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( modulo_modulo_nat @ A2 @ ( times_times_nat @ B2 @ C ) )
        = ( plus_plus_nat @ ( times_times_nat @ B2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C ) ) @ ( modulo_modulo_nat @ A2 @ B2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_7540_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A2 @ ( times_times_int @ B2 @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B2 @ ( modulo_modulo_int @ ( divide_divide_int @ A2 @ B2 ) @ C ) ) @ ( modulo_modulo_int @ A2 @ B2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_7541_even__iff__mod__2__eq__zero,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A2 )
      = ( ( modulo1504961113040953224l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
        = zero_z3563351764282998399l_num1 ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_7542_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_7543_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_7544_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_7545_even__iff__mod__2__eq__zero,axiom,
    ! [A2: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A2 )
      = ( ( modulo_modulo_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
        = zero_zero_uint32 ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_7546_odd__iff__mod__2__eq__one,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A2 ) )
      = ( ( modulo1504961113040953224l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
        = one_on7727431528512463931l_num1 ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_7547_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_7548_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_7549_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_7550_odd__iff__mod__2__eq__one,axiom,
    ! [A2: uint32] :
      ( ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A2 ) )
      = ( ( modulo_modulo_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
        = one_one_uint32 ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_7551_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X: nat > nat > nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X @ Xa @ Xb @ Xc )
        = Y )
     => ( ( ( ord_less_nat @ Xb @ Xa )
         => ( Y = Xc ) )
        & ( ~ ( ord_less_nat @ Xb @ Xa )
         => ( Y
            = ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
thf(fact_7552_fold__atLeastAtMost__nat_Osimps,axiom,
    ( set_fo2584398358068434914at_nat
    = ( ^ [F2: nat > nat > nat,A4: nat,B4: nat,Acc: nat] : ( if_nat @ ( ord_less_nat @ B4 @ A4 ) @ Acc @ ( set_fo2584398358068434914at_nat @ F2 @ ( plus_plus_nat @ A4 @ one_one_nat ) @ B4 @ ( F2 @ A4 @ Acc ) ) ) ) ) ).

% fold_atLeastAtMost_nat.simps
thf(fact_7553_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_7554_nmod2,axiom,
    ! [N: int] :
      ( ( ( modulo_modulo_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int )
      | ( ( modulo_modulo_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% nmod2
thf(fact_7555_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_7556_mod__exp__less__eq__exp,axiom,
    ! [A2: int,N: nat] : ( ord_less_int @ ( modulo_modulo_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% mod_exp_less_eq_exp
thf(fact_7557_mod__power__lem,axiom,
    ! [A2: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ( ( ord_less_eq_nat @ M @ N )
         => ( ( modulo_modulo_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ A2 @ M ) )
            = zero_zero_int ) )
        & ( ~ ( ord_less_eq_nat @ M @ N )
         => ( ( modulo_modulo_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ A2 @ M ) )
            = ( power_power_int @ A2 @ N ) ) ) ) ) ).

% mod_power_lem
thf(fact_7558_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_7559_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_7560_split__zmod,axiom,
    ! [P: int > $o,N: int,K: int] :
      ( ( P @ ( modulo_modulo_int @ N @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ N ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I4: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ J3 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I4: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ J3 ) ) ) ) ) ).

% split_zmod
thf(fact_7561_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
        = ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L ) ) ) ) ).

% minus_mod_int_eq
thf(fact_7562_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_7563_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_7564_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_7565_zmod__zmult2__eq,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A2 @ ( times_times_int @ B2 @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B2 @ ( modulo_modulo_int @ ( divide_divide_int @ A2 @ B2 ) @ C ) ) @ ( modulo_modulo_int @ A2 @ B2 ) ) ) ) ).

% zmod_zmult2_eq
thf(fact_7566_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_7567_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_7568_binomial__strict__antimono,axiom,
    ! [K: nat,K5: nat,N: nat] :
      ( ( ord_less_nat @ K @ K5 )
     => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
       => ( ( ord_less_eq_nat @ K5 @ N )
         => ( ord_less_nat @ ( binomial @ N @ K5 ) @ ( binomial @ N @ K ) ) ) ) ) ).

% binomial_strict_antimono
thf(fact_7569_binomial__strict__mono,axiom,
    ! [K: nat,K5: nat,N: nat] :
      ( ( ord_less_nat @ K @ K5 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K5 ) @ N )
       => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K5 ) ) ) ) ).

% binomial_strict_mono
thf(fact_7570_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_7571_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_7572_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_7573_bits__stable__imp__add__self,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( ( divide1791077408188789448l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
        = A2 )
     => ( ( plus_p361126936061061375l_num1 @ A2 @ ( modulo1504961113040953224l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) )
        = zero_z3563351764282998399l_num1 ) ) ).

% bits_stable_imp_add_self
thf(fact_7574_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_7575_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_7576_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_7577_bits__stable__imp__add__self,axiom,
    ! [A2: uint32] :
      ( ( ( divide_divide_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
        = A2 )
     => ( ( plus_plus_uint32 @ A2 @ ( modulo_modulo_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) )
        = zero_zero_uint32 ) ) ).

% bits_stable_imp_add_self
thf(fact_7578_parity__cases,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo1504961113040953224l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
         != zero_z3563351764282998399l_num1 ) )
     => ~ ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A2 )
         => ( ( modulo1504961113040953224l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
           != one_on7727431528512463931l_num1 ) ) ) ).

% parity_cases
thf(fact_7579_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_7580_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_7581_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_7582_parity__cases,axiom,
    ! [A2: uint32] :
      ( ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo_modulo_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
         != zero_zero_uint32 ) )
     => ~ ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A2 )
         => ( ( modulo_modulo_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
           != one_one_uint32 ) ) ) ).

% parity_cases
thf(fact_7583_mod2__eq__if,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo1504961113040953224l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
          = zero_z3563351764282998399l_num1 ) )
      & ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo1504961113040953224l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
          = one_on7727431528512463931l_num1 ) ) ) ).

% mod2_eq_if
thf(fact_7584_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_7585_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_7586_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_7587_mod2__eq__if,axiom,
    ! [A2: uint32] :
      ( ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo_modulo_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
          = zero_zero_uint32 ) )
      & ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo_modulo_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
          = one_one_uint32 ) ) ) ).

% mod2_eq_if
thf(fact_7588_div__exp__mod__exp__eq,axiom,
    ! [A2: word_N3645301735248828278l_num1,N: nat,M: nat] :
      ( ( modulo1504961113040953224l_num1 @ ( divide1791077408188789448l_num1 @ A2 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N ) ) @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ M ) )
      = ( divide1791077408188789448l_num1 @ ( modulo1504961113040953224l_num1 @ A2 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_7589_div__exp__mod__exp__eq,axiom,
    ! [A2: int,N: nat,M: nat] :
      ( ( modulo_modulo_int @ ( divide_divide_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( 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 @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_7590_div__exp__mod__exp__eq,axiom,
    ! [A2: nat,N: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( divide_divide_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( 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 @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_7591_div__exp__mod__exp__eq,axiom,
    ! [A2: code_integer,N: nat,M: nat] :
      ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
      = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_7592_div__exp__mod__exp__eq,axiom,
    ! [A2: uint32,N: nat,M: nat] :
      ( ( modulo_modulo_uint32 @ ( divide_divide_uint32 @ A2 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ M ) )
      = ( divide_divide_uint32 @ ( modulo_modulo_uint32 @ A2 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_7593_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_7594_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_7595_verit__le__mono__div__int,axiom,
    ! [A: int,B: int,N: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int
          @ ( plus_plus_int @ ( divide_divide_int @ A @ N )
            @ ( if_int
              @ ( ( modulo_modulo_int @ B @ N )
                = zero_zero_int )
              @ one_one_int
              @ zero_zero_int ) )
          @ ( divide_divide_int @ B @ N ) ) ) ) ).

% verit_le_mono_div_int
thf(fact_7596_split__neg__lemma,axiom,
    ! [K: int,P: int > int > $o,N: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
        = ( ! [I4: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ I4 @ J3 ) ) ) ) ) ).

% split_neg_lemma
thf(fact_7597_split__pos__lemma,axiom,
    ! [K: int,P: int > int > $o,N: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
        = ( ! [I4: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ I4 @ J3 ) ) ) ) ) ).

% split_pos_lemma
thf(fact_7598_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_7599_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_7600_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_7601_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A2: word_N3645301735248828278l_num1] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo1504961113040953224l_num1 @ ( times_7065122842183080059l_num1 @ A2 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ M ) ) @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N ) )
        = ( times_7065122842183080059l_num1 @ ( modulo1504961113040953224l_num1 @ A2 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_7602_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A2: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo_modulo_int @ ( times_times_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
        = ( times_times_int @ ( modulo_modulo_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_7603_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A2: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo_modulo_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
        = ( times_times_nat @ ( modulo_modulo_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_7604_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A2: code_integer] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
        = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_7605_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A2: uint32] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo_modulo_uint32 @ ( times_times_uint32 @ A2 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N ) )
        = ( times_times_uint32 @ ( modulo_modulo_uint32 @ A2 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_7606_p1mod22k_H,axiom,
    ! [B2: int,N: 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 ) ) @ N ) ) )
      = ( 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 ) ) @ N ) ) ) ) ) ).

% p1mod22k'
thf(fact_7607_p1mod22k,axiom,
    ! [B2: int,N: 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 ) ) @ N ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) @ one_one_int ) ) ).

% p1mod22k
thf(fact_7608_m1mod2k,axiom,
    ! [N: nat] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ).

% m1mod2k
thf(fact_7609_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_7610_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_7611_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_7612_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_7613_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_7614_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_7615_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_7616_eme1p,axiom,
    ! [N: int,D: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ D )
         => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ N ) @ D )
            = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ N @ D ) ) ) ) ) ) ).

% eme1p
thf(fact_7617_emep1,axiom,
    ! [N: int,D: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ D )
         => ( ( modulo_modulo_int @ ( plus_plus_int @ N @ one_one_int ) @ D )
            = ( plus_plus_int @ ( modulo_modulo_int @ N @ D ) @ one_one_int ) ) ) ) ) ).

% emep1
thf(fact_7618_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_7619_m1mod22k,axiom,
    ! [N: 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 ) ) @ N ) ) )
      = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ one_one_int ) ) ).

% m1mod22k
thf(fact_7620_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_7621_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_7622_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_7623_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_7624_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_7625_set__decode__plus__power__2,axiom,
    ! [N: nat,Z: nat] :
      ( ~ ( member_nat @ N @ ( nat_set_decode @ Z ) )
     => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z ) )
        = ( insert_nat @ N @ ( nat_set_decode @ Z ) ) ) ) ).

% set_decode_plus_power_2
thf(fact_7626_set__decode__def,axiom,
    ( nat_set_decode
    = ( ^ [X2: nat] :
          ( collect_nat
          @ ^ [N4: nat] :
              ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ) ).

% set_decode_def
thf(fact_7627_zero__less__binomial__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
      = ( ord_less_eq_nat @ K @ N ) ) ).

% zero_less_binomial_iff
thf(fact_7628_choose__two,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% choose_two
thf(fact_7629_binomial__n__0,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ zero_zero_nat )
      = one_one_nat ) ).

% binomial_n_0
thf(fact_7630_binomial__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ( binomial @ N @ K )
        = zero_zero_nat )
      = ( ord_less_nat @ N @ K ) ) ).

% binomial_eq_0_iff
thf(fact_7631_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
        = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% times_binomial_minus1_eq
thf(fact_7632_choose__reduce__nat,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( binomial @ N @ K )
          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).

% choose_reduce_nat
thf(fact_7633_mod__word__self,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( modulo1504961113040953224l_num1 @ W @ W )
      = zero_z3563351764282998399l_num1 ) ).

% mod_word_self
thf(fact_7634_nat__mod__eq_H,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_nat @ A2 @ N )
     => ( ( modulo_modulo_nat @ A2 @ N )
        = A2 ) ) ).

% nat_mod_eq'
thf(fact_7635_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( modulo_modulo_nat @ M @ N )
        = M ) ) ).

% mod_less
thf(fact_7636_binomial__n__n,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ N )
      = one_one_nat ) ).

% binomial_n_n
thf(fact_7637_word__mod__no,axiom,
    ! [A2: num,B2: num] :
      ( ( modulo1504961113040953224l_num1 @ ( numera7442385471795722001l_num1 @ A2 ) @ ( numera7442385471795722001l_num1 @ B2 ) )
      = ( ring_17408606157368542149l_num1 @ ( modulo_modulo_int @ ( semiri7338730514057886004m1_int @ ( numera7442385471795722001l_num1 @ A2 ) ) @ ( semiri7338730514057886004m1_int @ ( numera7442385471795722001l_num1 @ B2 ) ) ) ) ) ).

% word_mod_no
thf(fact_7638_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_7639_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_7640_nat__mod__eq,axiom,
    ! [B2: nat,N: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ N )
     => ( ( ( modulo_modulo_nat @ A2 @ N )
          = ( modulo_modulo_nat @ B2 @ N ) )
       => ( ( modulo_modulo_nat @ A2 @ N )
          = B2 ) ) ) ).

% nat_mod_eq
thf(fact_7641_word__mod__by__0,axiom,
    ! [K: word_N3645301735248828278l_num1] :
      ( ( modulo1504961113040953224l_num1 @ K @ zero_z3563351764282998399l_num1 )
      = K ) ).

% word_mod_by_0
thf(fact_7642_gcd__nat__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
     => ( ! [M4: nat,N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ( P @ N2 @ ( modulo_modulo_nat @ M4 @ N2 ) )
             => ( P @ M4 @ N2 ) ) )
       => ( P @ M @ N ) ) ) ).

% gcd_nat_induct
thf(fact_7643_nat__mod__lem,axiom,
    ! [N: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ B2 @ N )
        = ( ( modulo_modulo_nat @ B2 @ N )
          = B2 ) ) ) ).

% nat_mod_lem
thf(fact_7644_mod__less__divisor,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).

% mod_less_divisor
thf(fact_7645_word__rot__lem,axiom,
    ! [L: nat,K: nat,D: nat,N: nat] :
      ( ( ( plus_plus_nat @ L @ K )
        = ( plus_plus_nat @ D @ ( modulo_modulo_nat @ K @ L ) ) )
     => ( ( ord_less_nat @ N @ L )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ D @ N ) @ L )
          = N ) ) ) ).

% word_rot_lem
thf(fact_7646_nat__minus__mod,axiom,
    ! [N: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( minus_minus_nat @ N @ ( modulo_modulo_nat @ N @ M ) ) @ M )
      = zero_zero_nat ) ).

% nat_minus_mod
thf(fact_7647_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_7648_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M3: nat,N4: nat] : ( if_nat @ ( ord_less_nat @ M3 @ N4 ) @ M3 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M3 @ N4 ) @ N4 ) ) ) ) ).

% mod_if
thf(fact_7649_mod__geq,axiom,
    ! [M: nat,N: nat] :
      ( ~ ( ord_less_nat @ M @ N )
     => ( ( modulo_modulo_nat @ M @ N )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).

% mod_geq
thf(fact_7650_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( ( modulo_modulo_nat @ M @ D )
        = zero_zero_nat )
     => ? [Q5: nat] :
          ( M
          = ( times_times_nat @ D @ Q5 ) ) ) ).

% mod_eq_0D
thf(fact_7651_msrevs_I2_J,axiom,
    ! [K: nat,N: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) @ N )
      = ( modulo_modulo_nat @ M @ N ) ) ).

% msrevs(2)
thf(fact_7652_word__mod__less__divisor,axiom,
    ! [N: word_N3645301735248828278l_num1,M: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ N )
     => ( ord_le750835935415966154l_num1 @ ( modulo1504961113040953224l_num1 @ M @ N ) @ N ) ) ).

% word_mod_less_divisor
thf(fact_7653_uint__mod,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( semiri7338730514057886004m1_int @ ( modulo1504961113040953224l_num1 @ X @ Y ) )
      = ( modulo_modulo_int @ ( semiri7338730514057886004m1_int @ X ) @ ( semiri7338730514057886004m1_int @ Y ) ) ) ).

% uint_mod
thf(fact_7654_uint__mod__distrib,axiom,
    ! [V: word_N3645301735248828278l_num1,W: word_N3645301735248828278l_num1] :
      ( ( semiri7338730514057886004m1_int @ ( modulo1504961113040953224l_num1 @ V @ W ) )
      = ( modulo_modulo_int @ ( semiri7338730514057886004m1_int @ V ) @ ( semiri7338730514057886004m1_int @ W ) ) ) ).

% uint_mod_distrib
thf(fact_7655_zmod__int,axiom,
    ! [A2: nat,B2: nat] :
      ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A2 @ B2 ) )
      = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).

% zmod_int
thf(fact_7656_mod__le__divisor,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).

% mod_le_divisor
thf(fact_7657_div__less__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( modulo_modulo_nat @ A @ N )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B @ N )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A @ N ) @ ( divide_divide_nat @ B @ N ) ) ) ) ) ) ).

% div_less_mono
thf(fact_7658_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_7659_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
      = ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).

% mod_greater_zero_iff_not_dvd
thf(fact_7660_div__mod__decomp,axiom,
    ! [A: nat,N: nat] :
      ( A
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ N ) @ N ) @ ( modulo_modulo_nat @ A @ N ) ) ) ).

% div_mod_decomp
thf(fact_7661_word__mod__def,axiom,
    ( modulo1504961113040953224l_num1
    = ( ^ [A4: word_N3645301735248828278l_num1,B4: word_N3645301735248828278l_num1] : ( ring_17408606157368542149l_num1 @ ( modulo_modulo_int @ ( semiri7338730514057886004m1_int @ A4 ) @ ( semiri7338730514057886004m1_int @ B4 ) ) ) ) ) ).

% word_mod_def
thf(fact_7662_even__even__mod__4__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).

% even_even_mod_4_iff
thf(fact_7663_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N ) )
      = ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_7664_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_7665_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_7666_mod__lemma,axiom,
    ! [C: nat,R3: nat,B2: nat,Q3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( ( ord_less_nat @ R3 @ B2 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ ( modulo_modulo_nat @ Q3 @ C ) ) @ R3 ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).

% mod_lemma
thf(fact_7667_split__mod,axiom,
    ! [P: nat > $o,M: nat,N: nat] :
      ( ( P @ ( modulo_modulo_nat @ M @ N ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ M ) )
        & ( ( N != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
               => ( P @ J3 ) ) ) ) ) ) ).

% split_mod
thf(fact_7668_diff__mod__le,axiom,
    ! [A2: nat,D: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ D )
     => ( ( dvd_dvd_nat @ B2 @ D )
       => ( ord_less_eq_nat @ ( minus_minus_nat @ A2 @ ( modulo_modulo_nat @ A2 @ B2 ) ) @ ( minus_minus_nat @ D @ B2 ) ) ) ) ).

% diff_mod_le
thf(fact_7669_mod__nat__eqI,axiom,
    ! [R3: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ R3 @ N )
     => ( ( ord_less_eq_nat @ R3 @ M )
       => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R3 ) )
         => ( ( modulo_modulo_nat @ M @ N )
            = R3 ) ) ) ) ).

% mod_nat_eqI
thf(fact_7670_real__of__nat__div__aux,axiom,
    ! [X: nat,D: nat] :
      ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ D ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).

% real_of_nat_div_aux
thf(fact_7671_power__mod__div,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( 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 @ N @ M ) ) ) ) ).

% power_mod_div
thf(fact_7672_verit__le__mono__div,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A @ N )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B @ N )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B @ N ) ) ) ) ).

% verit_le_mono_div
thf(fact_7673_choose__one,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ one_one_nat )
      = N ) ).

% choose_one
thf(fact_7674_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( binomial @ N @ K )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_7675_binomial__le__pow,axiom,
    ! [R3: nat,N: nat] :
      ( ( ord_less_eq_nat @ R3 @ N )
     => ( ord_less_eq_nat @ ( binomial @ N @ R3 ) @ ( power_power_nat @ N @ R3 ) ) ) ).

% binomial_le_pow
thf(fact_7676_zero__less__binomial,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).

% zero_less_binomial
thf(fact_7677_binomial__absorb__comp,axiom,
    ! [N: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
      = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).

% binomial_absorb_comp
thf(fact_7678_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) ) ) ).

% binomial_ge_n_over_k_pow_k
thf(fact_7679_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) ) ) ).

% binomial_ge_n_over_k_pow_k
thf(fact_7680_binomial__le__pow2,axiom,
    ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% binomial_le_pow2
thf(fact_7681_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
            @ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N4 ) ) ) ) ) ) ) ).

% ln_series
thf(fact_7682_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_7683_signed__take__bit__rec,axiom,
    ( bit_ri1375673916561920181l_num1
    = ( ^ [N4: nat,A4: word_N3645301735248828278l_num1] : ( if_wor5778924947035936048l_num1 @ ( N4 = zero_zero_nat ) @ ( uminus8244633308260627903l_num1 @ ( modulo1504961113040953224l_num1 @ A4 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) @ ( plus_p361126936061061375l_num1 @ ( modulo1504961113040953224l_num1 @ A4 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( bit_ri1375673916561920181l_num1 @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide1791077408188789448l_num1 @ A4 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_7684_signed__take__bit__rec,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N4: nat,A4: code_integer] : ( if_Code_integer @ ( N4 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_7685_signed__take__bit__rec,axiom,
    ( bit_ri6224792872505173163uint32
    = ( ^ [N4: nat,A4: uint32] : ( if_uint32 @ ( N4 = zero_zero_nat ) @ ( uminus_uminus_uint32 @ ( modulo_modulo_uint32 @ A4 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) @ ( plus_plus_uint32 @ ( modulo_modulo_uint32 @ A4 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_ri6224792872505173163uint32 @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide_divide_uint32 @ A4 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_7686_signed__take__bit__rec,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N4: nat,A4: int] : ( if_int @ ( N4 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_7687_binomial__code,axiom,
    ( binomial
    = ( ^ [N4: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N4 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N4 @ ( minus_minus_nat @ N4 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N4 @ K3 ) @ one_one_nat ) @ N4 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).

% binomial_code
thf(fact_7688_uint32_Osize__eq,axiom,
    ( size_size_uint32
    = ( ^ [P6: uint32] : ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% uint32.size_eq
thf(fact_7689_add__scale__eq__noteq,axiom,
    ! [R3: real,A2: real,B2: real,C: real,D: real] :
      ( ( R3 != zero_zero_real )
     => ( ( ( A2 = B2 )
          & ( C != D ) )
       => ( ( plus_plus_real @ A2 @ ( times_times_real @ R3 @ C ) )
         != ( plus_plus_real @ B2 @ ( times_times_real @ R3 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_7690_add__scale__eq__noteq,axiom,
    ! [R3: rat,A2: rat,B2: rat,C: rat,D: rat] :
      ( ( R3 != zero_zero_rat )
     => ( ( ( A2 = B2 )
          & ( C != D ) )
       => ( ( plus_plus_rat @ A2 @ ( times_times_rat @ R3 @ C ) )
         != ( plus_plus_rat @ B2 @ ( times_times_rat @ R3 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_7691_add__scale__eq__noteq,axiom,
    ! [R3: nat,A2: nat,B2: nat,C: nat,D: nat] :
      ( ( R3 != zero_zero_nat )
     => ( ( ( A2 = B2 )
          & ( C != D ) )
       => ( ( plus_plus_nat @ A2 @ ( times_times_nat @ R3 @ C ) )
         != ( plus_plus_nat @ B2 @ ( times_times_nat @ R3 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_7692_add__scale__eq__noteq,axiom,
    ! [R3: int,A2: int,B2: int,C: int,D: int] :
      ( ( R3 != zero_zero_int )
     => ( ( ( A2 = B2 )
          & ( C != D ) )
       => ( ( plus_plus_int @ A2 @ ( times_times_int @ R3 @ C ) )
         != ( plus_plus_int @ B2 @ ( times_times_int @ R3 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_7693_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_7694_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% nat.inject
thf(fact_7695_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_less_eq
thf(fact_7696_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_7697_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_7698_add__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ M @ ( suc @ N ) )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc_right
thf(fact_7699_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N @ M ) ) ).

% Suc_le_mono
thf(fact_7700_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_Suc_Suc
thf(fact_7701_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).

% Suc_diff_diff
thf(fact_7702_signed__take__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_ri1375673916561920181l_num1 @ N @ zero_z3563351764282998399l_num1 )
      = zero_z3563351764282998399l_num1 ) ).

% signed_take_bit_of_0
thf(fact_7703_signed__take__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
      = zero_zero_int ) ).

% signed_take_bit_of_0
thf(fact_7704_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_2184487114949457152l_num1 @ zero_z3563351764282998399l_num1 @ ( suc @ N ) )
      = zero_z3563351764282998399l_num1 ) ).

% power_0_Suc
thf(fact_7705_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
      = zero_zero_rat ) ).

% power_0_Suc
thf(fact_7706_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_7707_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
      = zero_zero_real ) ).

% power_0_Suc
thf(fact_7708_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
      = zero_zero_int ) ).

% power_0_Suc
thf(fact_7709_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
      = zero_zero_complex ) ).

% power_0_Suc
thf(fact_7710_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ ( suc @ N ) )
      = zero_z3403309356797280102nteger ) ).

% power_0_Suc
thf(fact_7711_power__Suc0__right,axiom,
    ! [A2: nat] :
      ( ( power_power_nat @ A2 @ ( suc @ zero_zero_nat ) )
      = A2 ) ).

% power_Suc0_right
thf(fact_7712_power__Suc0__right,axiom,
    ! [A2: real] :
      ( ( power_power_real @ A2 @ ( suc @ zero_zero_nat ) )
      = A2 ) ).

% power_Suc0_right
thf(fact_7713_power__Suc0__right,axiom,
    ! [A2: int] :
      ( ( power_power_int @ A2 @ ( suc @ zero_zero_nat ) )
      = A2 ) ).

% power_Suc0_right
thf(fact_7714_power__Suc0__right,axiom,
    ! [A2: complex] :
      ( ( power_power_complex @ A2 @ ( suc @ zero_zero_nat ) )
      = A2 ) ).

% power_Suc0_right
thf(fact_7715_power__Suc0__right,axiom,
    ! [A2: code_integer] :
      ( ( power_8256067586552552935nteger @ A2 @ ( suc @ zero_zero_nat ) )
      = A2 ) ).

% power_Suc0_right
thf(fact_7716_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_7717_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_7718_div__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
      = M ) ).

% div_by_Suc_0
thf(fact_7719_one__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( times_times_nat @ M @ N ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% one_eq_mult_iff
thf(fact_7720_mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% mult_eq_1_iff
thf(fact_7721_power__Suc__0,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_7722_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_7723_diff__Suc__1,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
      = N ) ).

% diff_Suc_1
thf(fact_7724_mult__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( times_times_nat @ M @ ( suc @ N ) )
      = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).

% mult_Suc_right
thf(fact_7725_dvd__1__left,axiom,
    ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).

% dvd_1_left
thf(fact_7726_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_7727_signed__take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_ri1375673916561920181l_num1 @ ( suc @ N ) @ one_on7727431528512463931l_num1 )
      = one_on7727431528512463931l_num1 ) ).

% signed_take_bit_Suc_1
thf(fact_7728_signed__take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
      = one_one_int ) ).

% signed_take_bit_Suc_1
thf(fact_7729_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] :
      ( ( bit_ri1375673916561920181l_num1 @ ( numeral_numeral_nat @ K ) @ one_on7727431528512463931l_num1 )
      = one_on7727431528512463931l_num1 ) ).

% signed_take_bit_numeral_of_1
thf(fact_7730_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_7731_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] :
      ( ( bit_ri1375673916561920181l_num1 @ N @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
      = ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ).

% signed_take_bit_of_minus_1
thf(fact_7732_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] :
      ( ( bit_ri6224792872505173163uint32 @ N @ ( uminus_uminus_uint32 @ one_one_uint32 ) )
      = ( uminus_uminus_uint32 @ one_one_uint32 ) ) ).

% signed_take_bit_of_minus_1
thf(fact_7733_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] :
      ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% signed_take_bit_of_minus_1
thf(fact_7734_mod__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% mod_by_Suc_0
thf(fact_7735_fact__0,axiom,
    ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
    = one_one_rat ) ).

% fact_0
thf(fact_7736_fact__0,axiom,
    ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
    = one_one_int ) ).

% fact_0
thf(fact_7737_fact__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
    = one_one_nat ) ).

% fact_0
thf(fact_7738_fact__0,axiom,
    ( ( semiri2265585572941072030t_real @ zero_zero_nat )
    = one_one_real ) ).

% fact_0
thf(fact_7739_fact__0,axiom,
    ( ( semiri5044797733671781792omplex @ zero_zero_nat )
    = one_one_complex ) ).

% fact_0
thf(fact_7740_fact__1,axiom,
    ( ( semiri773545260158071498ct_rat @ one_one_nat )
    = one_one_rat ) ).

% fact_1
thf(fact_7741_fact__1,axiom,
    ( ( semiri1406184849735516958ct_int @ one_one_nat )
    = one_one_int ) ).

% fact_1
thf(fact_7742_fact__1,axiom,
    ( ( semiri1408675320244567234ct_nat @ one_one_nat )
    = one_one_nat ) ).

% fact_1
thf(fact_7743_fact__1,axiom,
    ( ( semiri2265585572941072030t_real @ one_one_nat )
    = one_one_real ) ).

% fact_1
thf(fact_7744_fact__1,axiom,
    ( ( semiri5044797733671781792omplex @ one_one_nat )
    = one_one_complex ) ).

% fact_1
thf(fact_7745_binomial__1,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
      = N ) ).

% binomial_1
thf(fact_7746_binomial__0__Suc,axiom,
    ! [K: nat] :
      ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% binomial_0_Suc
thf(fact_7747_of__nat__fact,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( semiri1408675320244567234ct_nat @ N ) )
      = ( semiri1406184849735516958ct_int @ N ) ) ).

% of_nat_fact
thf(fact_7748_of__nat__fact,axiom,
    ! [N: nat] :
      ( ( semiri4939895301339042750nteger @ ( semiri1408675320244567234ct_nat @ N ) )
      = ( semiri3624122377584611663nteger @ N ) ) ).

% of_nat_fact
thf(fact_7749_of__nat__fact,axiom,
    ! [N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( semiri1408675320244567234ct_nat @ N ) )
      = ( semiri1408675320244567234ct_nat @ N ) ) ).

% of_nat_fact
thf(fact_7750_of__nat__fact,axiom,
    ! [N: nat] :
      ( ( semiri5074537144036343181t_real @ ( semiri1408675320244567234ct_nat @ N ) )
      = ( semiri2265585572941072030t_real @ N ) ) ).

% of_nat_fact
thf(fact_7751_of__nat__fact,axiom,
    ! [N: nat] :
      ( ( semiri8010041392384452111omplex @ ( semiri1408675320244567234ct_nat @ N ) )
      = ( semiri5044797733671781792omplex @ N ) ) ).

% of_nat_fact
thf(fact_7752_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri8819519690708144855l_num1 @ ( suc @ M ) )
      = ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ ( semiri8819519690708144855l_num1 @ M ) ) ) ).

% of_nat_Suc
thf(fact_7753_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).

% of_nat_Suc
thf(fact_7754_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% of_nat_Suc
thf(fact_7755_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
      = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).

% of_nat_Suc
thf(fact_7756_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).

% of_nat_Suc
thf(fact_7757_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri4939895301339042750nteger @ ( suc @ M ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( semiri4939895301339042750nteger @ M ) ) ) ).

% of_nat_Suc
thf(fact_7758_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
      = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).

% of_nat_Suc
thf(fact_7759_Suc__pred,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        = N ) ) ).

% Suc_pred
thf(fact_7760_one__le__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
      = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
        & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).

% one_le_mult_iff
thf(fact_7761_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_7762_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
        = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_7763_Suc__diff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ one_one_nat @ M )
       => ( ( suc @ ( minus_minus_nat @ N @ M ) )
          = ( minus_minus_nat @ N @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ).

% Suc_diff
thf(fact_7764_fact__Suc__0,axiom,
    ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
    = one_one_rat ) ).

% fact_Suc_0
thf(fact_7765_fact__Suc__0,axiom,
    ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% fact_Suc_0
thf(fact_7766_fact__Suc__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% fact_Suc_0
thf(fact_7767_fact__Suc__0,axiom,
    ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
    = one_one_real ) ).

% fact_Suc_0
thf(fact_7768_fact__Suc__0,axiom,
    ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
    = one_one_complex ) ).

% fact_Suc_0
thf(fact_7769_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% fact_Suc
thf(fact_7770_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).

% fact_Suc
thf(fact_7771_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri3624122377584611663nteger @ ( suc @ N ) )
      = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( suc @ N ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).

% fact_Suc
thf(fact_7772_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% fact_Suc
thf(fact_7773_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri2265585572941072030t_real @ ( suc @ N ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).

% fact_Suc
thf(fact_7774_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri5044797733671781792omplex @ ( suc @ N ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).

% fact_Suc
thf(fact_7775_Suc__numeral,axiom,
    ! [N: num] :
      ( ( suc @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).

% Suc_numeral
thf(fact_7776_negative__zless,axiom,
    ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zless
thf(fact_7777_add__2__eq__Suc_H,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( suc @ ( suc @ N ) ) ) ).

% add_2_eq_Suc'
thf(fact_7778_add__2__eq__Suc,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
      = ( suc @ ( suc @ N ) ) ) ).

% add_2_eq_Suc
thf(fact_7779_Suc__1,axiom,
    ( ( suc @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% Suc_1
thf(fact_7780_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_7781_even__Suc__Suc__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% even_Suc_Suc_iff
thf(fact_7782_even__Suc,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% even_Suc
thf(fact_7783_signed__take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_bit0
thf(fact_7784_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_7785_Suc__diff__1,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
        = N ) ) ).

% Suc_diff_1
thf(fact_7786_fact__2,axiom,
    ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_7787_fact__2,axiom,
    ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_7788_fact__2,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_7789_fact__2,axiom,
    ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_7790_fact__2,axiom,
    ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_7791_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N: nat] :
      ( ( ( numeral_numeral_nat @ K )
       != one_one_nat )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
        = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
thf(fact_7792_Suc__unat__minus__one,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( X != zero_z3563351764282998399l_num1 )
     => ( ( suc @ ( semiri7341220984566936280m1_nat @ ( minus_4019991460397169231l_num1 @ X @ one_on7727431528512463931l_num1 ) ) )
        = ( semiri7341220984566936280m1_nat @ X ) ) ) ).

% Suc_unat_minus_one
thf(fact_7793_shiftl__Suc__0,axiom,
    ! [N: nat] :
      ( ( bit_Sh3965577149348748681tl_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% shiftl_Suc_0
thf(fact_7794_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N: nat] :
      ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != ( suc @ zero_zero_nat ) )
      = ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
thf(fact_7795_even__Suc__div__two,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_Suc_div_two
thf(fact_7796_odd__Suc__div__two,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% odd_Suc_div_two
thf(fact_7797_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_minus_bit0
thf(fact_7798_shiftl__of__Suc,axiom,
    ! [A2: word_N3645301735248828278l_num1,N: nat] :
      ( ( bit_Sh9074413540854191407l_num1 @ A2 @ ( suc @ N ) )
      = ( bit_Sh9074413540854191407l_num1 @ ( times_7065122842183080059l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) @ N ) ) ).

% shiftl_of_Suc
thf(fact_7799_shiftl__of__Suc,axiom,
    ! [A2: int,N: nat] :
      ( ( bit_Sh3963086678839698405tl_int @ A2 @ ( suc @ N ) )
      = ( bit_Sh3963086678839698405tl_int @ ( times_times_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N ) ) ).

% shiftl_of_Suc
thf(fact_7800_shiftl__of__Suc,axiom,
    ! [A2: nat,N: nat] :
      ( ( bit_Sh3965577149348748681tl_nat @ A2 @ ( suc @ N ) )
      = ( bit_Sh3965577149348748681tl_nat @ ( times_times_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N ) ) ).

% shiftl_of_Suc
thf(fact_7801_set__decode__Suc,axiom,
    ! [N: nat,X: nat] :
      ( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
      = ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_decode_Suc
thf(fact_7802_odd__Suc__minus__one,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        = N ) ) ).

% odd_Suc_minus_one
thf(fact_7803_signed__take__bit__0,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( bit_ri1375673916561920181l_num1 @ zero_zero_nat @ A2 )
      = ( uminus8244633308260627903l_num1 @ ( modulo1504961113040953224l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ).

% signed_take_bit_0
thf(fact_7804_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_7805_signed__take__bit__0,axiom,
    ! [A2: uint32] :
      ( ( bit_ri6224792872505173163uint32 @ zero_zero_nat @ A2 )
      = ( uminus_uminus_uint32 @ ( modulo_modulo_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ).

% signed_take_bit_0
thf(fact_7806_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_7807_fact__ge__Suc__0__nat,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_Suc_0_nat
thf(fact_7808_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_7809_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( suc @ X )
        = ( suc @ Y ) )
     => ( X = Y ) ) ).

% Suc_inject
thf(fact_7810_fact__nonzero,axiom,
    ! [N: nat] :
      ( ( semiri773545260158071498ct_rat @ N )
     != zero_zero_rat ) ).

% fact_nonzero
thf(fact_7811_fact__nonzero,axiom,
    ! [N: nat] :
      ( ( semiri1406184849735516958ct_int @ N )
     != zero_zero_int ) ).

% fact_nonzero
thf(fact_7812_fact__nonzero,axiom,
    ! [N: nat] :
      ( ( semiri1408675320244567234ct_nat @ N )
     != zero_zero_nat ) ).

% fact_nonzero
thf(fact_7813_fact__nonzero,axiom,
    ! [N: nat] :
      ( ( semiri2265585572941072030t_real @ N )
     != zero_zero_real ) ).

% fact_nonzero
thf(fact_7814_fact__nonzero,axiom,
    ! [N: nat] :
      ( ( semiri5044797733671781792omplex @ N )
     != zero_zero_complex ) ).

% fact_nonzero
thf(fact_7815_pi__neq__zero,axiom,
    pi != zero_zero_real ).

% pi_neq_zero
thf(fact_7816_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N2: nat] :
            ( X
           != ( suc @ N2 ) ) ) ).

% list_decode.cases
thf(fact_7817_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 ) )
       => ~ ! [Va2: nat] :
              ( X
             != ( suc @ ( suc @ Va2 ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
thf(fact_7818_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_12: nat] : ( P @ X_12 )
       => ? [N2: nat] :
            ( ~ ( P @ N2 )
            & ( P @ ( suc @ N2 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_7819_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M4: nat] :
          ( N
          = ( suc @ M4 ) ) ) ).

% not0_implies_Suc
thf(fact_7820_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_7821_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_7822_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_7823_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_7824_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
     => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
       => ( ! [X3: nat,Y4: nat] :
              ( ( P @ X3 @ Y4 )
             => ( P @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
         => ( P @ M @ N ) ) ) ) ).

% diff_induct
thf(fact_7825_nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( P @ N2 )
           => ( P @ ( suc @ N2 ) ) )
       => ( P @ N ) ) ) ).

% nat_induct
thf(fact_7826_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_7827_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_7828_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_7829_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_7830_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_7831_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_7832_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I3: nat] :
            ( ( J
              = ( suc @ I3 ) )
           => ( P @ I3 ) )
       => ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ J )
             => ( ( P @ ( suc @ I3 ) )
               => ( P @ I3 ) ) )
         => ( P @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_7833_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
       => ( ! [I3: nat,J2: nat,K2: nat] :
              ( ( ord_less_nat @ I3 @ J2 )
             => ( ( ord_less_nat @ J2 @ K2 )
               => ( ( P @ I3 @ J2 )
                 => ( ( P @ J2 @ K2 )
                   => ( P @ I3 @ K2 ) ) ) ) )
         => ( P @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_7834_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ J @ K )
       => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_7835_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_less_SucD
thf(fact_7836_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
       => ( M = N ) ) ) ).

% less_antisym
thf(fact_7837_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M )
      = ( ? [M5: nat] :
            ( ( M
              = ( suc @ M5 ) )
            & ( ord_less_nat @ N @ M5 ) ) ) ) ).

% Suc_less_eq2
thf(fact_7838_Nat_OAll__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
           => ( P @ I4 ) ) )
      = ( ( P @ N )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
           => ( P @ I4 ) ) ) ) ).

% Nat.All_less_Suc
thf(fact_7839_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_7840_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( M = N ) ) ) ).

% less_Suc_eq
thf(fact_7841_Ex__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
            & ( P @ I4 ) ) )
      = ( ( P @ N )
        | ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
            & ( P @ I4 ) ) ) ) ).

% Ex_less_Suc
thf(fact_7842_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_7843_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( M = N ) ) ) ).

% less_SucE
thf(fact_7844_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ( suc @ M )
         != N )
       => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).

% Suc_lessI
thf(fact_7845_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ K )
     => ~ ! [J2: nat] :
            ( ( ord_less_nat @ I @ J2 )
           => ( K
             != ( suc @ J2 ) ) ) ) ).

% Suc_lessE
thf(fact_7846_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_lessD
thf(fact_7847_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ I @ K )
     => ( ( K
         != ( suc @ I ) )
       => ~ ! [J2: nat] :
              ( ( ord_less_nat @ I @ J2 )
             => ( K
               != ( suc @ J2 ) ) ) ) ) ).

% Nat.lessE
thf(fact_7848_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_7849_add__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc
thf(fact_7850_add__Suc__shift,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).

% add_Suc_shift
thf(fact_7851_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_7852_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M @ N )
       => ( M
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_7853_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_7854_Suc__le__D,axiom,
    ! [N: nat,M6: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
     => ? [M4: nat] :
          ( M6
          = ( suc @ M4 ) ) ) ).

% Suc_le_D
thf(fact_7855_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M @ N )
        | ( M
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_7856_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_7857_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).

% not_less_eq_eq
thf(fact_7858_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
             => ( P @ M2 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_7859_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( P @ M )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ M @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_7860_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X3: nat] : ( R @ X3 @ X3 )
       => ( ! [X3: nat,Y4: nat,Z3: nat] :
              ( ( R @ X3 @ Y4 )
             => ( ( R @ Y4 @ Z3 )
               => ( R @ X3 @ Z3 ) ) )
         => ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
           => ( R @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_7861_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I: nat] :
      ( ( P @ K )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).

% zero_induct_lemma
thf(fact_7862_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ ( suc @ K ) @ M )
        = ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( M = N ) ) ).

% Suc_mult_cancel1
thf(fact_7863_fact__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ ( suc @ M ) )
     => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
        = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).

% fact_diff_Suc
thf(fact_7864_summable__Suc__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( F @ ( suc @ N4 ) ) )
      = ( summable_real @ F ) ) ).

% summable_Suc_iff
thf(fact_7865_summable__Suc__iff,axiom,
    ! [F: nat > complex] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( F @ ( suc @ N4 ) ) )
      = ( summable_complex @ F ) ) ).

% summable_Suc_iff
thf(fact_7866_fact__less__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).

% fact_less_mono_nat
thf(fact_7867_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).

% fact_ge_zero
thf(fact_7868_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).

% fact_ge_zero
thf(fact_7869_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_zero
thf(fact_7870_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).

% fact_ge_zero
thf(fact_7871_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).

% fact_not_neg
thf(fact_7872_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).

% fact_not_neg
thf(fact_7873_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).

% fact_not_neg
thf(fact_7874_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).

% fact_not_neg
thf(fact_7875_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).

% fact_gt_zero
thf(fact_7876_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).

% fact_gt_zero
thf(fact_7877_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_gt_zero
thf(fact_7878_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).

% fact_gt_zero
thf(fact_7879_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).

% fact_ge_1
thf(fact_7880_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).

% fact_ge_1
thf(fact_7881_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_1
thf(fact_7882_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).

% fact_ge_1
thf(fact_7883_pi__gt__zero,axiom,
    ord_less_real @ zero_zero_real @ pi ).

% pi_gt_zero
thf(fact_7884_pi__not__less__zero,axiom,
    ~ ( ord_less_real @ pi @ zero_zero_real ) ).

% pi_not_less_zero
thf(fact_7885_pi__ge__zero,axiom,
    ord_less_eq_real @ zero_zero_real @ pi ).

% pi_ge_zero
thf(fact_7886_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > real,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_7887_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > rat,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_7888_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > num,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_7889_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_7890_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > int,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_7891_lift__Suc__mono__less,axiom,
    ! [F: nat > real,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N7 )
       => ( ord_less_real @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_7892_lift__Suc__mono__less,axiom,
    ! [F: nat > rat,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N7 )
       => ( ord_less_rat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_7893_lift__Suc__mono__less,axiom,
    ! [F: nat > num,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N7 )
       => ( ord_less_num @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_7894_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N7 )
       => ( ord_less_nat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_7895_lift__Suc__mono__less,axiom,
    ! [F: nat > int,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N7 )
       => ( ord_less_int @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_7896_power__Suc2,axiom,
    ! [A2: complex,N: nat] :
      ( ( power_power_complex @ A2 @ ( suc @ N ) )
      = ( times_times_complex @ ( power_power_complex @ A2 @ N ) @ A2 ) ) ).

% power_Suc2
thf(fact_7897_power__Suc2,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ A2 @ ( suc @ N ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ A2 ) ) ).

% power_Suc2
thf(fact_7898_power__Suc2,axiom,
    ! [A2: real,N: nat] :
      ( ( power_power_real @ A2 @ ( suc @ N ) )
      = ( times_times_real @ ( power_power_real @ A2 @ N ) @ A2 ) ) ).

% power_Suc2
thf(fact_7899_power__Suc2,axiom,
    ! [A2: rat,N: nat] :
      ( ( power_power_rat @ A2 @ ( suc @ N ) )
      = ( times_times_rat @ ( power_power_rat @ A2 @ N ) @ A2 ) ) ).

% power_Suc2
thf(fact_7900_power__Suc2,axiom,
    ! [A2: nat,N: nat] :
      ( ( power_power_nat @ A2 @ ( suc @ N ) )
      = ( times_times_nat @ ( power_power_nat @ A2 @ N ) @ A2 ) ) ).

% power_Suc2
thf(fact_7901_power__Suc2,axiom,
    ! [A2: int,N: nat] :
      ( ( power_power_int @ A2 @ ( suc @ N ) )
      = ( times_times_int @ ( power_power_int @ A2 @ N ) @ A2 ) ) ).

% power_Suc2
thf(fact_7902_power__Suc,axiom,
    ! [A2: complex,N: nat] :
      ( ( power_power_complex @ A2 @ ( suc @ N ) )
      = ( times_times_complex @ A2 @ ( power_power_complex @ A2 @ N ) ) ) ).

% power_Suc
thf(fact_7903_power__Suc,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ A2 @ ( suc @ N ) )
      = ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N ) ) ) ).

% power_Suc
thf(fact_7904_power__Suc,axiom,
    ! [A2: real,N: nat] :
      ( ( power_power_real @ A2 @ ( suc @ N ) )
      = ( times_times_real @ A2 @ ( power_power_real @ A2 @ N ) ) ) ).

% power_Suc
thf(fact_7905_power__Suc,axiom,
    ! [A2: rat,N: nat] :
      ( ( power_power_rat @ A2 @ ( suc @ N ) )
      = ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N ) ) ) ).

% power_Suc
thf(fact_7906_power__Suc,axiom,
    ! [A2: nat,N: nat] :
      ( ( power_power_nat @ A2 @ ( suc @ N ) )
      = ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ).

% power_Suc
thf(fact_7907_power__Suc,axiom,
    ! [A2: int,N: nat] :
      ( ( power_power_int @ A2 @ ( suc @ N ) )
      = ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ).

% power_Suc
thf(fact_7908_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_nat,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_set_nat @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_7909_lift__Suc__antimono__le,axiom,
    ! [F: nat > rat,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_rat @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_7910_lift__Suc__antimono__le,axiom,
    ! [F: nat > num,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_num @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_7911_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_nat @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_7912_lift__Suc__antimono__le,axiom,
    ! [F: nat > int,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_int @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_7913_lift__Suc__mono__le,axiom,
    ! [F: nat > set_nat,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_7914_lift__Suc__mono__le,axiom,
    ! [F: nat > rat,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_7915_lift__Suc__mono__le,axiom,
    ! [F: nat > num,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_7916_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_7917_lift__Suc__mono__le,axiom,
    ! [F: nat > int,N: nat,N7: nat] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N7 )
       => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_7918_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
     != zero_zero_rat ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_7919_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
     != zero_zero_int ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_7920_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
     != zero_zero_real ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_7921_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
     != zero_zero_nat ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_7922_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri4939895301339042750nteger @ ( suc @ N ) )
     != zero_z3403309356797280102nteger ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_7923_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ N ) )
     != zero_zero_complex ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_7924_Comparator__Generator_OAll__less__Suc,axiom,
    ! [X: nat,P: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ X ) )
           => ( P @ I4 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ X )
           => ( P @ ( suc @ I4 ) ) ) ) ) ).

% Comparator_Generator.All_less_Suc
thf(fact_7925_forall__finite_I2_J,axiom,
    ! [P: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ zero_zero_nat ) )
           => ( P @ I4 ) ) )
      = ( P @ zero_zero_nat ) ) ).

% forall_finite(2)
thf(fact_7926_forall__finite_I3_J,axiom,
    ! [X: nat,P: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ ( suc @ X ) ) )
           => ( P @ I4 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ X ) )
           => ( P @ ( suc @ I4 ) ) ) ) ) ).

% forall_finite(3)
thf(fact_7927_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_7928_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M4: nat] :
          ( N
          = ( suc @ M4 ) ) ) ).

% gr0_implies_Suc
thf(fact_7929_All__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
           => ( P @ I4 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
           => ( P @ ( suc @ I4 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_7930_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M3: nat] :
            ( N
            = ( suc @ M3 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_7931_Ex__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N ) )
            & ( P @ I4 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ N )
            & ( P @ ( suc @ I4 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_7932_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M @ N ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_7933_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_7934_nat__compl__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq_nat @ Nn @ N2 )
               => ( P @ Nn ) )
           => ( P @ ( suc @ N2 ) ) )
       => ( P @ N ) ) ) ).

% nat_compl_induct
thf(fact_7935_nat__compl__induct_H,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq_nat @ Nn @ N2 )
               => ( P @ Nn ) )
           => ( P @ ( suc @ N2 ) ) )
       => ( P @ N ) ) ) ).

% nat_compl_induct'
thf(fact_7936_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_7937_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ~ ! [Q5: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M @ Q5 ) ) ) ) ).

% less_natE
thf(fact_7938_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).

% less_add_Suc1
thf(fact_7939_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).

% less_add_Suc2
thf(fact_7940_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N4: nat] :
        ? [K3: nat] :
          ( N4
          = ( suc @ ( plus_plus_nat @ M3 @ K3 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_7941_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ? [K2: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_7942_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_7943_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_le_eq
thf(fact_7944_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ N2 )
                 => ( P @ ( suc @ N2 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_7945_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ ( suc @ N2 ) )
                 => ( P @ N2 ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_7946_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_le_lessD
thf(fact_7947_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% le_less_Suc_eq
thf(fact_7948_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% less_Suc_eq_le
thf(fact_7949_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_7950_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_7951_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_7952_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_7953_Suc__to__right,axiom,
    ! [N: nat,M: nat] :
      ( ( ( suc @ N )
        = M )
     => ( N
        = ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ).

% Suc_to_right
thf(fact_7954_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_7955_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
        = ( minus_minus_nat @ M @ N ) ) ) ).

% Suc_diff_Suc
thf(fact_7956_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_7957_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_7958_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_7959_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_mult_less_cancel1
thf(fact_7960_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_7961_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_7962_mult__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ M ) @ N )
      = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).

% mult_Suc
thf(fact_7963_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_mult_le_cancel1
thf(fact_7964_pochhammer__fact,axiom,
    ( semiri773545260158071498ct_rat
    = ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).

% pochhammer_fact
thf(fact_7965_pochhammer__fact,axiom,
    ( semiri1406184849735516958ct_int
    = ( comm_s4660882817536571857er_int @ one_one_int ) ) ).

% pochhammer_fact
thf(fact_7966_pochhammer__fact,axiom,
    ( semiri1408675320244567234ct_nat
    = ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).

% pochhammer_fact
thf(fact_7967_pochhammer__fact,axiom,
    ( semiri2265585572941072030t_real
    = ( comm_s7457072308508201937r_real @ one_one_real ) ) ).

% pochhammer_fact
thf(fact_7968_pochhammer__fact,axiom,
    ( semiri5044797733671781792omplex
    = ( comm_s2602460028002588243omplex @ one_one_complex ) ) ).

% pochhammer_fact
thf(fact_7969_mod__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
          = N )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
          = zero_zero_nat ) )
      & ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
         != N )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
          = ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).

% mod_Suc
thf(fact_7970_mod__induct,axiom,
    ! [P: nat > $o,N: nat,P4: nat,M: nat] :
      ( ( P @ N )
     => ( ( ord_less_nat @ N @ P4 )
       => ( ( ord_less_nat @ M @ P4 )
         => ( ! [N2: nat] :
                ( ( ord_less_nat @ N2 @ P4 )
               => ( ( P @ N2 )
                 => ( P @ ( modulo_modulo_nat @ ( suc @ N2 ) @ P4 ) ) ) )
           => ( P @ M ) ) ) ) ) ).

% mod_induct
thf(fact_7971_int__cases,axiom,
    ! [Z: int] :
      ( ! [N2: nat] :
          ( Z
         != ( semiri1314217659103216013at_int @ N2 ) )
     => ~ ! [N2: nat] :
            ( Z
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).

% int_cases
thf(fact_7972_int__of__nat__induct,axiom,
    ! [P: int > $o,Z: int] :
      ( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
     => ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
       => ( P @ Z ) ) ) ).

% int_of_nat_induct
thf(fact_7973_dvd__fact,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).

% dvd_fact
thf(fact_7974_signed__take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
     => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).

% signed_take_bit_int_less_eq
thf(fact_7975_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ) ).

% fact_less_mono
thf(fact_7976_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).

% fact_less_mono
thf(fact_7977_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).

% fact_less_mono
thf(fact_7978_fact__less__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ).

% fact_less_mono
thf(fact_7979_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) )
        = zero_zero_int ) ) ).

% fact_mod
thf(fact_7980_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) )
        = zero_z3403309356797280102nteger ) ) ).

% fact_mod
thf(fact_7981_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) )
        = zero_zero_nat ) ) ).

% fact_mod
thf(fact_7982_fact__le__power,axiom,
    ! [N: nat] : ( ord_le3102999989581377725nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri4939895301339042750nteger @ ( power_power_nat @ N @ N ) ) ) ).

% fact_le_power
thf(fact_7983_fact__le__power,axiom,
    ! [N: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N @ N ) ) ) ).

% fact_le_power
thf(fact_7984_fact__le__power,axiom,
    ! [N: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N @ N ) ) ) ).

% fact_le_power
thf(fact_7985_fact__le__power,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N @ N ) ) ) ).

% fact_le_power
thf(fact_7986_fact__le__power,axiom,
    ! [N: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N @ N ) ) ) ).

% fact_le_power
thf(fact_7987_power__inject__base,axiom,
    ! [A2: real,N: nat,B2: real] :
      ( ( ( power_power_real @ A2 @ ( suc @ N ) )
        = ( power_power_real @ B2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
         => ( A2 = B2 ) ) ) ) ).

% power_inject_base
thf(fact_7988_power__inject__base,axiom,
    ! [A2: code_integer,N: nat,B2: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A2 @ ( suc @ N ) )
        = ( power_8256067586552552935nteger @ B2 @ ( suc @ N ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
         => ( A2 = B2 ) ) ) ) ).

% power_inject_base
thf(fact_7989_power__inject__base,axiom,
    ! [A2: rat,N: nat,B2: rat] :
      ( ( ( power_power_rat @ A2 @ ( suc @ N ) )
        = ( power_power_rat @ B2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
         => ( A2 = B2 ) ) ) ) ).

% power_inject_base
thf(fact_7990_power__inject__base,axiom,
    ! [A2: nat,N: nat,B2: nat] :
      ( ( ( power_power_nat @ A2 @ ( suc @ N ) )
        = ( power_power_nat @ B2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
         => ( A2 = B2 ) ) ) ) ).

% power_inject_base
thf(fact_7991_power__inject__base,axiom,
    ! [A2: int,N: nat,B2: int] :
      ( ( ( power_power_int @ A2 @ ( suc @ N ) )
        = ( power_power_int @ B2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
         => ( A2 = B2 ) ) ) ) ).

% power_inject_base
thf(fact_7992_power__le__imp__le__base,axiom,
    ! [A2: real,N: nat,B2: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A2 @ ( suc @ N ) ) @ ( power_power_real @ B2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_eq_real @ A2 @ B2 ) ) ) ).

% power_le_imp_le_base
thf(fact_7993_power__le__imp__le__base,axiom,
    ! [A2: code_integer,N: nat,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ ( suc @ N ) ) @ ( power_8256067586552552935nteger @ B2 @ ( suc @ N ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ) ).

% power_le_imp_le_base
thf(fact_7994_power__le__imp__le__base,axiom,
    ! [A2: rat,N: nat,B2: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ ( suc @ N ) ) @ ( power_power_rat @ B2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).

% power_le_imp_le_base
thf(fact_7995_power__le__imp__le__base,axiom,
    ! [A2: nat,N: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ ( suc @ N ) ) @ ( power_power_nat @ B2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).

% power_le_imp_le_base
thf(fact_7996_power__le__imp__le__base,axiom,
    ! [A2: int,N: nat,B2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A2 @ ( suc @ N ) ) @ ( power_power_int @ B2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ A2 @ B2 ) ) ) ).

% power_le_imp_le_base
thf(fact_7997_power__gt1,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
     => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A2 @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_7998_power__gt1,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ord_less_real @ one_one_real @ ( power_power_real @ A2 @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_7999_power__gt1,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A2 @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_8000_power__gt1,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A2 @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_8001_power__gt1,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ord_less_int @ one_one_int @ ( power_power_int @ A2 @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_8002_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_8003_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N )
            & ! [I2: nat] :
                ( ( ord_less_eq_nat @ I2 @ K2 )
               => ~ ( P @ I2 ) )
            & ( P @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_8004_nat__induct__non__zero,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( P @ one_one_nat )
       => ( ! [N2: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_non_zero
thf(fact_8005_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).

% diff_Suc_less
thf(fact_8006_one__less__mult,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).

% one_less_mult
thf(fact_8007_n__less__m__mult__n,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).

% n_less_m_mult_n
thf(fact_8008_n__less__n__mult__m,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).

% n_less_n_mult_m
thf(fact_8009_power__gt__expt,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).

% power_gt_expt
thf(fact_8010_nat__one__le__power,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
     => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).

% nat_one_le_power
thf(fact_8011_realpow__pos__nth2,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ? [R2: real] :
          ( ( ord_less_real @ zero_zero_real @ R2 )
          & ( ( power_power_real @ R2 @ ( suc @ N ) )
            = A2 ) ) ) ).

% realpow_pos_nth2
thf(fact_8012_unat__eq__1,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( ( semiri7341220984566936280m1_nat @ X )
        = ( suc @ zero_zero_nat ) )
      = ( X = one_on7727431528512463931l_num1 ) ) ).

% unat_eq_1
thf(fact_8013_int__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).

% int_Suc
thf(fact_8014_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_8015_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W2: int,Z4: int] :
        ? [N4: nat] :
          ( Z4
          = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_8016_Abs__fnat__hom__1,axiom,
    ( one_on7727431528512463931l_num1
    = ( semiri8819519690708144855l_num1 @ ( suc @ zero_zero_nat ) ) ) ).

% Abs_fnat_hom_1
thf(fact_8017_powser__split__head_I3_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) )
     => ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ ( suc @ N4 ) ) @ ( power_power_complex @ Z @ N4 ) ) ) ) ).

% powser_split_head(3)
thf(fact_8018_powser__split__head_I3_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) )
     => ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ ( suc @ N4 ) ) @ ( power_power_real @ Z @ N4 ) ) ) ) ).

% powser_split_head(3)
thf(fact_8019_summable__powser__split__head,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ ( suc @ N4 ) ) @ ( power_power_complex @ Z @ N4 ) ) )
      = ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) ) ) ).

% summable_powser_split_head
thf(fact_8020_summable__powser__split__head,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ ( suc @ N4 ) ) @ ( power_power_real @ Z @ N4 ) ) )
      = ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) ) ) ).

% summable_powser_split_head
thf(fact_8021_signed__take__bit__Suc,axiom,
    ! [N: nat,A2: word_N3645301735248828278l_num1] :
      ( ( bit_ri1375673916561920181l_num1 @ ( suc @ N ) @ A2 )
      = ( plus_p361126936061061375l_num1 @ ( modulo1504961113040953224l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( bit_ri1375673916561920181l_num1 @ N @ ( divide1791077408188789448l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_8022_signed__take__bit__Suc,axiom,
    ! [N: nat,A2: code_integer] :
      ( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A2 )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_8023_signed__take__bit__Suc,axiom,
    ! [N: nat,A2: uint32] :
      ( ( bit_ri6224792872505173163uint32 @ ( suc @ N ) @ A2 )
      = ( plus_plus_uint32 @ ( modulo_modulo_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_ri6224792872505173163uint32 @ N @ ( divide_divide_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_8024_signed__take__bit__Suc,axiom,
    ! [N: nat,A2: int] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A2 )
      = ( plus_plus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_8025_fact__div__fact__le__pow,axiom,
    ! [R3: nat,N: nat] :
      ( ( ord_less_eq_nat @ R3 @ N )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R3 ) ) ) @ ( power_power_nat @ N @ R3 ) ) ) ).

% fact_div_fact_le_pow
thf(fact_8026_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).

% signed_take_bit_int_greater_eq
thf(fact_8027_pi__less__4,axiom,
    ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).

% pi_less_4
thf(fact_8028_pi__ge__two,axiom,
    ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).

% pi_ge_two
thf(fact_8029_power__Suc__le__self,axiom,
    ! [A2: real,N: 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 @ N ) ) @ A2 ) ) ) ).

% power_Suc_le_self
thf(fact_8030_power__Suc__le__self,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ ( suc @ N ) ) @ A2 ) ) ) ).

% power_Suc_le_self
thf(fact_8031_power__Suc__le__self,axiom,
    ! [A2: rat,N: 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 @ N ) ) @ A2 ) ) ) ).

% power_Suc_le_self
thf(fact_8032_power__Suc__le__self,axiom,
    ! [A2: nat,N: 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 @ N ) ) @ A2 ) ) ) ).

% power_Suc_le_self
thf(fact_8033_power__Suc__le__self,axiom,
    ! [A2: int,N: 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 @ N ) ) @ A2 ) ) ) ).

% power_Suc_le_self
thf(fact_8034_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_8035_power__Suc__less__one,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer )
       => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ ( suc @ N ) ) @ one_one_Code_integer ) ) ) ).

% power_Suc_less_one
thf(fact_8036_power__Suc__less__one,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_real @ A2 @ one_one_real )
       => ( ord_less_real @ ( power_power_real @ A2 @ ( suc @ N ) ) @ one_one_real ) ) ) ).

% power_Suc_less_one
thf(fact_8037_power__Suc__less__one,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ A2 @ one_one_rat )
       => ( ord_less_rat @ ( power_power_rat @ A2 @ ( suc @ N ) ) @ one_one_rat ) ) ) ).

% power_Suc_less_one
thf(fact_8038_power__Suc__less__one,axiom,
    ! [A2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ A2 @ one_one_nat )
       => ( ord_less_nat @ ( power_power_nat @ A2 @ ( suc @ N ) ) @ one_one_nat ) ) ) ).

% power_Suc_less_one
thf(fact_8039_power__Suc__less__one,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ A2 @ one_one_int )
       => ( ord_less_int @ ( power_power_int @ A2 @ ( suc @ N ) ) @ one_one_int ) ) ) ).

% power_Suc_less_one
thf(fact_8040_numeral__2__eq__2,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ one ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% numeral_2_eq_2
thf(fact_8041_double__not__eq__Suc__double,axiom,
    ! [M: nat,N: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
     != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% double_not_eq_Suc_double
thf(fact_8042_Suc__double__not__eq__double,axiom,
    ! [M: nat,N: nat] :
      ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
     != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% Suc_double_not_eq_double
thf(fact_8043_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_8044_Suc__pred_H,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( N
        = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_8045_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_8046_div__geq,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( ( divide_divide_nat @ M @ N )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).

% div_geq
thf(fact_8047_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M3: nat,N4: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M3 @ N4 )
            | ( N4 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M3 @ N4 ) @ N4 ) ) ) ) ) ).

% div_if
thf(fact_8048_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M3: nat,N4: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N4 ) ) ) ) ) ).

% add_eq_if
thf(fact_8049_Suc__n__minus__m__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_nat @ one_one_nat @ M )
       => ( ( suc @ ( minus_minus_nat @ N @ M ) )
          = ( minus_minus_nat @ N @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ).

% Suc_n_minus_m_eq
thf(fact_8050_div__nat__eqI,axiom,
    ! [N: nat,Q3: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M )
     => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
       => ( ( divide_divide_nat @ M @ N )
          = Q3 ) ) ) ).

% div_nat_eqI
thf(fact_8051_pochhammer__rec,axiom,
    ! [A2: word_N3645301735248828278l_num1,N: nat] :
      ( ( comm_s6431939913906641691l_num1 @ A2 @ ( suc @ N ) )
      = ( times_7065122842183080059l_num1 @ A2 @ ( comm_s6431939913906641691l_num1 @ ( plus_p361126936061061375l_num1 @ A2 @ one_on7727431528512463931l_num1 ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_8052_pochhammer__rec,axiom,
    ! [A2: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ A2 @ ( suc @ N ) )
      = ( times_times_real @ A2 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A2 @ one_one_real ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_8053_pochhammer__rec,axiom,
    ! [A2: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ A2 @ ( suc @ N ) )
      = ( times_times_rat @ A2 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_8054_pochhammer__rec,axiom,
    ! [A2: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ A2 @ ( suc @ N ) )
      = ( times_times_nat @ A2 @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_8055_pochhammer__rec,axiom,
    ! [A2: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ A2 @ ( suc @ N ) )
      = ( times_times_int @ A2 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A2 @ one_one_int ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_8056_pochhammer__rec_H,axiom,
    ! [Z: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) )
      = ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) ) ).

% pochhammer_rec'
thf(fact_8057_pochhammer__rec_H,axiom,
    ! [Z: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N ) )
      = ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z @ N ) ) ) ).

% pochhammer_rec'
thf(fact_8058_pochhammer__rec_H,axiom,
    ! [Z: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) )
      = ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) ) ).

% pochhammer_rec'
thf(fact_8059_pochhammer__rec_H,axiom,
    ! [Z: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N ) )
      = ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N ) ) ) ).

% pochhammer_rec'
thf(fact_8060_pochhammer__rec_H,axiom,
    ! [Z: code_integer,N: nat] :
      ( ( comm_s8582702949713902594nteger @ Z @ ( suc @ N ) )
      = ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ Z @ ( semiri4939895301339042750nteger @ N ) ) @ ( comm_s8582702949713902594nteger @ Z @ N ) ) ) ).

% pochhammer_rec'
thf(fact_8061_pochhammer__rec_H,axiom,
    ! [Z: complex,N: nat] :
      ( ( comm_s2602460028002588243omplex @ Z @ ( suc @ N ) )
      = ( times_times_complex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N ) ) @ ( comm_s2602460028002588243omplex @ Z @ N ) ) ) ).

% pochhammer_rec'
thf(fact_8062_pochhammer__Suc,axiom,
    ! [A2: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ A2 @ ( suc @ N ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A2 @ N ) @ ( plus_plus_rat @ A2 @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_8063_pochhammer__Suc,axiom,
    ! [A2: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ A2 @ ( suc @ N ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ A2 @ N ) @ ( plus_plus_int @ A2 @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_8064_pochhammer__Suc,axiom,
    ! [A2: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ A2 @ ( suc @ N ) )
      = ( times_times_real @ ( comm_s7457072308508201937r_real @ A2 @ N ) @ ( plus_plus_real @ A2 @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_8065_pochhammer__Suc,axiom,
    ! [A2: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ A2 @ ( suc @ N ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A2 @ N ) @ ( plus_plus_nat @ A2 @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_8066_pochhammer__Suc,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( comm_s8582702949713902594nteger @ A2 @ ( suc @ N ) )
      = ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ A2 @ N ) @ ( plus_p5714425477246183910nteger @ A2 @ ( semiri4939895301339042750nteger @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_8067_pochhammer__Suc,axiom,
    ! [A2: complex,N: nat] :
      ( ( comm_s2602460028002588243omplex @ A2 @ ( suc @ N ) )
      = ( times_times_complex @ ( comm_s2602460028002588243omplex @ A2 @ N ) @ ( plus_plus_complex @ A2 @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_8068_Suc__nat__number__of__add,axiom,
    ! [V: num,N: nat] :
      ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).

% Suc_nat_number_of_add
thf(fact_8069_not__zle__0__negative,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).

% not_zle_0_negative
thf(fact_8070_negD,axiom,
    ! [X: int] :
      ( ( ord_less_int @ X @ zero_zero_int )
     => ? [N2: nat] :
          ( X
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).

% negD
thf(fact_8071_negative__zless__0,axiom,
    ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_8072_binomial__absorption,axiom,
    ! [K: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
      = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).

% binomial_absorption
thf(fact_8073_unatSuc2,axiom,
    ! [N: word_N3645301735248828278l_num1] :
      ( ( ( plus_p361126936061061375l_num1 @ N @ one_on7727431528512463931l_num1 )
       != zero_z3563351764282998399l_num1 )
     => ( ( semiri7341220984566936280m1_nat @ ( plus_p361126936061061375l_num1 @ N @ one_on7727431528512463931l_num1 ) )
        = ( suc @ ( semiri7341220984566936280m1_nat @ N ) ) ) ) ).

% unatSuc2
thf(fact_8074_unatSuc,axiom,
    ! [N: word_N3645301735248828278l_num1] :
      ( ( ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ N )
       != zero_z3563351764282998399l_num1 )
     => ( ( semiri7341220984566936280m1_nat @ ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ N ) )
        = ( suc @ ( semiri7341220984566936280m1_nat @ N ) ) ) ) ).

% unatSuc
thf(fact_8075_Suc__unat__diff__1,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ one_on7727431528512463931l_num1 @ X )
     => ( ( suc @ ( semiri7341220984566936280m1_nat @ ( minus_4019991460397169231l_num1 @ X @ one_on7727431528512463931l_num1 ) ) )
        = ( semiri7341220984566936280m1_nat @ X ) ) ) ).

% Suc_unat_diff_1
thf(fact_8076_unat__Suc2,axiom,
    ! [N: word_N3645301735248828278l_num1] :
      ( ( N
       != ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
     => ( ( semiri7341220984566936280m1_nat @ ( plus_p361126936061061375l_num1 @ N @ one_on7727431528512463931l_num1 ) )
        = ( suc @ ( semiri7341220984566936280m1_nat @ N ) ) ) ) ).

% unat_Suc2
thf(fact_8077_suminf__split__head,axiom,
    ! [F: nat > complex] :
      ( ( summable_complex @ F )
     => ( ( suminf_complex
          @ ^ [N4: nat] : ( F @ ( suc @ N4 ) ) )
        = ( minus_minus_complex @ ( suminf_complex @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).

% suminf_split_head
thf(fact_8078_suminf__split__head,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N4: nat] : ( F @ ( suc @ N4 ) ) )
        = ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).

% suminf_split_head
thf(fact_8079_signed__take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% signed_take_bit_int_less_exp
thf(fact_8080_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_8081_even__signed__take__bit__iff,axiom,
    ! [M: nat,A2: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_ri6224792872505173163uint32 @ M @ A2 ) )
      = ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A2 ) ) ).

% even_signed_take_bit_iff
thf(fact_8082_even__signed__take__bit__iff,axiom,
    ! [M: nat,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ M @ A2 ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 ) ) ).

% even_signed_take_bit_iff
thf(fact_8083_even__signed__take__bit__iff,axiom,
    ! [M: nat,A2: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( bit_ri1375673916561920181l_num1 @ M @ A2 ) )
      = ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A2 ) ) ).

% even_signed_take_bit_iff
thf(fact_8084_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_8085_pi__half__neq__zero,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% pi_half_neq_zero
thf(fact_8086_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_8087_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_8088_nat__approx__posE,axiom,
    ! [E2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ E2 )
     => ~ ! [N2: nat] :
            ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) ) @ E2 ) ) ).

% nat_approx_posE
thf(fact_8089_nat__approx__posE,axiom,
    ! [E2: real] :
      ( ( ord_less_real @ zero_zero_real @ E2 )
     => ~ ! [N2: nat] :
            ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ E2 ) ) ).

% nat_approx_posE
thf(fact_8090_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ( N = zero_zero_nat )
        | ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases_iff
thf(fact_8091_less__2__cases,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
     => ( ( N = zero_zero_nat )
        | ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases
thf(fact_8092_le__div__geq,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( divide_divide_nat @ M @ N )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).

% le_div_geq
thf(fact_8093_split__div_H,axiom,
    ! [P: nat > $o,M: nat,N: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N ) )
      = ( ( ( N = zero_zero_nat )
          & ( P @ zero_zero_nat ) )
        | ? [Q6: nat] :
            ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q6 ) @ M )
            & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q6 ) ) )
            & ( P @ Q6 ) ) ) ) ).

% split_div'
thf(fact_8094_Suc__times__mod__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
        = one_one_nat ) ) ).

% Suc_times_mod_eq
thf(fact_8095_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).

% signed_take_bit_int_less_self_iff
thf(fact_8096_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% signed_take_bit_int_greater_eq_self_iff
thf(fact_8097_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).

% signed_take_bit_int_less_eq_self_iff
thf(fact_8098_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ).

% signed_take_bit_int_greater_eq_minus_exp
thf(fact_8099_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
      = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% signed_take_bit_int_greater_self_iff
thf(fact_8100_fact__code,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [N4: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_8101_fact__code,axiom,
    ( semiri3624122377584611663nteger
    = ( ^ [N4: nat] : ( semiri4939895301339042750nteger @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_8102_fact__code,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [N4: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_8103_fact__code,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [N4: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_8104_fact__code,axiom,
    ( semiri5044797733671781792omplex
    = ( ^ [N4: nat] : ( semiri8010041392384452111omplex @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_8105_fact__num__eq__if,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M3 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8106_fact__num__eq__if,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8107_fact__num__eq__if,axiom,
    ( semiri3624122377584611663nteger
    = ( ^ [M3: nat] : ( if_Code_integer @ ( M3 = zero_zero_nat ) @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M3 ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8108_fact__num__eq__if,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8109_fact__num__eq__if,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [M3: nat] : ( if_real @ ( M3 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8110_fact__num__eq__if,axiom,
    ( semiri5044797733671781792omplex
    = ( ^ [M3: nat] : ( if_complex @ ( M3 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M3 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8111_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri773545260158071498ct_rat @ N )
        = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_8112_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri1406184849735516958ct_int @ N )
        = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_8113_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri3624122377584611663nteger @ N )
        = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_8114_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri1408675320244567234ct_nat @ N )
        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_8115_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri2265585572941072030t_real @ N )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_8116_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri5044797733671781792omplex @ N )
        = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_8117_pochhammer__same,axiom,
    ! [N: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ N )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% pochhammer_same
thf(fact_8118_pochhammer__same,axiom,
    ! [N: nat] :
      ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ N )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).

% pochhammer_same
thf(fact_8119_pochhammer__same,axiom,
    ! [N: nat] :
      ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ N )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).

% pochhammer_same
thf(fact_8120_pochhammer__same,axiom,
    ! [N: nat] :
      ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ N )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).

% pochhammer_same
thf(fact_8121_pochhammer__same,axiom,
    ! [N: nat] :
      ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ N )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).

% pochhammer_same
thf(fact_8122_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_8123_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_8124_binomial__fact,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
        = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).

% binomial_fact
thf(fact_8125_binomial__fact,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
        = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).

% binomial_fact
thf(fact_8126_binomial__fact,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
        = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).

% binomial_fact
thf(fact_8127_fact__binomial,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) )
        = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).

% fact_binomial
thf(fact_8128_fact__binomial,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) )
        = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).

% fact_binomial
thf(fact_8129_fact__binomial,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) ) )
        = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).

% fact_binomial
thf(fact_8130_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_8131_power__odd__eq,axiom,
    ! [A2: complex,N: nat] :
      ( ( power_power_complex @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_complex @ A2 @ ( power_power_complex @ ( power_power_complex @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_8132_power__odd__eq,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ ( power_8256067586552552935nteger @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_8133_power__odd__eq,axiom,
    ! [A2: real,N: nat] :
      ( ( power_power_real @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_real @ A2 @ ( power_power_real @ ( power_power_real @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_8134_power__odd__eq,axiom,
    ! [A2: rat,N: nat] :
      ( ( power_power_rat @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_rat @ A2 @ ( power_power_rat @ ( power_power_rat @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_8135_power__odd__eq,axiom,
    ! [A2: nat,N: nat] :
      ( ( power_power_nat @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_nat @ A2 @ ( power_power_nat @ ( power_power_nat @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_8136_power__odd__eq,axiom,
    ! [A2: int,N: nat] :
      ( ( power_power_int @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_int @ A2 @ ( power_power_int @ ( power_power_int @ A2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_8137_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% Suc_n_div_2_gt_zero
thf(fact_8138_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% div_2_gt_zero
thf(fact_8139_nat__bit__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( P @ N2 )
           => ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
       => ( ! [N2: nat] :
              ( ( P @ N2 )
             => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_bit_induct
thf(fact_8140_arctan__ubound,axiom,
    ! [Y: real] : ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arctan_ubound
thf(fact_8141_arctan__one,axiom,
    ( ( arctan @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% arctan_one
thf(fact_8142_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).

% binomial_less_binomial_Suc
thf(fact_8143_central__binomial__odd,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( binomial @ N @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% central_binomial_odd
thf(fact_8144_binomial__addition__formula,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( binomial @ N @ ( suc @ K ) )
        = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).

% binomial_addition_formula
thf(fact_8145_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ( ord_less_eq_nat @ M @ I3 )
            & ( ord_less_nat @ I3 @ N ) )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( ord_less_eq_int @ ( F @ M ) @ K )
         => ( ( ord_less_eq_int @ K @ ( F @ N ) )
           => ? [I3: nat] :
                ( ( ord_less_eq_nat @ M @ I3 )
                & ( ord_less_eq_nat @ I3 @ N )
                & ( ( F @ I3 )
                  = K ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_8146_add__0__iff,axiom,
    ! [B2: real,A2: real] :
      ( ( B2
        = ( plus_plus_real @ B2 @ A2 ) )
      = ( A2 = zero_zero_real ) ) ).

% add_0_iff
thf(fact_8147_add__0__iff,axiom,
    ! [B2: rat,A2: rat] :
      ( ( B2
        = ( plus_plus_rat @ B2 @ A2 ) )
      = ( A2 = zero_zero_rat ) ) ).

% add_0_iff
thf(fact_8148_add__0__iff,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2
        = ( plus_plus_nat @ B2 @ A2 ) )
      = ( A2 = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_8149_add__0__iff,axiom,
    ! [B2: int,A2: int] :
      ( ( B2
        = ( plus_plus_int @ B2 @ A2 ) )
      = ( A2 = zero_zero_int ) ) ).

% add_0_iff
thf(fact_8150_signed__take__bit__int__eq__self,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_ri631733984087533419it_int @ N @ K )
          = K ) ) ) ).

% signed_take_bit_int_eq_self
thf(fact_8151_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_ri631733984087533419it_int @ N @ K )
        = K )
      = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% signed_take_bit_int_eq_self_iff
thf(fact_8152_powser__split__head_I1_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) )
     => ( ( suminf_complex
          @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) )
        = ( plus_plus_complex @ ( F @ zero_zero_nat )
          @ ( times_times_complex
            @ ( suminf_complex
              @ ^ [N4: nat] : ( times_times_complex @ ( F @ ( suc @ N4 ) ) @ ( power_power_complex @ Z @ N4 ) ) )
            @ Z ) ) ) ) ).

% powser_split_head(1)
thf(fact_8153_powser__split__head_I1_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) )
     => ( ( suminf_real
          @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) )
        = ( plus_plus_real @ ( F @ zero_zero_nat )
          @ ( times_times_real
            @ ( suminf_real
              @ ^ [N4: nat] : ( times_times_real @ ( F @ ( suc @ N4 ) ) @ ( power_power_real @ Z @ N4 ) ) )
            @ Z ) ) ) ) ).

% powser_split_head(1)
thf(fact_8154_powser__split__head_I2_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) )
     => ( ( times_times_complex
          @ ( suminf_complex
            @ ^ [N4: nat] : ( times_times_complex @ ( F @ ( suc @ N4 ) ) @ ( power_power_complex @ Z @ N4 ) ) )
          @ Z )
        = ( minus_minus_complex
          @ ( suminf_complex
            @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_8155_powser__split__head_I2_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) )
     => ( ( times_times_real
          @ ( suminf_real
            @ ^ [N4: nat] : ( times_times_real @ ( F @ ( suc @ N4 ) ) @ ( power_power_real @ Z @ N4 ) ) )
          @ Z )
        = ( minus_minus_real
          @ ( suminf_real
            @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_8156_odd__0__le__power__imp__0__le,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_8157_odd__0__le__power__imp__0__le,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_8158_odd__0__le__power__imp__0__le,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_8159_odd__0__le__power__imp__0__le,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_8160_odd__power__less__zero,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_z3403309356797280102nteger ) ) ).

% odd_power_less_zero
thf(fact_8161_odd__power__less__zero,axiom,
    ! [A2: real,N: nat] :
      ( ( ord_less_real @ A2 @ zero_zero_real )
     => ( ord_less_real @ ( power_power_real @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).

% odd_power_less_zero
thf(fact_8162_odd__power__less__zero,axiom,
    ! [A2: rat,N: nat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ord_less_rat @ ( power_power_rat @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).

% odd_power_less_zero
thf(fact_8163_odd__power__less__zero,axiom,
    ! [A2: int,N: nat] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ord_less_int @ ( power_power_int @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).

% odd_power_less_zero
thf(fact_8164_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_8165_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_2184487114949457152l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ).

% power_minus1_odd
thf(fact_8166_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% power_minus1_odd
thf(fact_8167_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power_minus1_odd
thf(fact_8168_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus_uminus_uint32 @ one_one_uint32 ) ) ).

% power_minus1_odd
thf(fact_8169_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus_uminus_real @ one_one_real ) ) ).

% power_minus1_odd
thf(fact_8170_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus_uminus_rat @ one_one_rat ) ) ).

% power_minus1_odd
thf(fact_8171_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% power_minus1_odd
thf(fact_8172_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_8173_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_8174_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_8175_nat__ivt__aux,axiom,
    ! [N: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N ) )
         => ? [I3: nat] :
              ( ( ord_less_eq_nat @ I3 @ N )
              & ( ( F @ I3 )
                = K ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_8176_nat__div__eq__Suc__0__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( ( divide_divide_nat @ N @ M )
        = ( suc @ zero_zero_nat ) )
      = ( ( ord_less_eq_nat @ M @ N )
        & ( ord_less_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% nat_div_eq_Suc_0_iff
thf(fact_8177_unset__bit__Suc,axiom,
    ! [N: nat,A2: word_N3645301735248828278l_num1] :
      ( ( bit_se5331074070815623765l_num1 @ ( suc @ N ) @ A2 )
      = ( plus_p361126936061061375l_num1 @ ( modulo1504961113040953224l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( bit_se5331074070815623765l_num1 @ N @ ( divide1791077408188789448l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_8178_unset__bit__Suc,axiom,
    ! [N: nat,A2: code_integer] :
      ( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A2 )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_8179_unset__bit__Suc,axiom,
    ! [N: nat,A2: uint32] :
      ( ( bit_se4315839071623982667uint32 @ ( suc @ N ) @ A2 )
      = ( plus_plus_uint32 @ ( modulo_modulo_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_se4315839071623982667uint32 @ N @ ( divide_divide_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_8180_unset__bit__Suc,axiom,
    ! [N: nat,A2: int] :
      ( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A2 )
      = ( plus_plus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_8181_unset__bit__Suc,axiom,
    ! [N: nat,A2: nat] :
      ( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A2 )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_8182_set__bit__Suc,axiom,
    ! [N: nat,A2: word_N3645301735248828278l_num1] :
      ( ( bit_se4894374433684937756l_num1 @ ( suc @ N ) @ A2 )
      = ( plus_p361126936061061375l_num1 @ ( modulo1504961113040953224l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( bit_se4894374433684937756l_num1 @ N @ ( divide1791077408188789448l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_8183_set__bit__Suc,axiom,
    ! [N: nat,A2: code_integer] :
      ( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A2 )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_8184_set__bit__Suc,axiom,
    ! [N: nat,A2: uint32] :
      ( ( bit_se6647067497041451410uint32 @ ( suc @ N ) @ A2 )
      = ( plus_plus_uint32 @ ( modulo_modulo_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_se6647067497041451410uint32 @ N @ ( divide_divide_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_8185_set__bit__Suc,axiom,
    ! [N: nat,A2: int] :
      ( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A2 )
      = ( plus_plus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_8186_set__bit__Suc,axiom,
    ! [N: nat,A2: nat] :
      ( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A2 )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_8187_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( suc @ zero_zero_nat ) )
     => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_mod_4_div_2
thf(fact_8188_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_8189_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_8190_fact__double,axiom,
    ! [N: nat] :
      ( ( semiri773545260158071498ct_rat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% fact_double
thf(fact_8191_fact__double,axiom,
    ! [N: nat] :
      ( ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_real @ ( times_times_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s7457072308508201937r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).

% fact_double
thf(fact_8192_fact__double,axiom,
    ! [N: nat] :
      ( ( semiri5044797733671781792omplex @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s2602460028002588243omplex @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).

% fact_double
thf(fact_8193_lemma__termdiff3,axiom,
    ! [H2: real,Z: real,K4: real,N: nat] :
      ( ( H2 != zero_zero_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K4 )
       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H2 ) ) @ K4 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K4 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_8194_lemma__termdiff3,axiom,
    ! [H2: complex,Z: complex,K4: real,N: nat] :
      ( ( H2 != zero_zero_complex )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K4 )
       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H2 ) ) @ K4 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K4 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_8195_sin__cos__npi,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).

% sin_cos_npi
thf(fact_8196_sin__zero,axiom,
    ( ( sin_real @ zero_zero_real )
    = zero_zero_real ) ).

% sin_zero
thf(fact_8197_norm__eq__zero,axiom,
    ! [X: real] :
      ( ( ( real_V7735802525324610683m_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% norm_eq_zero
thf(fact_8198_norm__eq__zero,axiom,
    ! [X: complex] :
      ( ( ( real_V1022390504157884413omplex @ X )
        = zero_zero_real )
      = ( X = zero_zero_complex ) ) ).

% norm_eq_zero
thf(fact_8199_norm__zero,axiom,
    ( ( real_V7735802525324610683m_real @ zero_zero_real )
    = zero_zero_real ) ).

% norm_zero
thf(fact_8200_norm__zero,axiom,
    ( ( real_V1022390504157884413omplex @ zero_zero_complex )
    = zero_zero_real ) ).

% norm_zero
thf(fact_8201_norm__one,axiom,
    ( ( real_V7735802525324610683m_real @ one_one_real )
    = one_one_real ) ).

% norm_one
thf(fact_8202_norm__one,axiom,
    ( ( real_V1022390504157884413omplex @ one_one_complex )
    = one_one_real ) ).

% norm_one
thf(fact_8203_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_8204_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_8205_norm__of__nat,axiom,
    ! [N: nat] :
      ( ( real_V7735802525324610683m_real @ ( semiri5074537144036343181t_real @ N ) )
      = ( semiri5074537144036343181t_real @ N ) ) ).

% norm_of_nat
thf(fact_8206_norm__of__nat,axiom,
    ! [N: nat] :
      ( ( real_V1022390504157884413omplex @ ( semiri8010041392384452111omplex @ N ) )
      = ( semiri5074537144036343181t_real @ N ) ) ).

% norm_of_nat
thf(fact_8207_sin__pi,axiom,
    ( ( sin_real @ pi )
    = zero_zero_real ) ).

% sin_pi
thf(fact_8208_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_8209_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_8210_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_8211_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_8212_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_8213_norm__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_neg_numeral
thf(fact_8214_sin__of__real__pi,axiom,
    ( ( sin_real @ ( real_V1803761363581548252l_real @ pi ) )
    = zero_zero_real ) ).

% sin_of_real_pi
thf(fact_8215_sin__of__real__pi,axiom,
    ( ( sin_complex @ ( real_V4546457046886955230omplex @ pi ) )
    = zero_zero_complex ) ).

% sin_of_real_pi
thf(fact_8216_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_8217_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_8218_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_8219_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_8220_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_8221_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_8222_sin__npi,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% sin_npi
thf(fact_8223_sin__npi2,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
      = zero_zero_real ) ).

% sin_npi2
thf(fact_8224_sin__npi__int,axiom,
    ! [N: int] :
      ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
      = zero_zero_real ) ).

% sin_npi_int
thf(fact_8225_summable__geometric__iff,axiom,
    ! [C: real] :
      ( ( summable_real @ ( power_power_real @ C ) )
      = ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real ) ) ).

% summable_geometric_iff
thf(fact_8226_summable__geometric__iff,axiom,
    ! [C: complex] :
      ( ( summable_complex @ ( power_power_complex @ C ) )
      = ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real ) ) ).

% summable_geometric_iff
thf(fact_8227_sin__two__pi,axiom,
    ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = zero_zero_real ) ).

% sin_two_pi
thf(fact_8228_sin__pi__half,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = one_one_real ) ).

% sin_pi_half
thf(fact_8229_norm__of__real__add1,axiom,
    ! [X: real] :
      ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ one_one_real ) )
      = ( abs_abs_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).

% norm_of_real_add1
thf(fact_8230_norm__of__real__add1,axiom,
    ! [X: real] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ one_one_complex ) )
      = ( abs_abs_real @ ( plus_plus_real @ X @ one_one_real ) ) ) ).

% norm_of_real_add1
thf(fact_8231_norm__of__real__addn,axiom,
    ! [X: real,B2: num] :
      ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X ) @ ( numeral_numeral_real @ B2 ) ) )
      = ( abs_abs_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ B2 ) ) ) ) ).

% norm_of_real_addn
thf(fact_8232_norm__of__real__addn,axiom,
    ! [X: real,B2: num] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X ) @ ( numera6690914467698888265omplex @ B2 ) ) )
      = ( abs_abs_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ B2 ) ) ) ) ).

% norm_of_real_addn
thf(fact_8233_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_8234_sin__of__real__pi__half,axiom,
    ( ( sin_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = one_one_real ) ).

% sin_of_real_pi_half
thf(fact_8235_sin__of__real__pi__half,axiom,
    ( ( sin_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
    = one_one_complex ) ).

% sin_of_real_pi_half
thf(fact_8236_sin__2npi,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
      = zero_zero_real ) ).

% sin_2npi
thf(fact_8237_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_8238_sin__int__2pin,axiom,
    ! [N: int] :
      ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
      = zero_zero_real ) ).

% sin_int_2pin
thf(fact_8239_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_8240_sin__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( sin_real @ X ) @ one_one_real ) ).

% sin_le_one
thf(fact_8241_norm__not__less__zero,axiom,
    ! [X: complex] :
      ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real ) ).

% norm_not_less_zero
thf(fact_8242_norm__ge__zero,axiom,
    ! [X: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) ) ).

% norm_ge_zero
thf(fact_8243_norm__power,axiom,
    ! [X: real,N: nat] :
      ( ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) )
      = ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).

% norm_power
thf(fact_8244_norm__power,axiom,
    ! [X: complex,N: nat] :
      ( ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) )
      = ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).

% norm_power
thf(fact_8245_summable__norm__cancel,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( real_V7735802525324610683m_real @ ( F @ N4 ) ) )
     => ( summable_real @ F ) ) ).

% summable_norm_cancel
thf(fact_8246_summable__norm__cancel,axiom,
    ! [F: nat > complex] :
      ( ( summable_real
        @ ^ [N4: nat] : ( real_V1022390504157884413omplex @ ( F @ N4 ) ) )
     => ( summable_complex @ F ) ) ).

% summable_norm_cancel
thf(fact_8247_power__eq__imp__eq__norm,axiom,
    ! [W: real,N: nat,Z: real] :
      ( ( ( power_power_real @ W @ N )
        = ( power_power_real @ Z @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( real_V7735802525324610683m_real @ W )
          = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_8248_power__eq__imp__eq__norm,axiom,
    ! [W: complex,N: nat,Z: complex] :
      ( ( ( power_power_complex @ W @ N )
        = ( power_power_complex @ Z @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( real_V1022390504157884413omplex @ W )
          = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_8249_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_8250_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_8251_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_8252_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_8253_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_8254_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_8255_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_8256_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_8257_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_8258_norm__triangle__lt,axiom,
    ! [X: real,Y: real,E2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
     => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E2 ) ) ).

% norm_triangle_lt
thf(fact_8259_norm__triangle__lt,axiom,
    ! [X: complex,Y: complex,E2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
     => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E2 ) ) ).

% norm_triangle_lt
thf(fact_8260_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_8261_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_8262_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_8263_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_8264_norm__power__ineq,axiom,
    ! [X: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).

% norm_power_ineq
thf(fact_8265_norm__power__ineq,axiom,
    ! [X: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).

% norm_power_ineq
thf(fact_8266_summable__norm__comparison__test,axiom,
    ! [F: nat > complex,G: nat > real] :
      ( ? [N8: nat] :
        ! [N2: nat] :
          ( ( ord_less_eq_nat @ N8 @ N2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N4: nat] : ( real_V1022390504157884413omplex @ ( F @ N4 ) ) ) ) ) ).

% summable_norm_comparison_test
thf(fact_8267_power__eq__1__iff,axiom,
    ! [W: real,N: nat] :
      ( ( ( power_power_real @ W @ N )
        = one_one_real )
     => ( ( ( real_V7735802525324610683m_real @ W )
          = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_8268_power__eq__1__iff,axiom,
    ! [W: complex,N: nat] :
      ( ( ( power_power_complex @ W @ N )
        = one_one_complex )
     => ( ( ( real_V1022390504157884413omplex @ W )
          = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_8269_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_8270_norm__less__p1,axiom,
    ! [X: real] : ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ ( real_V7735802525324610683m_real @ X ) ) @ one_one_real ) ) ) ).

% norm_less_p1
thf(fact_8271_norm__less__p1,axiom,
    ! [X: complex] : ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ X ) ) @ one_one_complex ) ) ) ).

% norm_less_p1
thf(fact_8272_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_8273_sin__zero__iff__int2,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [I4: int] :
            ( X
            = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ pi ) ) ) ) ).

% sin_zero_iff_int2
thf(fact_8274_powser__inside,axiom,
    ! [F: nat > real,X: real,Z: real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ X @ N4 ) ) )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X ) )
       => ( summable_real
          @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) ) ) ) ).

% powser_inside
thf(fact_8275_powser__inside,axiom,
    ! [F: nat > complex,X: complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ X @ N4 ) ) )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X ) )
       => ( summable_complex
          @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) ) ) ) ).

% powser_inside
thf(fact_8276_complete__algebra__summable__geometric,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ one_one_real )
     => ( summable_real @ ( power_power_real @ X ) ) ) ).

% complete_algebra_summable_geometric
thf(fact_8277_complete__algebra__summable__geometric,axiom,
    ! [X: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ one_one_real )
     => ( summable_complex @ ( power_power_complex @ X ) ) ) ).

% complete_algebra_summable_geometric
thf(fact_8278_summable__geometric,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
     => ( summable_real @ ( power_power_real @ C ) ) ) ).

% summable_geometric
thf(fact_8279_summable__geometric,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( summable_complex @ ( power_power_complex @ C ) ) ) ).

% summable_geometric
thf(fact_8280_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_8281_summable__norm,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( real_V7735802525324610683m_real @ ( F @ N4 ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( suminf_real @ F ) )
        @ ( suminf_real
          @ ^ [N4: nat] : ( real_V7735802525324610683m_real @ ( F @ N4 ) ) ) ) ) ).

% summable_norm
thf(fact_8282_summable__norm,axiom,
    ! [F: nat > complex] :
      ( ( summable_real
        @ ^ [N4: nat] : ( real_V1022390504157884413omplex @ ( F @ N4 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( suminf_complex @ F ) )
        @ ( suminf_real
          @ ^ [N4: nat] : ( real_V1022390504157884413omplex @ ( F @ N4 ) ) ) ) ) ).

% summable_norm
thf(fact_8283_powser__insidea,axiom,
    ! [F: nat > real,X: real,Z: real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ X @ N4 ) ) )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X ) )
       => ( summable_real
          @ ^ [N4: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ Z @ N4 ) ) ) ) ) ) ).

% powser_insidea
thf(fact_8284_powser__insidea,axiom,
    ! [F: nat > complex,X: complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ X @ N4 ) ) )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X ) )
       => ( summable_real
          @ ^ [N4: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N4 ) @ ( power_power_complex @ Z @ N4 ) ) ) ) ) ) ).

% powser_insidea
thf(fact_8285_square__fact__le__2__fact,axiom,
    ! [N: nat] : ( ord_less_eq_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% square_fact_le_2_fact
thf(fact_8286_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_8287_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_8288_sin__pi__divide__n__ge__0,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% sin_pi_divide_n_ge_0
thf(fact_8289_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_8290_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_8291_suminf__exist__split,axiom,
    ! [R3: real,F: nat > real] :
      ( ( ord_less_real @ zero_zero_real @ R3 )
     => ( ( summable_real @ F )
       => ? [N9: nat] :
          ! [N10: nat] :
            ( ( ord_less_eq_nat @ N9 @ N10 )
           => ( ord_less_real
              @ ( real_V7735802525324610683m_real
                @ ( suminf_real
                  @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N10 ) ) ) )
              @ R3 ) ) ) ) ).

% suminf_exist_split
thf(fact_8292_suminf__exist__split,axiom,
    ! [R3: real,F: nat > complex] :
      ( ( ord_less_real @ zero_zero_real @ R3 )
     => ( ( summable_complex @ F )
       => ? [N9: nat] :
          ! [N10: nat] :
            ( ( ord_less_eq_nat @ N9 @ N10 )
           => ( ord_less_real
              @ ( real_V1022390504157884413omplex
                @ ( suminf_complex
                  @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N10 ) ) ) )
              @ R3 ) ) ) ) ).

% suminf_exist_split
thf(fact_8293_Abel__lemma,axiom,
    ! [R3: real,R0: real,A2: nat > complex,M7: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R3 )
     => ( ( ord_less_real @ R3 @ R0 )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A2 @ N2 ) ) @ ( power_power_real @ R0 @ N2 ) ) @ M7 )
         => ( summable_real
            @ ^ [N4: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A2 @ N4 ) ) @ ( power_power_real @ R3 @ N4 ) ) ) ) ) ) ).

% Abel_lemma
thf(fact_8294_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_8295_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_8296_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_8297_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_8298_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_8299_summable__ratio__test,axiom,
    ! [C: real,N3: nat,F: nat > real] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N2: nat] :
            ( ( ord_less_eq_nat @ N3 @ N2 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N2 ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N2 ) ) ) ) )
       => ( summable_real @ F ) ) ) ).

% summable_ratio_test
thf(fact_8300_summable__ratio__test,axiom,
    ! [C: real,N3: nat,F: nat > complex] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N2: nat] :
            ( ( ord_less_eq_nat @ N3 @ N2 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N2 ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) )
       => ( summable_complex @ F ) ) ) ).

% summable_ratio_test
thf(fact_8301_suminf__geometric,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
     => ( ( suminf_real @ ( power_power_real @ C ) )
        = ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).

% suminf_geometric
thf(fact_8302_suminf__geometric,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( ( suminf_complex @ ( power_power_complex @ C ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).

% suminf_geometric
thf(fact_8303_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_8304_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_8305_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_8306_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_8307_sin__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_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 )
              = Y )
            & ! [Y5: real] :
                ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
                  & ( ord_less_eq_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
                  & ( ( sin_real @ Y5 )
                    = Y ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% sin_total
thf(fact_8308_sin__pi__divide__n__gt__0,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% sin_pi_divide_n_gt_0
thf(fact_8309_sin__zero__iff__int,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [I4: int] :
            ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I4 )
            & ( X
              = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_iff_int
thf(fact_8310_sin__zero__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( sin_real @ X )
          = zero_zero_real )
       => ? [N2: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_lemma
thf(fact_8311_sin__zero__iff,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [N4: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N4: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
            & ( X
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% sin_zero_iff
thf(fact_8312_sin__coeff__def,axiom,
    ( sin_coeff
    = ( ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N4 ) ) ) ) ) ).

% sin_coeff_def
thf(fact_8313_cos__coeff__def,axiom,
    ( cos_coeff
    = ( ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N4 ) ) @ zero_zero_real ) ) ) ).

% cos_coeff_def
thf(fact_8314_flip__bit__Suc,axiom,
    ! [N: nat,A2: word_N3645301735248828278l_num1] :
      ( ( bit_se4491814353640558621l_num1 @ ( suc @ N ) @ A2 )
      = ( plus_p361126936061061375l_num1 @ ( modulo1504961113040953224l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( bit_se4491814353640558621l_num1 @ N @ ( divide1791077408188789448l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_8315_flip__bit__Suc,axiom,
    ! [N: nat,A2: code_integer] :
      ( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A2 )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_8316_flip__bit__Suc,axiom,
    ! [N: nat,A2: uint32] :
      ( ( bit_se7025624438249859091uint32 @ ( suc @ N ) @ A2 )
      = ( plus_plus_uint32 @ ( modulo_modulo_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_se7025624438249859091uint32 @ N @ ( divide_divide_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_8317_flip__bit__Suc,axiom,
    ! [N: nat,A2: int] :
      ( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A2 )
      = ( plus_plus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_8318_flip__bit__Suc,axiom,
    ! [N: nat,A2: nat] :
      ( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A2 )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_8319_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_minus_bit1
thf(fact_8320_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 ) ) )
         => ~ ! [N2: nat] :
                ( ( X
                  = ( suc @ ( suc @ N2 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                     => ( Y
                        = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( 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 @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                     => ( Y
                        = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( 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 @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.elims
thf(fact_8321_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( ( bit1 @ X32 )
        = ( bit1 @ Y32 ) )
      = ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
thf(fact_8322_semiring__norm_I90_J,axiom,
    ! [M: num,N: num] :
      ( ( ( bit1 @ M )
        = ( bit1 @ N ) )
      = ( M = N ) ) ).

% semiring_norm(90)
thf(fact_8323_semiring__norm_I88_J,axiom,
    ! [M: num,N: num] :
      ( ( bit0 @ M )
     != ( bit1 @ N ) ) ).

% semiring_norm(88)
thf(fact_8324_semiring__norm_I89_J,axiom,
    ! [M: num,N: num] :
      ( ( bit1 @ M )
     != ( bit0 @ N ) ) ).

% semiring_norm(89)
thf(fact_8325_semiring__norm_I84_J,axiom,
    ! [N: num] :
      ( one
     != ( bit1 @ N ) ) ).

% semiring_norm(84)
thf(fact_8326_semiring__norm_I86_J,axiom,
    ! [M: num] :
      ( ( bit1 @ M )
     != one ) ).

% semiring_norm(86)
thf(fact_8327_flip__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% flip_bit_nonnegative_int_iff
thf(fact_8328_flip__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% flip_bit_negative_int_iff
thf(fact_8329_semiring__norm_I73_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(73)
thf(fact_8330_semiring__norm_I80_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(80)
thf(fact_8331_semiring__norm_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).

% semiring_norm(9)
thf(fact_8332_semiring__norm_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).

% semiring_norm(7)
thf(fact_8333_semiring__norm_I15_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).

% semiring_norm(15)
thf(fact_8334_semiring__norm_I14_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).

% semiring_norm(14)
thf(fact_8335_semiring__norm_I72_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(72)
thf(fact_8336_semiring__norm_I81_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(81)
thf(fact_8337_semiring__norm_I70_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).

% semiring_norm(70)
thf(fact_8338_semiring__norm_I77_J,axiom,
    ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).

% semiring_norm(77)
thf(fact_8339_sin__coeff__0,axiom,
    ( ( sin_coeff @ zero_zero_nat )
    = zero_zero_real ) ).

% sin_coeff_0
thf(fact_8340_cos__coeff__0,axiom,
    ( ( cos_coeff @ zero_zero_nat )
    = one_one_real ) ).

% cos_coeff_0
thf(fact_8341_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu8115118780965096967l_num1 @ ( numera7442385471795722001l_num1 @ K ) )
      = ( numera7442385471795722001l_num1 @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_8342_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) )
      = ( numeral_numeral_real @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_8343_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) )
      = ( numeral_numeral_rat @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_8344_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_8345_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_8346_semiring__norm_I3_J,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bit0 @ N ) )
      = ( bit1 @ N ) ) ).

% semiring_norm(3)
thf(fact_8347_semiring__norm_I4_J,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bit1 @ N ) )
      = ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).

% semiring_norm(4)
thf(fact_8348_semiring__norm_I5_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ one )
      = ( bit1 @ M ) ) ).

% semiring_norm(5)
thf(fact_8349_semiring__norm_I8_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ one )
      = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).

% semiring_norm(8)
thf(fact_8350_semiring__norm_I10_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).

% semiring_norm(10)
thf(fact_8351_semiring__norm_I16_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).

% semiring_norm(16)
thf(fact_8352_semiring__norm_I79_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(79)
thf(fact_8353_semiring__norm_I74_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(74)
thf(fact_8354_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8115118780965096967l_num1 @ one_on7727431528512463931l_num1 )
    = ( numera7442385471795722001l_num1 @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_8355_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_8356_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_8357_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_8358_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_8359_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
      = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).

% div_Suc_eq_div_add3
thf(fact_8360_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_8361_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
      = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).

% mod_Suc_eq_mod_add3
thf(fact_8362_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu93272222329896523l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
    = ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_8363_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_8364_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu965353292909893953uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) )
    = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_8365_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_8366_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_8367_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_8368_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_8369_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_bit1
thf(fact_8370_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_8371_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X32: num] :
      ( ( bit0 @ X22 )
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(14)
thf(fact_8372_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] :
      ( one
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(12)
thf(fact_8373_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X23: num] :
            ( Y
           != ( bit0 @ X23 ) )
       => ~ ! [X33: num] :
              ( Y
             != ( bit1 @ X33 ) ) ) ) ).

% num.exhaust
thf(fact_8374_sin__coeff__Suc,axiom,
    ! [N: nat] :
      ( ( sin_coeff @ ( suc @ N ) )
      = ( divide_divide_real @ ( cos_coeff @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).

% sin_coeff_Suc
thf(fact_8375_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_8376_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_8377_VEBT__internal_OT__vebt__buildupi__gq__0,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( vEBT_V441764108873111860ildupi @ N ) ) ).

% VEBT_internal.T_vebt_buildupi_gq_0
thf(fact_8378_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numera7442385471795722001l_num1 @ ( bit1 @ N ) )
      = ( plus_p361126936061061375l_num1 @ ( plus_p361126936061061375l_num1 @ ( numera7442385471795722001l_num1 @ N ) @ ( numera7442385471795722001l_num1 @ N ) ) @ one_on7727431528512463931l_num1 ) ) ).

% numeral_Bit1
thf(fact_8379_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bit1 @ N ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).

% numeral_Bit1
thf(fact_8380_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).

% numeral_Bit1
thf(fact_8381_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).

% numeral_Bit1
thf(fact_8382_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).

% numeral_Bit1
thf(fact_8383_eval__nat__numeral_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N ) )
      = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).

% eval_nat_numeral(3)
thf(fact_8384_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q3: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_8385_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q3: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_8386_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q3: num,N: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_8387_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q3: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_8388_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q3: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_8389_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q3: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_8390_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q3: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_8391_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q3: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_8392_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q3: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_8393_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_8394_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_8395_power__minus__Bit1,axiom,
    ! [X: uint32,K: num] :
      ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_uint32 @ ( power_power_uint32 @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_8396_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_8397_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_8398_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_8399_cos__coeff__Suc,axiom,
    ! [N: nat] :
      ( ( cos_coeff @ ( suc @ N ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).

% cos_coeff_Suc
thf(fact_8400_VEBT__internal_OT__vebt__buildupi__univ,axiom,
    ! [U2: nat,N: nat] :
      ( ( U2
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ord_less_eq_nat @ ( vEBT_V441764108873111860ildupi @ N ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ U2 ) ) ) ).

% VEBT_internal.T_vebt_buildupi_univ
thf(fact_8401_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numera7442385471795722001l_num1 @ ( bit1 @ N ) )
      = ( plus_p361126936061061375l_num1 @ ( plus_p361126936061061375l_num1 @ ( numera7442385471795722001l_num1 @ N ) @ ( numera7442385471795722001l_num1 @ N ) ) @ one_on7727431528512463931l_num1 ) ) ).

% numeral_code(3)
thf(fact_8402_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bit1 @ N ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).

% numeral_code(3)
thf(fact_8403_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).

% numeral_code(3)
thf(fact_8404_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).

% numeral_code(3)
thf(fact_8405_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).

% numeral_code(3)
thf(fact_8406_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_8407_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_8408_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_8409_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_8410_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_8411_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_8412_numeral__Bit1__div__2,axiom,
    ! [N: num] :
      ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( numeral_numeral_nat @ N ) ) ).

% numeral_Bit1_div_2
thf(fact_8413_numeral__Bit1__div__2,axiom,
    ! [N: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( numeral_numeral_int @ N ) ) ).

% numeral_Bit1_div_2
thf(fact_8414_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q3: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != zero_zero_int ) ).

% cong_exp_iff_simps(3)
thf(fact_8415_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q3: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != zero_zero_nat ) ).

% cong_exp_iff_simps(3)
thf(fact_8416_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q3: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
     != zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(3)
thf(fact_8417_odd__numeral,axiom,
    ! [N: num] :
      ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( numera9087168376688890119uint32 @ ( bit1 @ N ) ) ) ).

% odd_numeral
thf(fact_8418_odd__numeral,axiom,
    ! [N: num] :
      ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ).

% odd_numeral
thf(fact_8419_odd__numeral,axiom,
    ! [N: num] :
      ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( numera7442385471795722001l_num1 @ ( bit1 @ N ) ) ) ).

% odd_numeral
thf(fact_8420_odd__numeral,axiom,
    ! [N: num] :
      ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).

% odd_numeral
thf(fact_8421_odd__numeral,axiom,
    ! [N: num] :
      ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).

% odd_numeral
thf(fact_8422_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_8423_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_8424_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_8425_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_8426_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_8427_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_8428_numeral__3__eq__3,axiom,
    ( ( numeral_numeral_nat @ ( bit1 @ one ) )
    = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).

% numeral_3_eq_3
thf(fact_8429_Suc3__eq__add__3,axiom,
    ! [N: nat] :
      ( ( suc @ ( suc @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).

% Suc3_eq_add_3
thf(fact_8430_num_Osize_I6_J,axiom,
    ! [X32: num] :
      ( ( size_size_num @ ( bit1 @ X32 ) )
      = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(6)
thf(fact_8431_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
thf(fact_8432_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
thf(fact_8433_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
thf(fact_8434_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_8435_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_8436_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_8437_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).

% Suc_div_eq_add3_div
thf(fact_8438_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).

% Suc_mod_eq_add3_mod
thf(fact_8439_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_8440_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_8441_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_8442_even__flip__bit__iff,axiom,
    ! [M: nat,A2: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_se7025624438249859091uint32 @ M @ A2 ) )
      = ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A2 )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_8443_even__flip__bit__iff,axiom,
    ! [M: nat,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A2 ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_8444_even__flip__bit__iff,axiom,
    ! [M: nat,A2: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( bit_se4491814353640558621l_num1 @ M @ A2 ) )
      = ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A2 )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_8445_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_8446_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_8447_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
     => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% odd_mod_4_div_2
thf(fact_8448_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_8449_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_8450_VEBT__internal_OT__vebt__buildupi_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ N ) ) )
          = ( 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 )
       => ( ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ N ) ) )
          = ( 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.simps(3)
thf(fact_8451_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 ) )
         => ~ ! [N2: nat] :
                ( ( X
                  = ( suc @ ( suc @ N2 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                     => ( Y
                        = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( 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 @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                     => ( Y
                        = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( 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 @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.elims
thf(fact_8452_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 ) ) ) ) )
         => ~ ! [Va2: nat] :
                ( ( X
                  = ( suc @ ( suc @ Va2 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                     => ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                     => ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
thf(fact_8453_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 ) ) ) )
         => ~ ! [Va2: nat] :
                ( ( X
                  = ( suc @ ( suc @ Va2 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                     => ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                     => ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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_8454_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 ) ) ) )
         => ~ ! [N2: nat] :
                ( ( X
                  = ( suc @ ( suc @ N2 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                     => ( Y
                        = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                     => ( Y
                        = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb.elims
thf(fact_8455_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_8456_VEBT__internal_Obuildup__build__time,axiom,
    ! [N: nat] : ( ord_less_nat @ ( vEBT_V8346862874174094_d_u_p @ N ) @ ( vEBT_V8646137997579335489_i_l_d @ N ) ) ).

% VEBT_internal.buildup_build_time
thf(fact_8457_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_8458_VEBT__internal_OTbuildupi__buildupi_H,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( vEBT_V441764108873111860ildupi @ N ) )
      = ( vEBT_V9176841429113362141ildupi @ N ) ) ).

% VEBT_internal.Tbuildupi_buildupi'
thf(fact_8459_VEBT__internal_OTb__T__vebt__buildupi_H,axiom,
    ! [N: nat] : ( ord_less_eq_int @ ( vEBT_V9176841429113362141ildupi @ N ) @ ( minus_minus_int @ ( vEBT_VEBT_Tb @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% VEBT_internal.Tb_T_vebt_buildupi'
thf(fact_8460_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_8461_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_8462_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_8463_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_8464_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_8465_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_8466_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_8467_VEBT__internal_OTb__T__vebt__buildupi,axiom,
    ! [N: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( vEBT_V441764108873111860ildupi @ N ) ) @ ( minus_minus_int @ ( vEBT_VEBT_Tb @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% VEBT_internal.Tb_T_vebt_buildupi
thf(fact_8468_VEBT__internal_OTb_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( vEBT_VEBT_Tb @ ( suc @ ( suc @ N ) ) )
          = ( 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 )
       => ( ( vEBT_VEBT_Tb @ ( suc @ ( suc @ N ) ) )
          = ( 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.simps(3)
thf(fact_8469_VEBT__internal_OT__vebt__buildupi_H_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ N ) ) )
          = ( 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 )
       => ( ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ N ) ) )
          = ( 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'.simps(3)
thf(fact_8470_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_8471_vebt__inst_Otime__vebt__buildup,axiom,
    ! [U2: nat,N: nat] :
      ( ( U2
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ord_less_eq_nat @ ( vEBT_V8346862874174094_d_u_p @ N ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) @ U2 ) ) ) ).

% vebt_inst.time_vebt_buildup
thf(fact_8472_vebt__buildupi__rule,axiom,
    ! [N: nat] : ( time_htt_VEBT_VEBTi @ ( pure_assn @ ( ord_less_nat @ zero_zero_nat @ N ) ) @ ( vEBT_vebt_buildupi @ N ) @ ( vEBT_Intf_vebt_assn @ N @ bot_bot_set_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% vebt_buildupi_rule
thf(fact_8473_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 ) ) ) )
         => ~ ! [N2: nat] :
                ( ( X
                  = ( suc @ ( suc @ N2 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                     => ( Y
                        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                     => ( Y
                        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.elims
thf(fact_8474_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_numeral_minus_bit1
thf(fact_8475_VEBT__internal_OTb_H_Osimps_I3_J,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ N ) ) )
          = ( 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 )
       => ( ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ N ) ) )
          = ( 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'.simps(3)
thf(fact_8476_pred__numeral__simps_I1_J,axiom,
    ( ( pred_numeral @ one )
    = zero_zero_nat ) ).

% pred_numeral_simps(1)
thf(fact_8477_eq__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ( numeral_numeral_nat @ K )
        = ( suc @ N ) )
      = ( ( pred_numeral @ K )
        = N ) ) ).

% eq_numeral_Suc
thf(fact_8478_Suc__eq__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ( suc @ N )
        = ( numeral_numeral_nat @ K ) )
      = ( N
        = ( pred_numeral @ K ) ) ) ).

% Suc_eq_numeral
thf(fact_8479_pred__numeral__simps_I3_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit1 @ K ) )
      = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).

% pred_numeral_simps(3)
thf(fact_8480_less__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).

% less_numeral_Suc
thf(fact_8481_less__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).

% less_Suc_numeral
thf(fact_8482_le__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).

% le_Suc_numeral
thf(fact_8483_le__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).

% le_numeral_Suc
thf(fact_8484_diff__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( minus_minus_nat @ ( pred_numeral @ K ) @ N ) ) ).

% diff_numeral_Suc
thf(fact_8485_diff__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( minus_minus_nat @ N @ ( pred_numeral @ K ) ) ) ).

% diff_Suc_numeral
thf(fact_8486_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_numeral_bit0
thf(fact_8487_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_numeral_minus_bit0
thf(fact_8488_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_numeral_bit1
thf(fact_8489_numeral__eq__Suc,axiom,
    ( numeral_numeral_nat
    = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).

% numeral_eq_Suc
thf(fact_8490_pred__numeral__def,axiom,
    ( pred_numeral
    = ( ^ [K3: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K3 ) @ one_one_nat ) ) ) ).

% pred_numeral_def
thf(fact_8491_VEBT__internal_OTb__Tb_H,axiom,
    ( vEBT_VEBT_Tb
    = ( ^ [T2: nat] : ( semiri1314217659103216013at_int @ ( vEBT_VEBT_Tb2 @ T2 ) ) ) ) ).

% VEBT_internal.Tb_Tb'
thf(fact_8492_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ K ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_8493_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ K ) )
      = ( times_times_int @ ( numeral_numeral_int @ K ) @ ( semiri1406184849735516958ct_int @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_8494_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ K ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_8495_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ K ) )
      = ( times_times_real @ ( numeral_numeral_real @ K ) @ ( semiri2265585572941072030t_real @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_8496_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ K ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_8497_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_8498_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_8499_VEBT__internal_OTb__T__vebt__buildupi_H_H,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ ( vEBT_V441764108873111860ildupi @ N ) @ ( minus_minus_nat @ ( vEBT_VEBT_Tb2 @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% VEBT_internal.Tb_T_vebt_buildupi''
thf(fact_8500_vebt__inserti__rule,axiom,
    ! [X: nat,N: nat,S2: set_nat,Ti: vEBT_VEBTi] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( time_htt_VEBT_VEBTi @ ( vEBT_Intf_vebt_assn @ N @ S2 @ Ti ) @ ( vEBT_vebt_inserti @ Ti @ X ) @ ( vEBT_Intf_vebt_assn @ N @ ( sup_sup_set_nat @ S2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( 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 @ N ) ) ) ) ) ) ) ) ).

% vebt_inserti_rule
thf(fact_8501_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 )
     => ~ ! [T3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T3 )
           => ( ( ord_less_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( ( X
                  = ( cos_real @ T3 ) )
               => ( Y
                 != ( sin_real @ T3 ) ) ) ) ) ) ).

% sincos_total_2pi
thf(fact_8502_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_8503_triangle__def,axiom,
    ( nat_triangle
    = ( ^ [N4: nat] : ( divide_divide_nat @ ( times_times_nat @ N4 @ ( suc @ N4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% triangle_def
thf(fact_8504_sin__paired,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
      @ ( sin_real @ X ) ) ).

% sin_paired
thf(fact_8505_nat__int,axiom,
    ! [N: nat] :
      ( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
      = N ) ).

% nat_int
thf(fact_8506_sums__zero,axiom,
    ( sums_complex
    @ ^ [N4: nat] : zero_zero_complex
    @ zero_zero_complex ) ).

% sums_zero
thf(fact_8507_sums__zero,axiom,
    ( sums_real
    @ ^ [N4: nat] : zero_zero_real
    @ zero_zero_real ) ).

% sums_zero
thf(fact_8508_sums__zero,axiom,
    ( sums_nat
    @ ^ [N4: nat] : zero_zero_nat
    @ zero_zero_nat ) ).

% sums_zero
thf(fact_8509_sums__zero,axiom,
    ( sums_int
    @ ^ [N4: nat] : zero_zero_int
    @ zero_zero_int ) ).

% sums_zero
thf(fact_8510_triangle__0,axiom,
    ( ( nat_triangle @ zero_zero_nat )
    = zero_zero_nat ) ).

% triangle_0
thf(fact_8511_cos__zero,axiom,
    ( ( cos_real @ zero_zero_real )
    = one_one_real ) ).

% cos_zero
thf(fact_8512_nat__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% nat_numeral
thf(fact_8513_nat__1,axiom,
    ( ( nat2 @ one_one_int )
    = ( suc @ zero_zero_nat ) ) ).

% nat_1
thf(fact_8514_nat__0__iff,axiom,
    ! [I: int] :
      ( ( ( nat2 @ I )
        = zero_zero_nat )
      = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).

% nat_0_iff
thf(fact_8515_nat__le__0,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ Z )
        = zero_zero_nat ) ) ).

% nat_le_0
thf(fact_8516_nat__neg__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = zero_zero_nat ) ).

% nat_neg_numeral
thf(fact_8517_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_8518_nat__zminus__int,axiom,
    ! [N: nat] :
      ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
      = zero_zero_nat ) ).

% nat_zminus_int
thf(fact_8519_int__nat__eq,axiom,
    ! [Z: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
          = Z ) )
      & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
          = zero_zero_int ) ) ) ).

% int_nat_eq
thf(fact_8520_cos__pi,axiom,
    ( ( cos_real @ pi )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% cos_pi
thf(fact_8521_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_8522_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_8523_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_8524_of__nat__nat,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri681578069525770553at_rat @ ( nat2 @ Z ) )
        = ( ring_1_of_int_rat @ Z ) ) ) ).

% of_nat_nat
thf(fact_8525_of__nat__nat,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri8819519690708144855l_num1 @ ( nat2 @ Z ) )
        = ( ring_17408606157368542149l_num1 @ Z ) ) ) ).

% of_nat_nat
thf(fact_8526_of__nat__nat,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
        = ( ring_1_of_int_int @ Z ) ) ) ).

% of_nat_nat
thf(fact_8527_of__nat__nat,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri5074537144036343181t_real @ ( nat2 @ Z ) )
        = ( ring_1_of_int_real @ Z ) ) ) ).

% of_nat_nat
thf(fact_8528_of__nat__nat,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri4939895301339042750nteger @ ( nat2 @ Z ) )
        = ( ring_18347121197199848620nteger @ Z ) ) ) ).

% of_nat_nat
thf(fact_8529_of__nat__nat,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri8010041392384452111omplex @ ( nat2 @ Z ) )
        = ( ring_17405671764205052669omplex @ Z ) ) ) ).

% of_nat_nat
thf(fact_8530_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( nat2 @ Y )
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% nat_eq_numeral_power_cancel_iff
thf(fact_8531_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = ( nat2 @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
thf(fact_8532_cos__of__real__pi,axiom,
    ( ( cos_real @ ( real_V1803761363581548252l_real @ pi ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% cos_of_real_pi
thf(fact_8533_cos__of__real__pi,axiom,
    ( ( cos_complex @ ( real_V4546457046886955230omplex @ pi ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% cos_of_real_pi
thf(fact_8534_dvd__nat__abs__iff,axiom,
    ! [N: nat,K: int] :
      ( ( dvd_dvd_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
      = ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ).

% dvd_nat_abs_iff
thf(fact_8535_nat__abs__dvd__iff,axiom,
    ! [K: int,N: nat] :
      ( ( dvd_dvd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
      = ( dvd_dvd_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% nat_abs_dvd_iff
thf(fact_8536_powser__sums__zero__iff,axiom,
    ! [A2: nat > complex,X: complex] :
      ( ( sums_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( A2 @ N4 ) @ ( power_power_complex @ zero_zero_complex @ N4 ) )
        @ X )
      = ( ( A2 @ zero_zero_nat )
        = X ) ) ).

% powser_sums_zero_iff
thf(fact_8537_powser__sums__zero__iff,axiom,
    ! [A2: nat > real,X: real] :
      ( ( sums_real
        @ ^ [N4: nat] : ( times_times_real @ ( A2 @ N4 ) @ ( power_power_real @ zero_zero_real @ N4 ) )
        @ X )
      = ( ( A2 @ zero_zero_nat )
        = X ) ) ).

% powser_sums_zero_iff
thf(fact_8538_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_8539_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_8540_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_8541_nat__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( ( ord_less_nat @ ( nat2 @ A2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% nat_less_numeral_power_cancel_iff
thf(fact_8542_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A2 ) ) ).

% numeral_power_less_nat_cancel_iff
thf(fact_8543_nat__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ A2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% nat_le_numeral_power_cancel_iff
thf(fact_8544_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A2 ) ) ).

% numeral_power_le_nat_cancel_iff
thf(fact_8545_cos__pi__half,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = zero_zero_real ) ).

% cos_pi_half
thf(fact_8546_cos__two__pi,axiom,
    ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_real ) ).

% cos_two_pi
thf(fact_8547_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_8548_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_8549_cos__npi2,axiom,
    ! [N: nat] :
      ( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).

% cos_npi2
thf(fact_8550_cos__npi,axiom,
    ! [N: nat] :
      ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).

% cos_npi
thf(fact_8551_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_8552_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_8553_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_8554_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_8555_cos__of__real__pi__half,axiom,
    ( ( cos_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = zero_zero_real ) ).

% cos_of_real_pi_half
thf(fact_8556_cos__of__real__pi__half,axiom,
    ( ( cos_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
    = zero_zero_complex ) ).

% cos_of_real_pi_half
thf(fact_8557_cos__2npi,axiom,
    ! [N: nat] :
      ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
      = one_one_real ) ).

% cos_2npi
thf(fact_8558_cos__int__2pin,axiom,
    ! [N: int] :
      ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
      = one_one_real ) ).

% cos_int_2pin
thf(fact_8559_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_8560_cos__npi__int,axiom,
    ! [N: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
       => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
          = one_one_real ) )
      & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
       => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% cos_npi_int
thf(fact_8561_sums__of__real,axiom,
    ! [X5: nat > real,A2: real] :
      ( ( sums_real @ X5 @ A2 )
     => ( sums_real
        @ ^ [N4: nat] : ( real_V1803761363581548252l_real @ ( X5 @ N4 ) )
        @ ( real_V1803761363581548252l_real @ A2 ) ) ) ).

% sums_of_real
thf(fact_8562_sums__of__real,axiom,
    ! [X5: nat > real,A2: real] :
      ( ( sums_real @ X5 @ A2 )
     => ( sums_complex
        @ ^ [N4: nat] : ( real_V4546457046886955230omplex @ ( X5 @ N4 ) )
        @ ( real_V4546457046886955230omplex @ A2 ) ) ) ).

% sums_of_real
thf(fact_8563_sums__of__real__iff,axiom,
    ! [F: nat > real,C: real] :
      ( ( sums_real
        @ ^ [N4: nat] : ( real_V1803761363581548252l_real @ ( F @ N4 ) )
        @ ( real_V1803761363581548252l_real @ C ) )
      = ( sums_real @ F @ C ) ) ).

% sums_of_real_iff
thf(fact_8564_sums__of__real__iff,axiom,
    ! [F: nat > real,C: real] :
      ( ( sums_complex
        @ ^ [N4: nat] : ( real_V4546457046886955230omplex @ ( F @ N4 ) )
        @ ( real_V4546457046886955230omplex @ C ) )
      = ( sums_real @ F @ C ) ) ).

% sums_of_real_iff
thf(fact_8565_sums__0,axiom,
    ! [F: nat > complex] :
      ( ! [N2: nat] :
          ( ( F @ N2 )
          = zero_zero_complex )
     => ( sums_complex @ F @ zero_zero_complex ) ) ).

% sums_0
thf(fact_8566_sums__0,axiom,
    ! [F: nat > real] :
      ( ! [N2: nat] :
          ( ( F @ N2 )
          = zero_zero_real )
     => ( sums_real @ F @ zero_zero_real ) ) ).

% sums_0
thf(fact_8567_sums__0,axiom,
    ! [F: nat > nat] :
      ( ! [N2: nat] :
          ( ( F @ N2 )
          = zero_zero_nat )
     => ( sums_nat @ F @ zero_zero_nat ) ) ).

% sums_0
thf(fact_8568_sums__0,axiom,
    ! [F: nat > int] :
      ( ! [N2: nat] :
          ( ( F @ N2 )
          = zero_zero_int )
     => ( sums_int @ F @ zero_zero_int ) ) ).

% sums_0
thf(fact_8569_sums__single,axiom,
    ! [I: nat,F: nat > complex] :
      ( sums_complex
      @ ^ [R5: nat] : ( if_complex @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_complex )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_8570_sums__single,axiom,
    ! [I: nat,F: nat > real] :
      ( sums_real
      @ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_8571_sums__single,axiom,
    ! [I: nat,F: nat > nat] :
      ( sums_nat
      @ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_8572_sums__single,axiom,
    ! [I: nat,F: nat > int] :
      ( sums_int
      @ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_8573_sums__add,axiom,
    ! [F: nat > complex,A2: complex,G: nat > complex,B2: complex] :
      ( ( sums_complex @ F @ A2 )
     => ( ( sums_complex @ G @ B2 )
       => ( sums_complex
          @ ^ [N4: nat] : ( plus_plus_complex @ ( F @ N4 ) @ ( G @ N4 ) )
          @ ( plus_plus_complex @ A2 @ B2 ) ) ) ) ).

% sums_add
thf(fact_8574_sums__add,axiom,
    ! [F: nat > real,A2: real,G: nat > real,B2: real] :
      ( ( sums_real @ F @ A2 )
     => ( ( sums_real @ G @ B2 )
       => ( sums_real
          @ ^ [N4: nat] : ( plus_plus_real @ ( F @ N4 ) @ ( G @ N4 ) )
          @ ( plus_plus_real @ A2 @ B2 ) ) ) ) ).

% sums_add
thf(fact_8575_sums__add,axiom,
    ! [F: nat > nat,A2: nat,G: nat > nat,B2: nat] :
      ( ( sums_nat @ F @ A2 )
     => ( ( sums_nat @ G @ B2 )
       => ( sums_nat
          @ ^ [N4: nat] : ( plus_plus_nat @ ( F @ N4 ) @ ( G @ N4 ) )
          @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% sums_add
thf(fact_8576_sums__add,axiom,
    ! [F: nat > int,A2: int,G: nat > int,B2: int] :
      ( ( sums_int @ F @ A2 )
     => ( ( sums_int @ G @ B2 )
       => ( sums_int
          @ ^ [N4: nat] : ( plus_plus_int @ ( F @ N4 ) @ ( G @ N4 ) )
          @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).

% sums_add
thf(fact_8577_sums__mult,axiom,
    ! [F: nat > complex,A2: complex,C: complex] :
      ( ( sums_complex @ F @ A2 )
     => ( sums_complex
        @ ^ [N4: nat] : ( times_times_complex @ C @ ( F @ N4 ) )
        @ ( times_times_complex @ C @ A2 ) ) ) ).

% sums_mult
thf(fact_8578_sums__mult,axiom,
    ! [F: nat > real,A2: real,C: real] :
      ( ( sums_real @ F @ A2 )
     => ( sums_real
        @ ^ [N4: nat] : ( times_times_real @ C @ ( F @ N4 ) )
        @ ( times_times_real @ C @ A2 ) ) ) ).

% sums_mult
thf(fact_8579_sums__mult2,axiom,
    ! [F: nat > complex,A2: complex,C: complex] :
      ( ( sums_complex @ F @ A2 )
     => ( sums_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ C )
        @ ( times_times_complex @ A2 @ C ) ) ) ).

% sums_mult2
thf(fact_8580_sums__mult2,axiom,
    ! [F: nat > real,A2: real,C: real] :
      ( ( sums_real @ F @ A2 )
     => ( sums_real
        @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ C )
        @ ( times_times_real @ A2 @ C ) ) ) ).

% sums_mult2
thf(fact_8581_sums__diff,axiom,
    ! [F: nat > complex,A2: complex,G: nat > complex,B2: complex] :
      ( ( sums_complex @ F @ A2 )
     => ( ( sums_complex @ G @ B2 )
       => ( sums_complex
          @ ^ [N4: nat] : ( minus_minus_complex @ ( F @ N4 ) @ ( G @ N4 ) )
          @ ( minus_minus_complex @ A2 @ B2 ) ) ) ) ).

% sums_diff
thf(fact_8582_sums__diff,axiom,
    ! [F: nat > real,A2: real,G: nat > real,B2: real] :
      ( ( sums_real @ F @ A2 )
     => ( ( sums_real @ G @ B2 )
       => ( sums_real
          @ ^ [N4: nat] : ( minus_minus_real @ ( F @ N4 ) @ ( G @ N4 ) )
          @ ( minus_minus_real @ A2 @ B2 ) ) ) ) ).

% sums_diff
thf(fact_8583_sums__divide,axiom,
    ! [F: nat > complex,A2: complex,C: complex] :
      ( ( sums_complex @ F @ A2 )
     => ( sums_complex
        @ ^ [N4: nat] : ( divide1717551699836669952omplex @ ( F @ N4 ) @ C )
        @ ( divide1717551699836669952omplex @ A2 @ C ) ) ) ).

% sums_divide
thf(fact_8584_sums__divide,axiom,
    ! [F: nat > real,A2: real,C: real] :
      ( ( sums_real @ F @ A2 )
     => ( sums_real
        @ ^ [N4: nat] : ( divide_divide_real @ ( F @ N4 ) @ C )
        @ ( divide_divide_real @ A2 @ C ) ) ) ).

% sums_divide
thf(fact_8585_sums__minus,axiom,
    ! [F: nat > complex,A2: complex] :
      ( ( sums_complex @ F @ A2 )
     => ( sums_complex
        @ ^ [N4: nat] : ( uminus1482373934393186551omplex @ ( F @ N4 ) )
        @ ( uminus1482373934393186551omplex @ A2 ) ) ) ).

% sums_minus
thf(fact_8586_sums__minus,axiom,
    ! [F: nat > real,A2: real] :
      ( ( sums_real @ F @ A2 )
     => ( sums_real
        @ ^ [N4: nat] : ( uminus_uminus_real @ ( F @ N4 ) )
        @ ( uminus_uminus_real @ A2 ) ) ) ).

% sums_minus
thf(fact_8587_nat__numeral__as__int,axiom,
    ( numeral_numeral_nat
    = ( ^ [I4: num] : ( nat2 @ ( numeral_numeral_int @ I4 ) ) ) ) ).

% nat_numeral_as_int
thf(fact_8588_nat__zero__as__int,axiom,
    ( zero_zero_nat
    = ( nat2 @ zero_zero_int ) ) ).

% nat_zero_as_int
thf(fact_8589_cos__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( cos_real @ X ) @ one_one_real ) ).

% cos_le_one
thf(fact_8590_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_8591_nat__one__as__int,axiom,
    ( one_one_nat
    = ( nat2 @ one_one_int ) ) ).

% nat_one_as_int
thf(fact_8592_eq__nat__nat__iff,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
       => ( ( ( nat2 @ Z )
            = ( nat2 @ Z6 ) )
          = ( Z = Z6 ) ) ) ) ).

% eq_nat_nat_iff
thf(fact_8593_all__nat,axiom,
    ( ( ^ [P2: nat > $o] :
        ! [X4: nat] : ( P2 @ X4 ) )
    = ( ^ [P3: nat > $o] :
        ! [X2: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X2 )
         => ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).

% all_nat
thf(fact_8594_ex__nat,axiom,
    ( ( ^ [P2: nat > $o] :
        ? [X4: nat] : ( P2 @ X4 ) )
    = ( ^ [P3: nat > $o] :
        ? [X2: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X2 )
          & ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).

% ex_nat
thf(fact_8595_sums__mult2__iff,axiom,
    ! [C: complex,F: nat > complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( ( sums_complex
          @ ^ [N4: nat] : ( times_times_complex @ ( F @ N4 ) @ C )
          @ ( times_times_complex @ D @ C ) )
        = ( sums_complex @ F @ D ) ) ) ).

% sums_mult2_iff
thf(fact_8596_sums__mult2__iff,axiom,
    ! [C: real,F: nat > real,D: real] :
      ( ( C != zero_zero_real )
     => ( ( sums_real
          @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ C )
          @ ( times_times_real @ D @ C ) )
        = ( sums_real @ F @ D ) ) ) ).

% sums_mult2_iff
thf(fact_8597_sums__mult__iff,axiom,
    ! [C: complex,F: nat > complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( ( sums_complex
          @ ^ [N4: nat] : ( times_times_complex @ C @ ( F @ N4 ) )
          @ ( times_times_complex @ C @ D ) )
        = ( sums_complex @ F @ D ) ) ) ).

% sums_mult_iff
thf(fact_8598_sums__mult__iff,axiom,
    ! [C: real,F: nat > real,D: real] :
      ( ( C != zero_zero_real )
     => ( ( sums_real
          @ ^ [N4: nat] : ( times_times_real @ C @ ( F @ N4 ) )
          @ ( times_times_real @ C @ D ) )
        = ( sums_real @ F @ D ) ) ) ).

% sums_mult_iff
thf(fact_8599_unat__eq__nat__uint,axiom,
    ( semiri7341220984566936280m1_nat
    = ( ^ [W2: word_N3645301735248828278l_num1] : ( nat2 @ ( semiri7338730514057886004m1_int @ W2 ) ) ) ) ).

% unat_eq_nat_uint
thf(fact_8600_cos__arctan__not__zero,axiom,
    ! [X: real] :
      ( ( cos_real @ ( arctan @ X ) )
     != zero_zero_real ) ).

% cos_arctan_not_zero
thf(fact_8601_unset__bit__nat__def,axiom,
    ( bit_se4205575877204974255it_nat
    = ( ^ [M3: nat,N4: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M3 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).

% unset_bit_nat_def
thf(fact_8602_cos__one__sin__zero,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = one_one_real )
     => ( ( sin_real @ X )
        = zero_zero_real ) ) ).

% cos_one_sin_zero
thf(fact_8603_sums__mult__D,axiom,
    ! [C: complex,F: nat > complex,A2: complex] :
      ( ( sums_complex
        @ ^ [N4: nat] : ( times_times_complex @ C @ ( F @ N4 ) )
        @ A2 )
     => ( ( C != zero_zero_complex )
       => ( sums_complex @ F @ ( divide1717551699836669952omplex @ A2 @ C ) ) ) ) ).

% sums_mult_D
thf(fact_8604_sums__mult__D,axiom,
    ! [C: real,F: nat > real,A2: real] :
      ( ( sums_real
        @ ^ [N4: nat] : ( times_times_real @ C @ ( F @ N4 ) )
        @ A2 )
     => ( ( C != zero_zero_real )
       => ( sums_real @ F @ ( divide_divide_real @ A2 @ C ) ) ) ) ).

% sums_mult_D
thf(fact_8605_sums__Suc__imp,axiom,
    ! [F: nat > complex,S2: complex] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_complex )
     => ( ( sums_complex
          @ ^ [N4: nat] : ( F @ ( suc @ N4 ) )
          @ S2 )
       => ( sums_complex @ F @ S2 ) ) ) ).

% sums_Suc_imp
thf(fact_8606_sums__Suc__imp,axiom,
    ! [F: nat > real,S2: real] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_real )
     => ( ( sums_real
          @ ^ [N4: nat] : ( F @ ( suc @ N4 ) )
          @ S2 )
       => ( sums_real @ F @ S2 ) ) ) ).

% sums_Suc_imp
thf(fact_8607_sums__Suc,axiom,
    ! [F: nat > complex,L: complex] :
      ( ( sums_complex
        @ ^ [N4: nat] : ( F @ ( suc @ N4 ) )
        @ L )
     => ( sums_complex @ F @ ( plus_plus_complex @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_8608_sums__Suc,axiom,
    ! [F: nat > real,L: real] :
      ( ( sums_real
        @ ^ [N4: nat] : ( F @ ( suc @ N4 ) )
        @ L )
     => ( sums_real @ F @ ( plus_plus_real @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_8609_sums__Suc,axiom,
    ! [F: nat > nat,L: nat] :
      ( ( sums_nat
        @ ^ [N4: nat] : ( F @ ( suc @ N4 ) )
        @ L )
     => ( sums_nat @ F @ ( plus_plus_nat @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_8610_sums__Suc,axiom,
    ! [F: nat > int,L: int] :
      ( ( sums_int
        @ ^ [N4: nat] : ( F @ ( suc @ N4 ) )
        @ L )
     => ( sums_int @ F @ ( plus_plus_int @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_8611_sums__Suc__iff,axiom,
    ! [F: nat > complex,S2: complex] :
      ( ( sums_complex
        @ ^ [N4: nat] : ( F @ ( suc @ N4 ) )
        @ S2 )
      = ( sums_complex @ F @ ( plus_plus_complex @ S2 @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc_iff
thf(fact_8612_sums__Suc__iff,axiom,
    ! [F: nat > real,S2: real] :
      ( ( sums_real
        @ ^ [N4: nat] : ( F @ ( suc @ N4 ) )
        @ S2 )
      = ( sums_real @ F @ ( plus_plus_real @ S2 @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc_iff
thf(fact_8613_sums__zero__iff__shift,axiom,
    ! [N: nat,F: nat > complex,S2: complex] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N )
         => ( ( F @ I3 )
            = zero_zero_complex ) )
     => ( ( sums_complex
          @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
          @ S2 )
        = ( sums_complex @ F @ S2 ) ) ) ).

% sums_zero_iff_shift
thf(fact_8614_sums__zero__iff__shift,axiom,
    ! [N: nat,F: nat > real,S2: real] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N )
         => ( ( F @ I3 )
            = zero_zero_real ) )
     => ( ( sums_real
          @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
          @ S2 )
        = ( sums_real @ F @ S2 ) ) ) ).

% sums_zero_iff_shift
thf(fact_8615_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_8616_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_8617_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_8618_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_8619_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_8620_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_8621_of__nat__ceiling,axiom,
    ! [R3: real] : ( ord_less_eq_real @ R3 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ R3 ) ) ) ) ).

% of_nat_ceiling
thf(fact_8622_of__nat__ceiling,axiom,
    ! [R3: rat] : ( ord_less_eq_rat @ R3 @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim2889992004027027881ng_rat @ R3 ) ) ) ) ).

% of_nat_ceiling
thf(fact_8623_nat__le__iff,axiom,
    ! [X: int,N: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
      = ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% nat_le_iff
thf(fact_8624_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_8625_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = Z )
      = ( ( M
          = ( nat2 @ Z ) )
        & ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).

% int_eq_iff
thf(fact_8626_nat__0__le,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
        = Z ) ) ).

% nat_0_le
thf(fact_8627_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_8628_nat__uint__less__helper,axiom,
    ! [Y: word_N3645301735248828278l_num1,Z: nat,X: word_N3645301735248828278l_num1] :
      ( ( ( nat2 @ ( semiri7338730514057886004m1_int @ Y ) )
        = Z )
     => ( ( ord_le750835935415966154l_num1 @ X @ Y )
       => ( ord_less_nat @ ( nat2 @ ( semiri7338730514057886004m1_int @ X ) ) @ Z ) ) ) ).

% nat_uint_less_helper
thf(fact_8629_int__minus,axiom,
    ! [N: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
      = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).

% int_minus
thf(fact_8630_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_8631_real__nat__ceiling__ge,axiom,
    ! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).

% real_nat_ceiling_ge
thf(fact_8632_nat__plus__as__int,axiom,
    ( plus_plus_nat
    = ( ^ [A4: nat,B4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_plus_as_int
thf(fact_8633_nat__times__as__int,axiom,
    ( times_times_nat
    = ( ^ [A4: nat,B4: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_times_as_int
thf(fact_8634_nat__minus__as__int,axiom,
    ( minus_minus_nat
    = ( ^ [A4: nat,B4: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_minus_as_int
thf(fact_8635_nat__div__as__int,axiom,
    ( divide_divide_nat
    = ( ^ [A4: nat,B4: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_div_as_int
thf(fact_8636_nat__mod__as__int,axiom,
    ( modulo_modulo_nat
    = ( ^ [A4: nat,B4: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_mod_as_int
thf(fact_8637_powser__sums__if,axiom,
    ! [M: nat,Z: complex] :
      ( sums_complex
      @ ^ [N4: nat] : ( times_times_complex @ ( if_complex @ ( N4 = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z @ N4 ) )
      @ ( power_power_complex @ Z @ M ) ) ).

% powser_sums_if
thf(fact_8638_powser__sums__if,axiom,
    ! [M: nat,Z: real] :
      ( sums_real
      @ ^ [N4: nat] : ( times_times_real @ ( if_real @ ( N4 = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z @ N4 ) )
      @ ( power_power_real @ Z @ M ) ) ).

% powser_sums_if
thf(fact_8639_powser__sums__if,axiom,
    ! [M: nat,Z: int] :
      ( sums_int
      @ ^ [N4: nat] : ( times_times_int @ ( if_int @ ( N4 = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z @ N4 ) )
      @ ( power_power_int @ Z @ M ) ) ).

% powser_sums_if
thf(fact_8640_powser__sums__zero,axiom,
    ! [A2: nat > complex] :
      ( sums_complex
      @ ^ [N4: nat] : ( times_times_complex @ ( A2 @ N4 ) @ ( power_power_complex @ zero_zero_complex @ N4 ) )
      @ ( A2 @ zero_zero_nat ) ) ).

% powser_sums_zero
thf(fact_8641_powser__sums__zero,axiom,
    ! [A2: nat > real] :
      ( sums_real
      @ ^ [N4: nat] : ( times_times_real @ ( A2 @ N4 ) @ ( power_power_real @ zero_zero_real @ N4 ) )
      @ ( A2 @ zero_zero_nat ) ) ).

% powser_sums_zero
thf(fact_8642_sin__zero__norm__cos__one,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( cos_real @ X ) )
        = one_one_real ) ) ).

% sin_zero_norm_cos_one
thf(fact_8643_sin__zero__norm__cos__one,axiom,
    ! [X: complex] :
      ( ( ( sin_complex @ X )
        = zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X ) )
        = one_one_real ) ) ).

% sin_zero_norm_cos_one
thf(fact_8644_cos__two__neq__zero,axiom,
    ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% cos_two_neq_zero
thf(fact_8645_of__nat__floor,axiom,
    ! [R3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R3 )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim6058952711729229775r_real @ R3 ) ) ) @ R3 ) ) ).

% of_nat_floor
thf(fact_8646_of__nat__floor,axiom,
    ! [R3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ R3 )
     => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim3151403230148437115or_rat @ R3 ) ) ) @ R3 ) ) ).

% of_nat_floor
thf(fact_8647_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_8648_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_8649_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_8650_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_8651_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_8652_split__nat,axiom,
    ! [P: nat > $o,I: int] :
      ( ( P @ ( nat2 @ I ) )
      = ( ! [N4: nat] :
            ( ( I
              = ( semiri1314217659103216013at_int @ N4 ) )
           => ( P @ N4 ) )
        & ( ( ord_less_int @ I @ zero_zero_int )
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_nat
thf(fact_8653_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_8654_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_8655_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_8656_le__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
        = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).

% le_nat_iff
thf(fact_8657_Suc__as__int,axiom,
    ( suc
    = ( ^ [A4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ one_one_int ) ) ) ) ).

% Suc_as_int
thf(fact_8658_le__mult__nat__floor,axiom,
    ! [A2: real,B2: real] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim6058952711729229775r_real @ A2 ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ B2 ) ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ ( times_times_real @ A2 @ B2 ) ) ) ) ).

% le_mult_nat_floor
thf(fact_8659_le__mult__nat__floor,axiom,
    ! [A2: rat,B2: rat] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim3151403230148437115or_rat @ A2 ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ B2 ) ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ) ).

% le_mult_nat_floor
thf(fact_8660_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_8661_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_8662_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_8663_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).

% nat_abs_triangle_ineq
thf(fact_8664_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_8665_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_8666_nat__power__eq,axiom,
    ! [Z: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( power_power_int @ Z @ N ) )
        = ( power_power_nat @ ( nat2 @ Z ) @ N ) ) ) ).

% nat_power_eq
thf(fact_8667_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_8668_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_8669_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] :
      ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).

% div_abs_eq_div_nat
thf(fact_8670_word__of__int__nat,axiom,
    ! [X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ring_17408606157368542149l_num1 @ X )
        = ( semiri8819519690708144855l_num1 @ ( nat2 @ X ) ) ) ) ).

% word_of_int_nat
thf(fact_8671_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_8672_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] :
      ( ( modulo_modulo_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
      = ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).

% mod_abs_eq_div_nat
thf(fact_8673_floor__eq3,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X )
     => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
          = N ) ) ) ).

% floor_eq3
thf(fact_8674_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_8675_sin__double,axiom,
    ! [X: real] :
      ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X ) ) @ ( cos_real @ X ) ) ) ).

% sin_double
thf(fact_8676_cos__paired,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) @ ( power_power_real @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
      @ ( cos_real @ X ) ) ).

% cos_paired
thf(fact_8677_nat__2,axiom,
    ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% nat_2
thf(fact_8678_cos__two__less__zero,axiom,
    ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_less_zero
thf(fact_8679_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_8680_cos__is__zero,axiom,
    ? [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 )
      & ! [Y5: real] :
          ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
            & ( ord_less_eq_real @ Y5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ Y5 )
              = zero_zero_real ) )
         => ( Y5 = X3 ) ) ) ).

% cos_is_zero
thf(fact_8681_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_8682_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_8683_cos__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ? [X3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X3 )
            & ( ord_less_eq_real @ X3 @ pi )
            & ( ( cos_real @ X3 )
              = Y )
            & ! [Y5: real] :
                ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
                  & ( ord_less_eq_real @ Y5 @ pi )
                  & ( ( cos_real @ Y5 )
                    = Y ) )
               => ( Y5 = X3 ) ) ) ) ) ).

% cos_total
thf(fact_8684_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_8685_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_8686_sincos__principal__value,axiom,
    ! [X: real] :
    ? [Y4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y4 )
      & ( ord_less_eq_real @ Y4 @ pi )
      & ( ( sin_real @ Y4 )
        = ( sin_real @ X ) )
      & ( ( cos_real @ Y4 )
        = ( cos_real @ X ) ) ) ).

% sincos_principal_value
thf(fact_8687_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_8688_floor__eq4,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X )
     => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
          = N ) ) ) ).

% floor_eq4
thf(fact_8689_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_8690_of__int__of__nat,axiom,
    ( ring_17408606157368542149l_num1
    = ( ^ [K3: int] : ( if_wor5778924947035936048l_num1 @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus8244633308260627903l_num1 @ ( semiri8819519690708144855l_num1 @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri8819519690708144855l_num1 @ ( nat2 @ K3 ) ) ) ) ) ).

% of_int_of_nat
thf(fact_8691_of__int__of__nat,axiom,
    ( ring_1_of_int_uint32
    = ( ^ [K3: int] : ( if_uint32 @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_uint32 @ ( semiri2565882477558803405uint32 @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri2565882477558803405uint32 @ ( nat2 @ K3 ) ) ) ) ) ).

% of_int_of_nat
thf(fact_8692_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_8693_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_8694_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_8695_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_8696_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_8697_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_8698_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_8699_cos__times__cos,axiom,
    ! [W: real,Z: real] :
      ( ( times_times_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( cos_real @ ( minus_minus_real @ W @ Z ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_times_cos
thf(fact_8700_cos__plus__cos,axiom,
    ! [W: real,Z: real] :
      ( ( plus_plus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_plus_cos
thf(fact_8701_cos__squared__eq,axiom,
    ! [X: complex] :
      ( ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_squared_eq
thf(fact_8702_cos__squared__eq,axiom,
    ! [X: real] :
      ( ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_squared_eq
thf(fact_8703_sin__squared__eq,axiom,
    ! [X: complex] :
      ( ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sin_squared_eq
thf(fact_8704_sin__squared__eq,axiom,
    ! [X: real] :
      ( ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sin_squared_eq
thf(fact_8705_geometric__sums,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
     => ( sums_real @ ( power_power_real @ C ) @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).

% geometric_sums
thf(fact_8706_geometric__sums,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( sums_complex @ ( power_power_complex @ C ) @ ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).

% geometric_sums
thf(fact_8707_power__half__series,axiom,
    ( sums_real
    @ ^ [N4: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N4 ) )
    @ one_one_real ) ).

% power_half_series
thf(fact_8708_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_8709_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_8710_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_8711_cos__one__2pi__int,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = one_one_real )
      = ( ? [X2: int] :
            ( X
            = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).

% cos_one_2pi_int
thf(fact_8712_cos__double__cos,axiom,
    ! [W: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).

% cos_double_cos
thf(fact_8713_cos__double__cos,axiom,
    ! [W: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( cos_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_real ) ) ).

% cos_double_cos
thf(fact_8714_cos__treble__cos,axiom,
    ! [X: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ ( cos_complex @ X ) ) ) ) ).

% cos_treble_cos
thf(fact_8715_cos__treble__cos,axiom,
    ! [X: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( cos_real @ X ) ) ) ) ).

% cos_treble_cos
thf(fact_8716_sin__times__sin,axiom,
    ! [W: real,Z: real] :
      ( ( times_times_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( cos_real @ ( minus_minus_real @ W @ Z ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_times_sin
thf(fact_8717_sin__times__cos,axiom,
    ! [W: real,Z: real] :
      ( ( times_times_real @ ( sin_real @ W ) @ ( cos_real @ Z ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( sin_real @ ( plus_plus_real @ W @ Z ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_times_cos
thf(fact_8718_cos__times__sin,axiom,
    ! [W: real,Z: real] :
      ( ( times_times_real @ ( cos_real @ W ) @ ( sin_real @ Z ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( sin_real @ ( plus_plus_real @ W @ Z ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_times_sin
thf(fact_8719_sin__plus__sin,axiom,
    ! [W: real,Z: real] :
      ( ( plus_plus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_plus_sin
thf(fact_8720_sin__diff__sin,axiom,
    ! [W: real,Z: real] :
      ( ( minus_minus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_diff_sin
thf(fact_8721_cos__diff__cos,axiom,
    ! [W: real,Z: real] :
      ( ( minus_minus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ Z @ W ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_diff_cos
thf(fact_8722_cos__double,axiom,
    ! [X: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
      = ( minus_minus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_double
thf(fact_8723_cos__double,axiom,
    ! [X: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
      = ( minus_minus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_double
thf(fact_8724_sums__if_H,axiom,
    ! [G: nat > real,X: real] :
      ( ( sums_real @ G @ X )
     => ( sums_real
        @ ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        @ X ) ) ).

% sums_if'
thf(fact_8725_cos__sin__eq,axiom,
    ( cos_real
    = ( ^ [X2: real] : ( sin_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) ) ) ) ).

% cos_sin_eq
thf(fact_8726_cos__sin__eq,axiom,
    ( cos_complex
    = ( ^ [X2: complex] : ( sin_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X2 ) ) ) ) ).

% cos_sin_eq
thf(fact_8727_sin__cos__eq,axiom,
    ( sin_real
    = ( ^ [X2: real] : ( cos_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) ) ) ) ).

% sin_cos_eq
thf(fact_8728_sin__cos__eq,axiom,
    ( sin_complex
    = ( ^ [X2: complex] : ( cos_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X2 ) ) ) ) ).

% sin_cos_eq
thf(fact_8729_sums__if,axiom,
    ! [G: nat > real,X: real,F: nat > real,Y: real] :
      ( ( sums_real @ G @ X )
     => ( ( sums_real @ F @ Y )
       => ( sums_real
          @ ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ ( F @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          @ ( plus_plus_real @ X @ Y ) ) ) ) ).

% sums_if
thf(fact_8730_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_8731_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_8732_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_8733_cos__one__2pi,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = one_one_real )
      = ( ? [X2: nat] :
            ( X
            = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
        | ? [X2: nat] :
            ( X
            = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).

% cos_one_2pi
thf(fact_8734_cos__double__sin,axiom,
    ! [W: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
      = ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_double_sin
thf(fact_8735_cos__double__sin,axiom,
    ! [W: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
      = ( minus_minus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( sin_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_double_sin
thf(fact_8736_minus__sin__cos__eq,axiom,
    ! [X: real] :
      ( ( uminus_uminus_real @ ( sin_real @ X ) )
      = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% minus_sin_cos_eq
thf(fact_8737_minus__sin__cos__eq,axiom,
    ! [X: complex] :
      ( ( uminus1482373934393186551omplex @ ( sin_complex @ X ) )
      = ( cos_complex @ ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% minus_sin_cos_eq
thf(fact_8738_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 )
       => ? [T3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T3 )
            & ( ord_less_eq_real @ T3 @ pi )
            & ( X
              = ( cos_real @ T3 ) )
            & ( Y
              = ( sin_real @ T3 ) ) ) ) ) ).

% sincos_total_pi
thf(fact_8739_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_8740_cos__zero__iff__int,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = zero_zero_real )
      = ( ? [I4: int] :
            ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I4 )
            & ( X
              = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_iff_int
thf(fact_8741_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_8742_cos__zero__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( cos_real @ X )
          = zero_zero_real )
       => ? [N2: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_lemma
thf(fact_8743_cos__zero__iff,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = zero_zero_real )
      = ( ? [N4: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N4: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
            & ( X
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% cos_zero_iff
thf(fact_8744_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_8745_geometric__deriv__sums,axiom,
    ! [Z: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
     => ( sums_real
        @ ^ [N4: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) @ ( power_power_real @ Z @ N4 ) )
        @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_8746_geometric__deriv__sums,axiom,
    ! [Z: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
     => ( sums_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N4 ) ) @ ( power_power_complex @ Z @ N4 ) )
        @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_8747_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 )
         => ? [T3: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ T3 )
              & ( ord_less_eq_real @ T3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( X
                = ( cos_real @ T3 ) )
              & ( Y
                = ( sin_real @ T3 ) ) ) ) ) ) ).

% sincos_total_pi_half
thf(fact_8748_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 )
     => ? [T3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ T3 )
          & ( ord_less_eq_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
          & ( X
            = ( cos_real @ T3 ) )
          & ( Y
            = ( sin_real @ T3 ) ) ) ) ).

% sincos_total_2pi_le
thf(fact_8749_vebt__deletei__rule,axiom,
    ! [N: nat,S2: set_nat,Ti: vEBT_VEBTi,X: nat] : ( time_htt_VEBT_VEBTi @ ( vEBT_Intf_vebt_assn @ N @ S2 @ Ti ) @ ( vEBT_vebt_deletei @ Ti @ X ) @ ( vEBT_Intf_vebt_assn @ N @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( nat2 @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ) ).

% vebt_deletei_rule
thf(fact_8750_diffs__equiv,axiom,
    ! [C: nat > real,X: real] :
      ( ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ ( diffs_real @ C @ N4 ) @ ( power_power_real @ X @ N4 ) ) )
     => ( sums_real
        @ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( C @ N4 ) ) @ ( power_power_real @ X @ ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) ) ) )
        @ ( suminf_real
          @ ^ [N4: nat] : ( times_times_real @ ( diffs_real @ C @ N4 ) @ ( power_power_real @ X @ N4 ) ) ) ) ) ).

% diffs_equiv
thf(fact_8751_diffs__equiv,axiom,
    ! [C: nat > complex,X: complex] :
      ( ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( diffs_complex @ C @ N4 ) @ ( power_power_complex @ X @ N4 ) ) )
     => ( sums_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N4 ) @ ( C @ N4 ) ) @ ( power_power_complex @ X @ ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) ) ) )
        @ ( suminf_complex
          @ ^ [N4: nat] : ( times_times_complex @ ( diffs_complex @ C @ N4 ) @ ( power_power_complex @ X @ N4 ) ) ) ) ) ).

% diffs_equiv
thf(fact_8752_tan__double,axiom,
    ! [X: complex] :
      ( ( ( cos_complex @ X )
       != zero_zero_complex )
     => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
         != zero_zero_complex )
       => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
          = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% tan_double
thf(fact_8753_tan__double,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
       != zero_zero_real )
     => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
         != zero_zero_real )
       => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
          = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% tan_double
thf(fact_8754_vebt__memberi__rule,axiom,
    ! [N: nat,S2: set_nat,Ti: vEBT_VEBTi,X: nat] :
      ( time_htt_o @ ( vEBT_Intf_vebt_assn @ N @ S2 @ Ti ) @ ( vEBT_vebt_memberi @ Ti @ X )
      @ ^ [R5: $o] :
          ( times_times_assn @ ( vEBT_Intf_vebt_assn @ N @ S2 @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( member_nat @ X @ S2 ) ) ) )
      @ ( 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 @ N ) ) ) ) ) ) ) ).

% vebt_memberi_rule
thf(fact_8755_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( ( real_V1022390504157884413omplex @ Z )
        = one_one_real )
     => ~ ! [T3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T3 )
           => ( ( ord_less_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( Z
               != ( complex2 @ ( cos_real @ T3 ) @ ( sin_real @ T3 ) ) ) ) ) ) ).

% complex_unimodular_polar
thf(fact_8756_tan__pi,axiom,
    ( ( tan_real @ pi )
    = zero_zero_real ) ).

% tan_pi
thf(fact_8757_tan__zero,axiom,
    ( ( tan_real @ zero_zero_real )
    = zero_zero_real ) ).

% tan_zero
thf(fact_8758_tan__npi,axiom,
    ! [N: nat] :
      ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% tan_npi
thf(fact_8759_tan__periodic__n,axiom,
    ! [X: real,N: num] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ N ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_n
thf(fact_8760_tan__periodic__nat,axiom,
    ! [X: real,N: nat] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_nat
thf(fact_8761_norm__cos__sin,axiom,
    ! [T: real] :
      ( ( real_V1022390504157884413omplex @ ( complex2 @ ( cos_real @ T ) @ ( sin_real @ T ) ) )
      = one_one_real ) ).

% norm_cos_sin
thf(fact_8762_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_8763_zero__complex_Ocode,axiom,
    ( zero_zero_complex
    = ( complex2 @ zero_zero_real @ zero_zero_real ) ) ).

% zero_complex.code
thf(fact_8764_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_8765_diffs__of__real,axiom,
    ! [F: nat > real] :
      ( ( diffs_complex
        @ ^ [N4: nat] : ( real_V4546457046886955230omplex @ ( F @ N4 ) ) )
      = ( ^ [N4: nat] : ( real_V4546457046886955230omplex @ ( diffs_real @ F @ N4 ) ) ) ) ).

% diffs_of_real
thf(fact_8766_one__complex_Ocode,axiom,
    ( one_one_complex
    = ( complex2 @ one_one_real @ zero_zero_real ) ) ).

% one_complex.code
thf(fact_8767_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_8768_diffs__minus,axiom,
    ! [C: nat > complex] :
      ( ( diffs_complex
        @ ^ [N4: nat] : ( uminus1482373934393186551omplex @ ( C @ N4 ) ) )
      = ( ^ [N4: nat] : ( uminus1482373934393186551omplex @ ( diffs_complex @ C @ N4 ) ) ) ) ).

% diffs_minus
thf(fact_8769_diffs__minus,axiom,
    ! [C: nat > uint32] :
      ( ( diffs_uint32
        @ ^ [N4: nat] : ( uminus_uminus_uint32 @ ( C @ N4 ) ) )
      = ( ^ [N4: nat] : ( uminus_uminus_uint32 @ ( diffs_uint32 @ C @ N4 ) ) ) ) ).

% diffs_minus
thf(fact_8770_diffs__minus,axiom,
    ! [C: nat > real] :
      ( ( diffs_real
        @ ^ [N4: nat] : ( uminus_uminus_real @ ( C @ N4 ) ) )
      = ( ^ [N4: nat] : ( uminus_uminus_real @ ( diffs_real @ C @ N4 ) ) ) ) ).

% diffs_minus
thf(fact_8771_diffs__minus,axiom,
    ! [C: nat > rat] :
      ( ( diffs_rat
        @ ^ [N4: nat] : ( uminus_uminus_rat @ ( C @ N4 ) ) )
      = ( ^ [N4: nat] : ( uminus_uminus_rat @ ( diffs_rat @ C @ N4 ) ) ) ) ).

% diffs_minus
thf(fact_8772_diffs__minus,axiom,
    ! [C: nat > int] :
      ( ( diffs_int
        @ ^ [N4: nat] : ( uminus_uminus_int @ ( C @ N4 ) ) )
      = ( ^ [N4: nat] : ( uminus_uminus_int @ ( diffs_int @ C @ N4 ) ) ) ) ).

% diffs_minus
thf(fact_8773_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_8774_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_8775_complex__of__real__code,axiom,
    ( real_V4546457046886955230omplex
    = ( ^ [X2: real] : ( complex2 @ X2 @ zero_zero_real ) ) ) ).

% complex_of_real_code
thf(fact_8776_complex__of__real__def,axiom,
    ( real_V4546457046886955230omplex
    = ( ^ [R5: real] : ( complex2 @ R5 @ zero_zero_real ) ) ) ).

% complex_of_real_def
thf(fact_8777_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_8778_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_8779_diffs__def,axiom,
    ( diffs_rat
    = ( ^ [C5: nat > rat,N4: nat] : ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N4 ) ) @ ( C5 @ ( suc @ N4 ) ) ) ) ) ).

% diffs_def
thf(fact_8780_diffs__def,axiom,
    ( diffs_int
    = ( ^ [C5: nat > int,N4: nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) @ ( C5 @ ( suc @ N4 ) ) ) ) ) ).

% diffs_def
thf(fact_8781_diffs__def,axiom,
    ( diffs_real
    = ( ^ [C5: nat > real,N4: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) @ ( C5 @ ( suc @ N4 ) ) ) ) ) ).

% diffs_def
thf(fact_8782_diffs__def,axiom,
    ( diffs_Code_integer
    = ( ^ [C5: nat > code_integer,N4: nat] : ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( suc @ N4 ) ) @ ( C5 @ ( suc @ N4 ) ) ) ) ) ).

% diffs_def
thf(fact_8783_diffs__def,axiom,
    ( diffs_complex
    = ( ^ [C5: nat > complex,N4: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N4 ) ) @ ( C5 @ ( suc @ N4 ) ) ) ) ) ).

% diffs_def
thf(fact_8784_termdiff__converges__all,axiom,
    ! [C: nat > complex,X: complex] :
      ( ! [X3: complex] :
          ( summable_complex
          @ ^ [N4: nat] : ( times_times_complex @ ( C @ N4 ) @ ( power_power_complex @ X3 @ N4 ) ) )
     => ( summable_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( diffs_complex @ C @ N4 ) @ ( power_power_complex @ X @ N4 ) ) ) ) ).

% termdiff_converges_all
thf(fact_8785_termdiff__converges__all,axiom,
    ! [C: nat > real,X: real] :
      ( ! [X3: real] :
          ( summable_real
          @ ^ [N4: nat] : ( times_times_real @ ( C @ N4 ) @ ( power_power_real @ X3 @ N4 ) ) )
     => ( summable_real
        @ ^ [N4: nat] : ( times_times_real @ ( diffs_real @ C @ N4 ) @ ( power_power_real @ X @ N4 ) ) ) ) ).

% termdiff_converges_all
thf(fact_8786_tan__45,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = one_one_real ) ).

% tan_45
thf(fact_8787_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_8788_lemma__tan__total,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ? [X3: real] :
          ( ( ord_less_real @ zero_zero_real @ X3 )
          & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ord_less_real @ Y @ ( tan_real @ X3 ) ) ) ) ).

% lemma_tan_total
thf(fact_8789_tan__total,axiom,
    ! [Y: 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 )
        = Y )
      & ! [Y5: real] :
          ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
            & ( ord_less_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( tan_real @ Y5 )
              = Y ) )
         => ( Y5 = X3 ) ) ) ).

% tan_total
thf(fact_8790_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_8791_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_8792_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_8793_lemma__tan__total1,axiom,
    ! [Y: 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 )
        = Y ) ) ).

% lemma_tan_total1
thf(fact_8794_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_8795_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_8796_vebt__heap__rules_I2_J,axiom,
    ! [N: nat,S2: set_nat,Ti: vEBT_VEBTi,X: nat] :
      ( hoare_hoare_triple_o @ ( vEBT_Intf_vebt_assn @ N @ S2 @ Ti ) @ ( vEBT_vebt_memberi @ Ti @ X )
      @ ^ [R5: $o] :
          ( times_times_assn @ ( vEBT_Intf_vebt_assn @ N @ S2 @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( member_nat @ X @ S2 ) ) ) ) ) ).

% vebt_heap_rules(2)
thf(fact_8797_add__tan__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ( cos_real @ X )
       != zero_zero_real )
     => ( ( ( cos_real @ Y )
         != zero_zero_real )
       => ( ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
          = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).

% add_tan_eq
thf(fact_8798_termdiff__converges,axiom,
    ! [X: real,K4: real,C: nat > real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ K4 )
     => ( ! [X3: real] :
            ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ K4 )
           => ( summable_real
              @ ^ [N4: nat] : ( times_times_real @ ( C @ N4 ) @ ( power_power_real @ X3 @ N4 ) ) ) )
       => ( summable_real
          @ ^ [N4: nat] : ( times_times_real @ ( diffs_real @ C @ N4 ) @ ( power_power_real @ X @ N4 ) ) ) ) ) ).

% termdiff_converges
thf(fact_8799_termdiff__converges,axiom,
    ! [X: complex,K4: real,C: nat > complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ K4 )
     => ( ! [X3: complex] :
            ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ K4 )
           => ( summable_complex
              @ ^ [N4: nat] : ( times_times_complex @ ( C @ N4 ) @ ( power_power_complex @ X3 @ N4 ) ) ) )
       => ( summable_complex
          @ ^ [N4: nat] : ( times_times_complex @ ( diffs_complex @ C @ N4 ) @ ( power_power_complex @ X @ N4 ) ) ) ) ) ).

% termdiff_converges
thf(fact_8800_tan__total__pos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ? [X3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X3 )
          & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( tan_real @ X3 )
            = Y ) ) ) ).

% tan_total_pos
thf(fact_8801_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_8802_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_8803_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_8804_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_8805_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_8806_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_8807_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_8808_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_8809_lemma__tan__add1,axiom,
    ! [X: real,Y: real] :
      ( ( ( cos_real @ X )
       != zero_zero_real )
     => ( ( ( cos_real @ Y )
         != zero_zero_real )
       => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) )
          = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).

% lemma_tan_add1
thf(fact_8810_tan__diff,axiom,
    ! [X: real,Y: real] :
      ( ( ( cos_real @ X )
       != zero_zero_real )
     => ( ( ( cos_real @ Y )
         != zero_zero_real )
       => ( ( ( cos_real @ ( minus_minus_real @ X @ Y ) )
           != zero_zero_real )
         => ( ( tan_real @ ( minus_minus_real @ X @ Y ) )
            = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).

% tan_diff
thf(fact_8811_tan__add,axiom,
    ! [X: real,Y: real] :
      ( ( ( cos_real @ X )
       != zero_zero_real )
     => ( ( ( cos_real @ Y )
         != zero_zero_real )
       => ( ( ( cos_real @ ( plus_plus_real @ X @ Y ) )
           != zero_zero_real )
         => ( ( tan_real @ ( plus_plus_real @ X @ Y ) )
            = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).

% tan_add
thf(fact_8812_tan__total__pi4,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ? [Z3: real] :
          ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z3 )
          & ( ord_less_real @ Z3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
          & ( ( tan_real @ Z3 )
            = X ) ) ) ).

% tan_total_pi4
thf(fact_8813_vebt__heap__rules_I8_J,axiom,
    ! [N: nat,S2: set_nat,Ti: vEBT_VEBTi,X: nat] : ( hoare_1429296392585015714_VEBTi @ ( vEBT_Intf_vebt_assn @ N @ S2 @ Ti ) @ ( vEBT_vebt_deletei @ Ti @ X ) @ ( vEBT_Intf_vebt_assn @ N @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% vebt_heap_rules(8)
thf(fact_8814_tan__half,axiom,
    ( tan_real
    = ( ^ [X2: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) @ one_one_real ) ) ) ) ).

% tan_half
thf(fact_8815_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_8816_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_8817_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_8818_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_8819_real__sqrt__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( sqrt @ X )
        = ( sqrt @ Y ) )
      = ( X = Y ) ) ).

% real_sqrt_eq_iff
thf(fact_8820_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_8821_real__sqrt__zero,axiom,
    ( ( sqrt @ zero_zero_real )
    = zero_zero_real ) ).

% real_sqrt_zero
thf(fact_8822_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_8823_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_8824_real__sqrt__eq__1__iff,axiom,
    ! [X: real] :
      ( ( ( sqrt @ X )
        = one_one_real )
      = ( X = one_one_real ) ) ).

% real_sqrt_eq_1_iff
thf(fact_8825_real__sqrt__one,axiom,
    ( ( sqrt @ one_one_real )
    = one_one_real ) ).

% real_sqrt_one
thf(fact_8826_cot__zero,axiom,
    ( ( cot_real @ zero_zero_real )
    = zero_zero_real ) ).

% cot_zero
thf(fact_8827_signed__0,axiom,
    ( ( ring_14059547012839848151l_num1 @ zero_z3563351764282998399l_num1 )
    = zero_z3563351764282998399l_num1 ) ).

% signed_0
thf(fact_8828_signed__0,axiom,
    ( ( ring_130596761880696677251_real @ zero_z3563351764282998399l_num1 )
    = zero_zero_real ) ).

% signed_0
thf(fact_8829_signed__0,axiom,
    ( ( ring_17861625399634564921m1_rat @ zero_z3563351764282998399l_num1 )
    = zero_zero_rat ) ).

% signed_0
thf(fact_8830_signed__0,axiom,
    ( ( ring_18494264989212010381m1_int @ zero_z3563351764282998399l_num1 )
    = zero_zero_int ) ).

% signed_0
thf(fact_8831_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_8832_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_8833_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_8834_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_8835_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_8836_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_8837_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_8838_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_8839_More__Word_Osint__0,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( ( ring_18494264989212010381m1_int @ X )
        = zero_zero_int )
      = ( X = zero_z3563351764282998399l_num1 ) ) ).

% More_Word.sint_0
thf(fact_8840_real__sqrt__abs2,axiom,
    ! [X: real] :
      ( ( sqrt @ ( times_times_real @ X @ X ) )
      = ( abs_abs_real @ X ) ) ).

% real_sqrt_abs2
thf(fact_8841_real__sqrt__mult__self,axiom,
    ! [A2: real] :
      ( ( times_times_real @ ( sqrt @ A2 ) @ ( sqrt @ A2 ) )
      = ( abs_abs_real @ A2 ) ) ).

% real_sqrt_mult_self
thf(fact_8842_arccos__1,axiom,
    ( ( arccos @ one_one_real )
    = zero_zero_real ) ).

% arccos_1
thf(fact_8843_cot__pi,axiom,
    ( ( cot_real @ pi )
    = zero_zero_real ) ).

% cot_pi
thf(fact_8844_real__sqrt__four,axiom,
    ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% real_sqrt_four
thf(fact_8845_arccos__minus__1,axiom,
    ( ( arccos @ ( uminus_uminus_real @ one_one_real ) )
    = pi ) ).

% arccos_minus_1
thf(fact_8846_signed__minus__1,axiom,
    ( ( ring_14059547012839848151l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
    = ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ).

% signed_minus_1
thf(fact_8847_signed__minus__1,axiom,
    ( ( ring_17006344825680464911omplex @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% signed_minus_1
thf(fact_8848_signed__minus__1,axiom,
    ( ( ring_12341578652349764045uint32 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
    = ( uminus_uminus_uint32 @ one_one_uint32 ) ) ).

% signed_minus_1
thf(fact_8849_signed__minus__1,axiom,
    ( ( ring_130596761880696677251_real @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% signed_minus_1
thf(fact_8850_signed__minus__1,axiom,
    ( ( ring_17861625399634564921m1_rat @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% signed_minus_1
thf(fact_8851_signed__minus__1,axiom,
    ( ( ring_18494264989212010381m1_int @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% signed_minus_1
thf(fact_8852_sint__minus1,axiom,
    ! [X: word_N3645301735248828278l_num1] :
      ( ( ( ring_18494264989212010381m1_int @ X )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( X
        = ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ) ).

% sint_minus1
thf(fact_8853_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_8854_cot__npi,axiom,
    ! [N: nat] :
      ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% cot_npi
thf(fact_8855_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_8856_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_8857_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_8858_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_8859_arccos__0,axiom,
    ( ( arccos @ zero_zero_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arccos_0
thf(fact_8860_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_8861_More__Word_Oof__int__sint,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( ring_17408606157368542149l_num1 @ ( ring_18494264989212010381m1_int @ A2 ) )
      = A2 ) ).

% More_Word.of_int_sint
thf(fact_8862_real__sqrt__minus,axiom,
    ! [X: real] :
      ( ( sqrt @ ( uminus_uminus_real @ X ) )
      = ( uminus_uminus_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_minus
thf(fact_8863_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_8864_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_8865_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_8866_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_8867_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_8868_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_8869_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_8870_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_8871_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_8872_Word_Osint__0,axiom,
    ( ( ring_18494264989212010381m1_int @ zero_z3563351764282998399l_num1 )
    = zero_zero_int ) ).

% Word.sint_0
thf(fact_8873_signed__eq__0__iff,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( ( ring_130596761880696677251_real @ W )
        = zero_zero_real )
      = ( W = zero_z3563351764282998399l_num1 ) ) ).

% signed_eq_0_iff
thf(fact_8874_signed__eq__0__iff,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( ( ring_17861625399634564921m1_rat @ W )
        = zero_zero_rat )
      = ( W = zero_z3563351764282998399l_num1 ) ) ).

% signed_eq_0_iff
thf(fact_8875_signed__eq__0__iff,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( ( ring_18494264989212010381m1_int @ W )
        = zero_zero_int )
      = ( W = zero_z3563351764282998399l_num1 ) ) ).

% signed_eq_0_iff
thf(fact_8876_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_8877_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_8878_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_8879_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_8880_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_8881_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_8882_sqrt2__less__2,axiom,
    ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% sqrt2_less_2
thf(fact_8883_sint__n1,axiom,
    ( ( ring_18494264989212010381m1_int @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% sint_n1
thf(fact_8884_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_8885_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_8886_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_8887_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_8888_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_8889_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_8890_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_8891_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_8892_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_8893_tan__60,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
    = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).

% tan_60
thf(fact_8894_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_8895_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_8896_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_8897_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_8898_lemma__real__divide__sqrt__less,axiom,
    ! [U2: real] :
      ( ( ord_less_real @ zero_zero_real @ U2 )
     => ( ord_less_real @ ( divide_divide_real @ U2 @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U2 ) ) ).

% lemma_real_divide_sqrt_less
thf(fact_8899_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_8900_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_8901_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_8902_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_8903_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A2: real,C: real,B2: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A2 @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B2 @ D ) @ ( 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 @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_triangle_ineq
thf(fact_8904_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_8905_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_8906_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_8907_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_8908_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_8909_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_8910_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_8911_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_8912_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_8913_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_8914_sqrt__even__pow2,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% sqrt_even_pow2
thf(fact_8915_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_8916_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_8917_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_8918_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_8919_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_8920_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_8921_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_8922_arsinh__real__def,axiom,
    ( arsinh_real
    = ( ^ [X2: real] : ( ln_ln_real @ ( plus_plus_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arsinh_real_def
thf(fact_8923_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_8924_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_8925_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_8926_real__sqrt__power__even,axiom,
    ! [N: nat,X: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( sqrt @ X ) @ N )
          = ( power_power_real @ X @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_power_even
thf(fact_8927_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_8928_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_8929_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_8930_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_8931_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_8932_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_8933_real__sqrt__sum__squares__less,axiom,
    ! [X: real,U2: real,Y: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ U2 @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
     => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ ( divide_divide_real @ U2 @ ( 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 ) ) ) ) ) @ U2 ) ) ) ).

% real_sqrt_sum_squares_less
thf(fact_8934_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_8935_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_8936_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_8937_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_8938_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_8939_sqrt__sum__squares__half__less,axiom,
    ! [X: real,U2: real,Y: real] :
      ( ( ord_less_real @ X @ ( divide_divide_real @ U2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_real @ Y @ ( divide_divide_real @ U2 @ ( 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 ) ) ) ) ) @ U2 ) ) ) ) ) ).

% sqrt_sum_squares_half_less
thf(fact_8940_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_8941_arctan__half,axiom,
    ( arctan
    = ( ^ [X2: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X2 @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% arctan_half
thf(fact_8942_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_8943_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_8944_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_8945_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_8946_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_8947_arcsin__0,axiom,
    ( ( arcsin @ zero_zero_real )
    = zero_zero_real ) ).

% arcsin_0
thf(fact_8948_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_8949_arcsin__1,axiom,
    ( ( arcsin @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arcsin_1
thf(fact_8950_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_8951_scast__0,axiom,
    ( ( ring_14059547012839848151l_num1 @ zero_z3563351764282998399l_num1 )
    = zero_z3563351764282998399l_num1 ) ).

% scast_0
thf(fact_8952_scast__n1,axiom,
    ( ( ring_14059547012839848151l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
    = ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ).

% scast_n1
thf(fact_8953_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_8954_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_8955_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_8956_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_8957_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_8958_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_8959_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_8960_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_8961_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_8962_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_8963_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_8964_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_8965_gbinomial__code,axiom,
    ( gbinomial_rat
    = ( ^ [A4: rat,K3: nat] :
          ( if_rat @ ( K3 = zero_zero_nat ) @ one_one_rat
          @ ( divide_divide_rat
            @ ( set_fo1949268297981939178at_rat
              @ ^ [L2: nat] : ( times_times_rat @ ( minus_minus_rat @ A4 @ ( semiri681578069525770553at_rat @ L2 ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K3 @ one_one_nat )
              @ one_one_rat )
            @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ) ).

% gbinomial_code
thf(fact_8966_gbinomial__code,axiom,
    ( gbinomial_real
    = ( ^ [A4: real,K3: nat] :
          ( if_real @ ( K3 = zero_zero_nat ) @ one_one_real
          @ ( divide_divide_real
            @ ( set_fo3111899725591712190t_real
              @ ^ [L2: nat] : ( times_times_real @ ( minus_minus_real @ A4 @ ( semiri5074537144036343181t_real @ L2 ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K3 @ one_one_nat )
              @ one_one_real )
            @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ) ).

% gbinomial_code
thf(fact_8967_gbinomial__code,axiom,
    ( gbinomial_complex
    = ( ^ [A4: complex,K3: nat] :
          ( if_complex @ ( K3 = zero_zero_nat ) @ one_one_complex
          @ ( divide1717551699836669952omplex
            @ ( set_fo1517530859248394432omplex
              @ ^ [L2: nat] : ( times_times_complex @ ( minus_minus_complex @ A4 @ ( semiri8010041392384452111omplex @ L2 ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K3 @ one_one_nat )
              @ one_one_complex )
            @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ) ).

% gbinomial_code
thf(fact_8968_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_8969_flip__bit__0,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( bit_se4491814353640558621l_num1 @ zero_zero_nat @ A2 )
      = ( plus_p361126936061061375l_num1 @ ( zero_n2087535428495186613l_num1 @ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A2 ) ) @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( divide1791077408188789448l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_8970_flip__bit__0,axiom,
    ! [A2: uint32] :
      ( ( bit_se7025624438249859091uint32 @ zero_zero_nat @ A2 )
      = ( plus_plus_uint32 @ ( zero_n412250872926760619uint32 @ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A2 ) ) @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( divide_divide_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_8971_flip__bit__0,axiom,
    ! [A2: code_integer] :
      ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A2 )
      = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_8972_flip__bit__0,axiom,
    ! [A2: int] :
      ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A2 )
      = ( plus_plus_int @ ( zero_n2684676970156552555ol_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 ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_8973_flip__bit__0,axiom,
    ! [A2: nat] :
      ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A2 )
      = ( plus_plus_nat @ ( zero_n2687167440665602831ol_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 ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_8974_take__bit__word__Bit1__eq,axiom,
    ! [N: num,M: num] :
      ( ( bit_se6195711425208868931l_num1 @ ( numeral_numeral_nat @ N ) @ ( numera7442385471795722001l_num1 @ ( bit1 @ M ) ) )
      = ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( bit_se6195711425208868931l_num1 @ ( pred_numeral @ N ) @ ( numera7442385471795722001l_num1 @ M ) ) ) ) ) ).

% take_bit_word_Bit1_eq
thf(fact_8975_nat__of__bool,axiom,
    ! [P: $o] :
      ( ( nat2 @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n2687167440665602831ol_nat @ P ) ) ).

% nat_of_bool
thf(fact_8976_take__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_se6195711425208868931l_num1 @ N @ zero_z3563351764282998399l_num1 )
      = zero_z3563351764282998399l_num1 ) ).

% take_bit_of_0
thf(fact_8977_take__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ zero_zero_nat )
      = zero_zero_nat ) ).

% take_bit_of_0
thf(fact_8978_take__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ N @ zero_zero_int )
      = zero_zero_int ) ).

% take_bit_of_0
thf(fact_8979_of__bool__eq_I1_J,axiom,
    ( ( zero_n2087535428495186613l_num1 @ $false )
    = zero_z3563351764282998399l_num1 ) ).

% of_bool_eq(1)
thf(fact_8980_of__bool__eq_I1_J,axiom,
    ( ( zero_n3304061248610475627l_real @ $false )
    = zero_zero_real ) ).

% of_bool_eq(1)
thf(fact_8981_of__bool__eq_I1_J,axiom,
    ( ( zero_n2052037380579107095ol_rat @ $false )
    = zero_zero_rat ) ).

% of_bool_eq(1)
thf(fact_8982_of__bool__eq_I1_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $false )
    = zero_zero_int ) ).

% of_bool_eq(1)
thf(fact_8983_of__bool__eq_I1_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $false )
    = zero_zero_nat ) ).

% of_bool_eq(1)
thf(fact_8984_of__bool__eq_I1_J,axiom,
    ( ( zero_n356916108424825756nteger @ $false )
    = zero_z3403309356797280102nteger ) ).

% of_bool_eq(1)
thf(fact_8985_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2087535428495186613l_num1 @ P )
        = zero_z3563351764282998399l_num1 )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_8986_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n3304061248610475627l_real @ P )
        = zero_zero_real )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_8987_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2052037380579107095ol_rat @ P )
        = zero_zero_rat )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_8988_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = zero_zero_int )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_8989_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = zero_zero_nat )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_8990_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n356916108424825756nteger @ P )
        = zero_z3403309356797280102nteger )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_8991_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
      = ( P
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_8992_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
      = ( P
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_8993_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
      = ( P
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_8994_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
      = ( P
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_8995_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_8996_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_8997_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_8998_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_8999_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_9000_of__bool__eq_I2_J,axiom,
    ( ( zero_n2087535428495186613l_num1 @ $true )
    = one_on7727431528512463931l_num1 ) ).

% of_bool_eq(2)
thf(fact_9001_of__bool__eq_I2_J,axiom,
    ( ( zero_n3304061248610475627l_real @ $true )
    = one_one_real ) ).

% of_bool_eq(2)
thf(fact_9002_of__bool__eq_I2_J,axiom,
    ( ( zero_n2052037380579107095ol_rat @ $true )
    = one_one_rat ) ).

% of_bool_eq(2)
thf(fact_9003_of__bool__eq_I2_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $true )
    = one_one_int ) ).

% of_bool_eq(2)
thf(fact_9004_of__bool__eq_I2_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $true )
    = one_one_nat ) ).

% of_bool_eq(2)
thf(fact_9005_of__bool__eq_I2_J,axiom,
    ( ( zero_n356916108424825756nteger @ $true )
    = one_one_Code_integer ) ).

% of_bool_eq(2)
thf(fact_9006_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2087535428495186613l_num1 @ P )
        = one_on7727431528512463931l_num1 )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_9007_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n3304061248610475627l_real @ P )
        = one_one_real )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_9008_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2052037380579107095ol_rat @ P )
        = one_one_rat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_9009_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = one_one_int )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_9010_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = one_one_nat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_9011_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n356916108424825756nteger @ P )
        = one_one_Code_integer )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_9012_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% of_nat_of_bool
thf(fact_9013_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri8010041392384452111omplex @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n1201886186963655149omplex @ P ) ) ).

% of_nat_of_bool
thf(fact_9014_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% of_nat_of_bool
thf(fact_9015_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2687167440665602831ol_nat @ P ) ) ).

% of_nat_of_bool
thf(fact_9016_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n356916108424825756nteger @ P ) ) ).

% of_nat_of_bool
thf(fact_9017_of__int__of__bool,axiom,
    ! [P: $o] :
      ( ( ring_1_of_int_real @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% of_int_of_bool
thf(fact_9018_of__int__of__bool,axiom,
    ! [P: $o] :
      ( ( ring_1_of_int_rat @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n2052037380579107095ol_rat @ P ) ) ).

% of_int_of_bool
thf(fact_9019_of__int__of__bool,axiom,
    ! [P: $o] :
      ( ( ring_17405671764205052669omplex @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n1201886186963655149omplex @ P ) ) ).

% of_int_of_bool
thf(fact_9020_of__int__of__bool,axiom,
    ! [P: $o] :
      ( ( ring_17408606157368542149l_num1 @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n2087535428495186613l_num1 @ P ) ) ).

% of_int_of_bool
thf(fact_9021_of__int__of__bool,axiom,
    ! [P: $o] :
      ( ( ring_1_of_int_int @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% of_int_of_bool
thf(fact_9022_of__int__of__bool,axiom,
    ! [P: $o] :
      ( ( ring_18347121197199848620nteger @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n356916108424825756nteger @ P ) ) ).

% of_int_of_bool
thf(fact_9023_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% abs_bool_eq
thf(fact_9024_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
      = ( zero_n2052037380579107095ol_rat @ P ) ) ).

% abs_bool_eq
thf(fact_9025_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% abs_bool_eq
thf(fact_9026_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
      = ( zero_n356916108424825756nteger @ P ) ) ).

% abs_bool_eq
thf(fact_9027_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_9028_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_9029_div__word__one,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( divide1791077408188789448l_num1 @ one_on7727431528512463931l_num1 @ W )
      = ( zero_n2087535428495186613l_num1 @ ( W = one_on7727431528512463931l_num1 ) ) ) ).

% div_word_one
thf(fact_9030_take__bit__0,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( bit_se6195711425208868931l_num1 @ zero_zero_nat @ A2 )
      = zero_z3563351764282998399l_num1 ) ).

% take_bit_0
thf(fact_9031_take__bit__0,axiom,
    ! [A2: nat] :
      ( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% take_bit_0
thf(fact_9032_take__bit__0,axiom,
    ! [A2: int] :
      ( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A2 )
      = zero_zero_int ) ).

% take_bit_0
thf(fact_9033_take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ one_one_nat )
      = one_one_nat ) ).

% take_bit_Suc_1
thf(fact_9034_take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ one_one_int )
      = one_one_int ) ).

% take_bit_Suc_1
thf(fact_9035_take__bit__numeral__1,axiom,
    ! [L: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ one_one_nat )
      = one_one_nat ) ).

% take_bit_numeral_1
thf(fact_9036_take__bit__numeral__1,axiom,
    ! [L: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ one_one_int )
      = one_one_int ) ).

% take_bit_numeral_1
thf(fact_9037_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_9038_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_9039_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_9040_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_9041_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_9042_exp__zero,axiom,
    ( ( exp_complex @ zero_zero_complex )
    = one_one_complex ) ).

% exp_zero
thf(fact_9043_exp__zero,axiom,
    ( ( exp_real @ zero_zero_real )
    = one_one_real ) ).

% exp_zero
thf(fact_9044_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_9045_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_9046_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_9047_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_9048_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_9049_Suc__0__mod__eq,axiom,
    ! [N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( zero_n2687167440665602831ol_nat
        @ ( N
         != ( suc @ zero_zero_nat ) ) ) ) ).

% Suc_0_mod_eq
thf(fact_9050_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n3304061248610475627l_real @ ~ P )
      = ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P ) ) ) ).

% of_bool_not_iff
thf(fact_9051_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2052037380579107095ol_rat @ ~ P )
      = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).

% of_bool_not_iff
thf(fact_9052_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2684676970156552555ol_int @ ~ P )
      = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).

% of_bool_not_iff
thf(fact_9053_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n356916108424825756nteger @ ~ P )
      = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P ) ) ) ).

% of_bool_not_iff
thf(fact_9054_exp__eq__one__iff,axiom,
    ! [X: real] :
      ( ( ( exp_real @ X )
        = one_one_real )
      = ( X = zero_zero_real ) ) ).

% exp_eq_one_iff
thf(fact_9055_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_9056_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_9057_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_9058_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_9059_exp__ln,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( exp_real @ ( ln_ln_real @ X ) )
        = X ) ) ).

% exp_ln
thf(fact_9060_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_9061_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_9062_not__exp__less__zero,axiom,
    ! [X: real] :
      ~ ( ord_less_real @ ( exp_real @ X ) @ zero_zero_real ) ).

% not_exp_less_zero
thf(fact_9063_exp__gt__zero,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X ) ) ).

% exp_gt_zero
thf(fact_9064_exp__total,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ? [X3: real] :
          ( ( exp_real @ X3 )
          = Y ) ) ).

% exp_total
thf(fact_9065_not__exp__le__zero,axiom,
    ! [X: real] :
      ~ ( ord_less_eq_real @ ( exp_real @ X ) @ zero_zero_real ) ).

% not_exp_le_zero
thf(fact_9066_exp__ge__zero,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X ) ) ).

% exp_ge_zero
thf(fact_9067_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_9068_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_9069_log__ln,axiom,
    ( ln_ln_real
    = ( log @ ( exp_real @ one_one_real ) ) ) ).

% log_ln
thf(fact_9070_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_9071_lemma__exp__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ one_one_real @ Y )
     => ? [X3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X3 )
          & ( ord_less_eq_real @ X3 @ ( minus_minus_real @ Y @ one_one_real ) )
          & ( ( exp_real @ X3 )
            = Y ) ) ) ).

% lemma_exp_total
thf(fact_9072_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_9073_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_9074_exp__le,axiom,
    ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).

% exp_le
thf(fact_9075_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_9076_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_9077_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_9078_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X ) ) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
thf(fact_9079_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
thf(fact_9080_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_9081_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_9082_tanh__real__altdef,axiom,
    ( tanh_real
    = ( ^ [X2: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) ) ) ) ) ) ).

% tanh_real_altdef
thf(fact_9083_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_9084_and__int_Osimps,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L2: int] :
          ( if_int
          @ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
          @ ( uminus_uminus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
          @ ( plus_plus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_int.simps
thf(fact_9085_arctan__def,axiom,
    ( arctan
    = ( ^ [Y2: real] :
          ( the_real
          @ ^ [X2: real] :
              ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
              & ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( tan_real @ X2 )
                = Y2 ) ) ) ) ) ).

% arctan_def
thf(fact_9086_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        | ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% and_nonnegative_int_iff
thf(fact_9087_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
        & ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% and_negative_int_iff
thf(fact_9088_pred__numeral__inc,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( inc @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% pred_numeral_inc
thf(fact_9089_and__minus__numerals_I2_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = one_one_int ) ).

% and_minus_numerals(2)
thf(fact_9090_and__minus__numerals_I6_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
      = one_one_int ) ).

% and_minus_numerals(6)
thf(fact_9091_take__bit__of__Suc__0,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% take_bit_of_Suc_0
thf(fact_9092_and__minus__numerals_I5_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_minus_numerals(5)
thf(fact_9093_and__minus__numerals_I1_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = zero_zero_int ) ).

% and_minus_numerals(1)
thf(fact_9094_num__induct,axiom,
    ! [P: num > $o,X: num] :
      ( ( P @ one )
     => ( ! [X3: num] :
            ( ( P @ X3 )
           => ( P @ ( inc @ X3 ) ) )
       => ( P @ X ) ) ) ).

% num_induct
thf(fact_9095_nat__take__bit__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) )
        = ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) ) ) ) ).

% nat_take_bit_eq
thf(fact_9096_take__bit__nat__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) )
        = ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).

% take_bit_nat_eq
thf(fact_9097_add__inc,axiom,
    ! [X: num,Y: num] :
      ( ( plus_plus_num @ X @ ( inc @ Y ) )
      = ( inc @ ( plus_plus_num @ X @ Y ) ) ) ).

% add_inc
thf(fact_9098_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_9099_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_9100_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_9101_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_9102_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_9103_take__bit__nonnegative,axiom,
    ! [N: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ).

% take_bit_nonnegative
thf(fact_9104_take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% take_bit_int_less_eq_self_iff
thf(fact_9105_not__take__bit__negative,axiom,
    ! [N: nat,K: int] :
      ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).

% not_take_bit_negative
thf(fact_9106_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% take_bit_int_greater_self_iff
thf(fact_9107_inc_Osimps_I1_J,axiom,
    ( ( inc @ one )
    = ( bit0 @ one ) ) ).

% inc.simps(1)
thf(fact_9108_inc_Osimps_I2_J,axiom,
    ! [X: num] :
      ( ( inc @ ( bit0 @ X ) )
      = ( bit1 @ X ) ) ).

% inc.simps(2)
thf(fact_9109_inc_Osimps_I3_J,axiom,
    ! [X: num] :
      ( ( inc @ ( bit1 @ X ) )
      = ( bit0 @ ( inc @ X ) ) ) ).

% inc.simps(3)
thf(fact_9110_add__One,axiom,
    ! [X: num] :
      ( ( plus_plus_num @ X @ one )
      = ( inc @ X ) ) ).

% add_One
thf(fact_9111_ln__real__def,axiom,
    ( ln_ln_real
    = ( ^ [X2: real] :
          ( the_real
          @ ^ [U: real] :
              ( ( exp_real @ U )
              = X2 ) ) ) ) ).

% ln_real_def
thf(fact_9112_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_9113_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_9114_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_9115_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).

% and_less_eq
thf(fact_9116_take__bit__decr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
       != zero_zero_int )
     => ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ one_one_int ) )
        = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ one_one_int ) ) ) ).

% take_bit_decr_eq
thf(fact_9117_ln__neg__is__const,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ln_ln_real @ X )
        = ( the_real
          @ ^ [X2: real] : $false ) ) ) ).

% ln_neg_is_const
thf(fact_9118_even__and__iff__int,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ).

% even_and_iff_int
thf(fact_9119_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N @ M )
        = M )
      = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_nat_eq_self_iff
thf(fact_9120_take__bit__nat__less__exp,axiom,
    ! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% take_bit_nat_less_exp
thf(fact_9121_take__bit__nat__eq__self,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( bit_se2925701944663578781it_nat @ N @ M )
        = M ) ) ).

% take_bit_nat_eq_self
thf(fact_9122_take__bit__nat__def,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N4: nat,M3: nat] : ( modulo_modulo_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% take_bit_nat_def
thf(fact_9123_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_bit1
thf(fact_9124_take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% take_bit_int_less_exp
thf(fact_9125_take__bit__int__def,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N4: nat,K3: int] : ( modulo_modulo_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% take_bit_int_def
thf(fact_9126_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_numeral_minus_bit1
thf(fact_9127_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).

% take_bit_nat_less_self_iff
thf(fact_9128_arccos__def,axiom,
    ( arccos
    = ( ^ [Y2: real] :
          ( the_real
          @ ^ [X2: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ X2 )
              & ( ord_less_eq_real @ X2 @ pi )
              & ( ( cos_real @ X2 )
                = Y2 ) ) ) ) ) ).

% arccos_def
thf(fact_9129_take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_minus_bit0
thf(fact_9130_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).

% take_bit_int_less_self_iff
thf(fact_9131_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_int_greater_eq_self_iff
thf(fact_9132_and__int__rec,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L2: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
              & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_int_rec
thf(fact_9133_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
        = K )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% take_bit_int_eq_self_iff
thf(fact_9134_take__bit__int__eq__self,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_se2923211474154528505it_int @ N @ K )
          = K ) ) ) ).

% take_bit_int_eq_self
thf(fact_9135_take__bit__incr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
       != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
     => ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
        = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).

% take_bit_incr_eq
thf(fact_9136_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_minus_bit0
thf(fact_9137_take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% take_bit_int_less_eq
thf(fact_9138_take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% take_bit_int_greater_eq
thf(fact_9139_signed__take__bit__eq__take__bit__shift,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N4: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% signed_take_bit_eq_take_bit_shift
thf(fact_9140_pi__half,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
    = ( the_real
      @ ^ [X2: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X2 )
          & ( ord_less_eq_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
          & ( ( cos_real @ X2 )
            = zero_zero_real ) ) ) ) ).

% pi_half
thf(fact_9141_pi__def,axiom,
    ( pi
    = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
      @ ( the_real
        @ ^ [X2: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X2 )
            & ( ord_less_eq_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ X2 )
              = zero_zero_real ) ) ) ) ) ).

% pi_def
thf(fact_9142_take__bit__minus__small__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
          = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).

% take_bit_minus_small_eq
thf(fact_9143_and__int__unfold,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L2: int] :
          ( if_int
          @ ( ( K3 = zero_zero_int )
            | ( L2 = zero_zero_int ) )
          @ zero_zero_int
          @ ( if_int
            @ ( K3
              = ( uminus_uminus_int @ one_one_int ) )
            @ L2
            @ ( if_int
              @ ( L2
                = ( uminus_uminus_int @ one_one_int ) )
              @ K3
              @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% and_int_unfold
thf(fact_9144_arcsin__def,axiom,
    ( arcsin
    = ( ^ [Y2: real] :
          ( the_real
          @ ^ [X2: real] :
              ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
              & ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( sin_real @ X2 )
                = Y2 ) ) ) ) ) ).

% arcsin_def
thf(fact_9145_modulo__int__def,axiom,
    ( modulo_modulo_int
    = ( ^ [K3: int,L2: int] :
          ( if_int @ ( L2 = zero_zero_int ) @ K3
          @ ( if_int
            @ ( ( sgn_sgn_int @ K3 )
              = ( sgn_sgn_int @ L2 ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L2 )
              @ ( minus_minus_int
                @ ( times_times_int @ ( abs_abs_int @ L2 )
                  @ ( zero_n2684676970156552555ol_int
                    @ ~ ( dvd_dvd_int @ L2 @ K3 ) ) )
                @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ) ) ) ).

% modulo_int_def
thf(fact_9146_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R3: int] :
      ( ( dvd_dvd_int @ L @ ( times_times_int @ K @ ( sgn_sgn_int @ R3 ) ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( R3 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
thf(fact_9147_dvd__sgn__mult__iff,axiom,
    ! [L: int,R3: int,K: int] :
      ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R3 ) @ K ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( R3 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
thf(fact_9148_mult__sgn__dvd__iff,axiom,
    ! [L: int,R3: int,K: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R3 ) ) @ K )
      = ( ( dvd_dvd_int @ L @ K )
        & ( ( R3 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
thf(fact_9149_sgn__mult__dvd__iff,axiom,
    ! [R3: int,L: int,K: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R3 ) @ L ) @ K )
      = ( ( dvd_dvd_int @ L @ K )
        & ( ( R3 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
thf(fact_9150_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_9151_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_9152_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_9153_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_9154_Suc__0__and__eq,axiom,
    ! [N: nat] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Suc_0_and_eq
thf(fact_9155_and__Suc__0__eq,axiom,
    ! [N: nat] :
      ( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% and_Suc_0_eq
thf(fact_9156_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N2: nat,L3: int] :
          ( K
         != ( times_times_int @ ( sgn_sgn_int @ L3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% int_sgnE
thf(fact_9157_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ~ ( dvd_dvd_int @ L @ K )
       => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L ) )
          = ( sgn_sgn_int @ L ) ) ) ) ).

% sgn_mod
thf(fact_9158_and__nat__def,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M3: nat,N4: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).

% and_nat_def
thf(fact_9159_zsgn__def,axiom,
    ( sgn_sgn_int
    = ( ^ [I4: int] : ( if_int @ ( I4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zsgn_def
thf(fact_9160_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L ) ) )
        = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).

% div_sgn_abs_cancel
thf(fact_9161_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ( ( sgn_sgn_int @ K )
         != ( sgn_sgn_int @ L ) )
       => ( ( divide_divide_int @ K @ L )
          = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) )
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( dvd_dvd_int @ L @ K ) ) ) ) ) ) ).

% div_noneq_sgn_abs
thf(fact_9162_floor__real__def,axiom,
    ( archim6058952711729229775r_real
    = ( ^ [X2: real] :
          ( the_int
          @ ^ [Z4: int] :
              ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z4 ) @ X2 )
              & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ) ) ) ).

% floor_real_def
thf(fact_9163_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M3: nat,N4: nat] :
          ( if_nat
          @ ( ( M3 = zero_zero_nat )
            | ( N4 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_9164_and__nat__rec,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M3: nat,N4: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
              & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_nat_rec
thf(fact_9165_divide__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
          = zero_zero_int ) )
      & ( ~ ( ( ( sgn_sgn_int @ L )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K )
              = ( sgn_sgn_int @ L ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( uminus_uminus_int
                @ ( semiri1314217659103216013at_int
                  @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
                    @ ( zero_n2687167440665602831ol_nat
                      @ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).

% divide_int_unfold
thf(fact_9166_modulo__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
          = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
      & ( ~ ( ( ( sgn_sgn_int @ L )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K )
              = ( sgn_sgn_int @ L ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L )
                @ ( minus_minus_int
                  @ ( semiri1314217659103216013at_int
                    @ ( times_times_nat @ N
                      @ ( zero_n2687167440665602831ol_nat
                        @ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
                  @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).

% modulo_int_unfold
thf(fact_9167_divide__int__def,axiom,
    ( divide_divide_int
    = ( ^ [K3: int,L2: int] :
          ( if_int @ ( L2 = zero_zero_int ) @ zero_zero_int
          @ ( if_int
            @ ( ( sgn_sgn_int @ K3 )
              = ( sgn_sgn_int @ L2 ) )
            @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) )
            @ ( uminus_uminus_int
              @ ( semiri1314217659103216013at_int
                @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) )
                  @ ( zero_n2687167440665602831ol_nat
                    @ ~ ( dvd_dvd_int @ L2 @ K3 ) ) ) ) ) ) ) ) ) ).

% divide_int_def
thf(fact_9168_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_9169_signed__take__bit__eq__take__bit__minus,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N4: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N4 ) ) ) ) ) ) ).

% signed_take_bit_eq_take_bit_minus
thf(fact_9170_num_Osize__gen_I3_J,axiom,
    ! [X32: num] :
      ( ( size_num @ ( bit1 @ X32 ) )
      = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(3)
thf(fact_9171_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_9172_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_9173_signed__take__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K ) )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).

% signed_take_bit_nonnegative_iff
thf(fact_9174_signed__take__bit__negative__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
      = ( bit_se1146084159140164899it_int @ K @ N ) ) ).

% signed_take_bit_negative_iff
thf(fact_9175_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N ) ) ).

% bit_minus_numeral_Bit0_Suc_iff
thf(fact_9176_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N ) ) ) ).

% bit_minus_numeral_Bit1_Suc_iff
thf(fact_9177_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,N: num] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N ) ) ) ).

% bit_minus_numeral_int(1)
thf(fact_9178_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,N: num] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ) ).

% bit_minus_numeral_int(2)
thf(fact_9179_bin__nth__minus__Bit0,axiom,
    ! [N: nat,W: num] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) @ N )
        = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% bin_nth_minus_Bit0
thf(fact_9180_bin__nth__minus__Bit1,axiom,
    ! [N: nat,W: num] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) @ N )
        = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% bin_nth_minus_Bit1
thf(fact_9181_bit__not__int__iff_H,axiom,
    ! [K: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).

% bit_not_int_iff'
thf(fact_9182_sgn__real__def,axiom,
    ( sgn_sgn_real
    = ( ^ [A4: real] : ( if_real @ ( A4 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A4 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).

% sgn_real_def
thf(fact_9183_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M: nat,K: int] :
      ( ( ord_less_nat @ N @ M )
     => ( ( bit_se1146084159140164899it_int @ K @ N )
       => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).

% bit_imp_take_bit_positive
thf(fact_9184_sgn__power__injE,axiom,
    ! [A2: real,N: nat,X: real,B2: real] :
      ( ( ( times_times_real @ ( sgn_sgn_real @ A2 ) @ ( power_power_real @ ( abs_abs_real @ A2 ) @ N ) )
        = X )
     => ( ( X
          = ( times_times_real @ ( sgn_sgn_real @ B2 ) @ ( power_power_real @ ( abs_abs_real @ B2 ) @ N ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( A2 = B2 ) ) ) ) ).

% sgn_power_injE
thf(fact_9185_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N2: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ N2 @ M2 )
             => ( ( bit_se1146084159140164899it_int @ K @ M2 )
                = ( bit_se1146084159140164899it_int @ K @ N2 ) ) )
         => ~ ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ one_one_nat ) )
                = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ) ) ).

% int_bit_bound
thf(fact_9186_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one )
    = zero_zero_nat ) ).

% num.size_gen(1)
thf(fact_9187_bit__int__def,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [K3: int,N4: nat] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).

% bit_int_def
thf(fact_9188_Bit__Operations_Oset__bit__eq,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N4: nat,K3: int] :
          ( plus_plus_int @ K3
          @ ( times_times_int
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( bit_se1146084159140164899it_int @ K3 @ N4 ) )
            @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).

% Bit_Operations.set_bit_eq
thf(fact_9189_unset__bit__eq,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N4: nat,K3: int] : ( minus_minus_int @ K3 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N4 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).

% unset_bit_eq
thf(fact_9190_take__bit__Suc__from__most,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K )
      = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N ) ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% take_bit_Suc_from_most
thf(fact_9191_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_9192_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_9193_floor__rat__def,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [X2: rat] :
          ( the_int
          @ ^ [Z4: int] :
              ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z4 ) @ X2 )
              & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ) ) ) ).

% floor_rat_def
thf(fact_9194_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 ) ) ) )
           => ~ ! [N2: nat] :
                  ( ( X
                    = ( suc @ ( suc @ N2 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                       => ( Y
                          = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( 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 @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                       => ( Y
                          = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( 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 @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_V3352910403632780892pi_rel @ ( suc @ ( suc @ N2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.pelims
thf(fact_9195_VEBT__internal_OTBOUND__buildupi,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( time_T5737551269749752165_VEBTi @ ( vEBT_vebt_buildupi @ N ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% VEBT_internal.TBOUND_buildupi
thf(fact_9196_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% mask_nat_positive_iff
thf(fact_9197_sgn__rat__def,axiom,
    ( sgn_sgn_rat
    = ( ^ [A4: rat] : ( if_rat @ ( A4 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A4 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_rat_def
thf(fact_9198_less__eq__rat__def,axiom,
    ( ord_less_eq_rat
    = ( ^ [X2: rat,Y2: rat] :
          ( ( ord_less_rat @ X2 @ Y2 )
          | ( X2 = Y2 ) ) ) ) ).

% less_eq_rat_def
thf(fact_9199_obtain__pos__sum,axiom,
    ! [R3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ R3 )
     => ~ ! [S3: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ S3 )
           => ! [T3: rat] :
                ( ( ord_less_rat @ zero_zero_rat @ T3 )
               => ( R3
                 != ( plus_plus_rat @ S3 @ T3 ) ) ) ) ) ).

% obtain_pos_sum
thf(fact_9200_abs__rat__def,axiom,
    ( abs_abs_rat
    = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).

% abs_rat_def
thf(fact_9201_bit__Suc__0__iff,axiom,
    ! [N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( N = zero_zero_nat ) ) ).

% bit_Suc_0_iff
thf(fact_9202_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).

% not_bit_Suc_0_Suc
thf(fact_9203_mask__nonnegative__int,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).

% mask_nonnegative_int
thf(fact_9204_not__mask__negative__int,axiom,
    ! [N: nat] :
      ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).

% not_mask_negative_int
thf(fact_9205_not__bit__Suc__0__numeral,axiom,
    ! [N: num] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).

% not_bit_Suc_0_numeral
thf(fact_9206_less__mask,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).

% less_mask
thf(fact_9207_bit__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).

% bit_nat_iff
thf(fact_9208_take__bit__eq__mask__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
        = ( bit_se2000444600071755411sk_int @ N ) )
      = ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
        = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
thf(fact_9209_Suc__mask__eq__exp,axiom,
    ! [N: nat] :
      ( ( suc @ ( bit_se2002935070580805687sk_nat @ N ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% Suc_mask_eq_exp
thf(fact_9210_mask__nat__less__exp,axiom,
    ! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% mask_nat_less_exp
thf(fact_9211_bit__nat__def,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [M3: nat,N4: nat] :
          ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).

% bit_nat_def
thf(fact_9212_mask__half__int,axiom,
    ! [N: nat] :
      ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).

% mask_half_int
thf(fact_9213_mask__int__def,axiom,
    ( bit_se2000444600071755411sk_int
    = ( ^ [N4: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) @ one_one_int ) ) ) ).

% mask_int_def
thf(fact_9214_mask__nat__def,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N4: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) ) ).

% mask_nat_def
thf(fact_9215_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
        = ( bit_se2000444600071755411sk_int @ N ) )
      = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).

% take_bit_eq_mask_iff_exp_dvd
thf(fact_9216_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 ) ) ) )
           => ~ ! [Va2: nat] :
                  ( ( X
                    = ( suc @ ( suc @ Va2 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                       => ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                       => ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
thf(fact_9217_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 ) ) ) )
           => ~ ! [Va2: nat] :
                  ( ( X
                    = ( suc @ ( suc @ Va2 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                       => ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                       => ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
thf(fact_9218_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 ) ) ) )
           => ~ ! [N2: nat] :
                  ( ( X
                    = ( suc @ ( suc @ N2 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                       => ( Y
                          = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                       => ( Y
                          = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_VEBT_Tb_rel2 @ ( suc @ ( suc @ N2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb.pelims
thf(fact_9219_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 ) ) ) )
           => ~ ! [N2: nat] :
                  ( ( X
                    = ( suc @ ( suc @ N2 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                       => ( Y
                          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                       => ( Y
                          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_VEBT_Tb_rel @ ( suc @ ( suc @ N2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.pelims
thf(fact_9220_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_9221_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_9222_not__int__def,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K3: int] : ( minus_minus_int @ ( uminus_uminus_int @ K3 ) @ one_one_int ) ) ) ).

% not_int_def
thf(fact_9223_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_9224_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_9225_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_9226_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_9227_and__not__numerals_I2_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = one_one_int ) ).

% and_not_numerals(2)
thf(fact_9228_bit__minus__int__iff,axiom,
    ! [K: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N )
      = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N ) ) ).

% bit_minus_int_iff
thf(fact_9229_and__not__numerals_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% and_not_numerals(5)
thf(fact_9230_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_9231_and__not__numerals_I3_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = zero_zero_int ) ).

% and_not_numerals(3)
thf(fact_9232_and__not__numerals_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% and_not_numerals(9)
thf(fact_9233_and__not__numerals_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% and_not_numerals(6)
thf(fact_9234_and__not__numerals_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).

% and_not_numerals(8)
thf(fact_9235_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_9236_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 ) ) ) )
           => ~ ! [N2: nat] :
                  ( ( X
                    = ( suc @ ( suc @ N2 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                       => ( Y
                          = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( 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 @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
                       => ( Y
                          = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( 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 @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_V2957053500504383685pi_rel @ ( suc @ ( suc @ N2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.pelims
thf(fact_9237_int__not__code_I1_J,axiom,
    ( ( bit_ri7919022796975470100ot_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% int_not_code(1)
thf(fact_9238_xor__int__unfold,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L2: int] :
          ( if_int
          @ ( K3
            = ( uminus_uminus_int @ one_one_int ) )
          @ ( bit_ri7919022796975470100ot_int @ L2 )
          @ ( if_int
            @ ( L2
              = ( uminus_uminus_int @ one_one_int ) )
            @ ( bit_ri7919022796975470100ot_int @ K3 )
            @ ( if_int @ ( K3 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
thf(fact_9239_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        = ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% xor_nonnegative_int_iff
thf(fact_9240_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
       != ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% xor_negative_int_iff
thf(fact_9241_int__xor__code_I2_J,axiom,
    ! [I: int] :
      ( ( bit_se6526347334894502574or_int @ I @ zero_zero_int )
      = I ) ).

% int_xor_code(2)
thf(fact_9242_int__xor__code_I1_J,axiom,
    ! [J: int] :
      ( ( bit_se6526347334894502574or_int @ zero_zero_int @ J )
      = J ) ).

% int_xor_code(1)
thf(fact_9243_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_9244_XOR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
         => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% XOR_upper
thf(fact_9245_test__bit__int__code_I1_J,axiom,
    ! [N: nat] :
      ~ ( bit_se1146084159140164899it_int @ zero_zero_int @ N ) ).

% test_bit_int_code(1)
thf(fact_9246_int__and__code_I1_J,axiom,
    ! [J: int] :
      ( ( bit_se725231765392027082nd_int @ zero_zero_int @ J )
      = zero_zero_int ) ).

% int_and_code(1)
thf(fact_9247_int__and__code_I2_J,axiom,
    ! [I: int] :
      ( ( bit_se725231765392027082nd_int @ I @ zero_zero_int )
      = zero_zero_int ) ).

% int_and_code(2)
thf(fact_9248_xor__int__rec,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L2: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) )
             != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_int_rec
thf(fact_9249_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_9250_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_9251_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_9252_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_9253_xor__nat__def,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M3: nat,N4: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).

% xor_nat_def
thf(fact_9254_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M3: nat,N4: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_9255_xor__nat__rec,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M3: nat,N4: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
             != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_nat_rec
thf(fact_9256_xor__Suc__0__eq,axiom,
    ! [N: nat] :
      ( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% xor_Suc_0_eq
thf(fact_9257_Suc__0__xor__eq,axiom,
    ! [N: nat] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% Suc_0_xor_eq
thf(fact_9258_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% or_nonnegative_int_iff
thf(fact_9259_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
        | ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% or_negative_int_iff
thf(fact_9260_or__minus__numerals_I6_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).

% or_minus_numerals(6)
thf(fact_9261_or__minus__numerals_I2_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).

% or_minus_numerals(2)
thf(fact_9262_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_9263_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_9264_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_9265_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_9266_and__minus__minus__numerals,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).

% and_minus_minus_numerals
thf(fact_9267_or__minus__minus__numerals,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).

% or_minus_minus_numerals
thf(fact_9268_int__or__code_I2_J,axiom,
    ! [I: int] :
      ( ( bit_se1409905431419307370or_int @ I @ zero_zero_int )
      = I ) ).

% int_or_code(2)
thf(fact_9269_int__or__code_I1_J,axiom,
    ! [J: int] :
      ( ( bit_se1409905431419307370or_int @ zero_zero_int @ J )
      = J ) ).

% int_or_code(1)
thf(fact_9270_or__nat__def,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M3: nat,N4: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).

% or_nat_def
thf(fact_9271_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L )
     => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L ) ) ) ).

% or_greater_eq
thf(fact_9272_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_9273_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_9274_or__not__numerals_I2_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).

% or_not_numerals(2)
thf(fact_9275_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_9276_or__not__numerals_I3_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).

% or_not_numerals(3)
thf(fact_9277_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_9278_or__not__numerals_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% or_not_numerals(6)
thf(fact_9279_OR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
         => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% OR_upper
thf(fact_9280_or__not__numerals_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).

% or_not_numerals(5)
thf(fact_9281_or__Suc__0__eq,axiom,
    ! [N: nat] :
      ( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% or_Suc_0_eq
thf(fact_9282_Suc__0__or__eq,axiom,
    ! [N: nat] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% Suc_0_or_eq
thf(fact_9283_or__nat__rec,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M3: nat,N4: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
              | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_nat_rec
thf(fact_9284_or__not__numerals_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).

% or_not_numerals(9)
thf(fact_9285_or__not__numerals_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).

% or_not_numerals(8)
thf(fact_9286_or__int__rec,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L2: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
              | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_int_rec
thf(fact_9287_or__int__unfold,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L2: int] :
          ( if_int
          @ ( ( K3
              = ( uminus_uminus_int @ one_one_int ) )
            | ( L2
              = ( uminus_uminus_int @ one_one_int ) ) )
          @ ( uminus_uminus_int @ one_one_int )
          @ ( if_int @ ( K3 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% or_int_unfold
thf(fact_9288_int__lsb__numeral_I1_J,axiom,
    ~ ( least_4859182151741483524sb_int @ zero_zero_int ) ).

% int_lsb_numeral(1)
thf(fact_9289_int__lsb__numeral_I2_J,axiom,
    least_4859182151741483524sb_int @ one_one_int ).

% int_lsb_numeral(2)
thf(fact_9290_int__lsb__numeral_I6_J,axiom,
    ! [W: num] :
      ~ ( least_4859182151741483524sb_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) ).

% int_lsb_numeral(6)
thf(fact_9291_int__lsb__numeral_I3_J,axiom,
    least_4859182151741483524sb_int @ ( numeral_numeral_int @ one ) ).

% int_lsb_numeral(3)
thf(fact_9292_int__lsb__numeral_I7_J,axiom,
    ! [W: num] : ( least_4859182151741483524sb_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) ).

% int_lsb_numeral(7)
thf(fact_9293_int__lsb__numeral_I4_J,axiom,
    least_4859182151741483524sb_int @ ( uminus_uminus_int @ one_one_int ) ).

% int_lsb_numeral(4)
thf(fact_9294_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_9295_int__lsb__numeral_I5_J,axiom,
    least_4859182151741483524sb_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) ).

% int_lsb_numeral(5)
thf(fact_9296_int__lsb__numeral_I9_J,axiom,
    ! [W: num] : ( least_4859182151741483524sb_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) ) ).

% int_lsb_numeral(9)
thf(fact_9297_sup__int__def,axiom,
    sup_sup_int = ord_max_int ).

% sup_int_def
thf(fact_9298_lsb__int__def,axiom,
    ( least_4859182151741483524sb_int
    = ( ^ [I4: int] : ( bit_se1146084159140164899it_int @ I4 @ zero_zero_nat ) ) ) ).

% lsb_int_def
thf(fact_9299_bin__last__conv__lsb,axiom,
    ( ( ^ [A4: int] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) )
    = least_4859182151741483524sb_int ) ).

% bin_last_conv_lsb
thf(fact_9300_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_9301_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_9302_powr__real__of__int,axiom,
    ! [X: real,N: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
            = ( power_power_real @ X @ ( nat2 @ N ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
            = ( inverse_inverse_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).

% powr_real_of_int
thf(fact_9303_max__enat__simps_I2_J,axiom,
    ! [Q3: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ Q3 @ zero_z5237406670263579293d_enat )
      = Q3 ) ).

% max_enat_simps(2)
thf(fact_9304_max__enat__simps_I3_J,axiom,
    ! [Q3: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q3 )
      = Q3 ) ).

% max_enat_simps(3)
thf(fact_9305_max__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( suc @ ( ord_max_nat @ M @ N ) ) ) ).

% max_Suc_Suc
thf(fact_9306_max__0R,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ N @ zero_zero_nat )
      = N ) ).

% max_0R
thf(fact_9307_max__0L,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N )
      = N ) ).

% max_0L
thf(fact_9308_max__nat_Oright__neutral,axiom,
    ! [A2: nat] :
      ( ( ord_max_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% max_nat.right_neutral
thf(fact_9309_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_9310_max__nat_Oleft__neutral,axiom,
    ! [A2: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A2 )
      = A2 ) ).

% max_nat.left_neutral
thf(fact_9311_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_9312_norm__ii,axiom,
    ( ( real_V1022390504157884413omplex @ imaginary_unit )
    = one_one_real ) ).

% norm_ii
thf(fact_9313_max__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K ) ) ) ) ).

% max_Suc_numeral
thf(fact_9314_max__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N ) ) ) ).

% max_numeral_Suc
thf(fact_9315_i__squared,axiom,
    ( ( times_times_complex @ imaginary_unit @ imaginary_unit )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% i_squared
thf(fact_9316_power2__i,axiom,
    ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power2_i
thf(fact_9317_i__even__power,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) ) ).

% i_even_power
thf(fact_9318_exp__pi__i,axiom,
    ( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% exp_pi_i
thf(fact_9319_exp__pi__i_H,axiom,
    ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% exp_pi_i'
thf(fact_9320_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q3: nat] :
      ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q3 ) )
      = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q3 ) ) ) ).

% nat_mult_max_right
thf(fact_9321_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q3: nat] :
      ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q3 )
      = ( ord_max_nat @ ( times_times_nat @ M @ Q3 ) @ ( times_times_nat @ N @ Q3 ) ) ) ).

% nat_mult_max_left
thf(fact_9322_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
thf(fact_9323_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q3: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q3 )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ Q3 ) @ ( plus_plus_nat @ N @ Q3 ) ) ) ).

% nat_add_max_left
thf(fact_9324_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q3: nat] :
      ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q3 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q3 ) ) ) ).

% nat_add_max_right
thf(fact_9325_complex__i__not__one,axiom,
    imaginary_unit != one_one_complex ).

% complex_i_not_one
thf(fact_9326_real__sqrt__inverse,axiom,
    ! [X: real] :
      ( ( sqrt @ ( inverse_inverse_real @ X ) )
      = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_inverse
thf(fact_9327_complex__i__not__zero,axiom,
    imaginary_unit != zero_zero_complex ).

% complex_i_not_zero
thf(fact_9328_nat__minus__add__max,axiom,
    ! [N: nat,M: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
      = ( ord_max_nat @ N @ M ) ) ).

% nat_minus_add_max
thf(fact_9329_divide__real__def,axiom,
    ( divide_divide_real
    = ( ^ [X2: real,Y2: real] : ( times_times_real @ X2 @ ( inverse_inverse_real @ Y2 ) ) ) ) ).

% divide_real_def
thf(fact_9330_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_9331_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_9332_imaginary__unit_Ocode,axiom,
    ( imaginary_unit
    = ( complex2 @ zero_zero_real @ one_one_real ) ) ).

% imaginary_unit.code
thf(fact_9333_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_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_forall__pos__mono__1,axiom,
    ! [P: real > $o,E2: real] :
      ( ! [D6: real,E: real] :
          ( ( ord_less_real @ D6 @ E )
         => ( ( P @ D6 )
           => ( P @ E ) ) )
     => ( ! [N2: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( P @ E2 ) ) ) ) ).

% forall_pos_mono_1
thf(fact_9336_real__arch__inverse,axiom,
    ! [E2: real] :
      ( ( ord_less_real @ zero_zero_real @ E2 )
      = ( ? [N4: nat] :
            ( ( N4 != zero_zero_nat )
            & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) )
            & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) @ E2 ) ) ) ) ).

% real_arch_inverse
thf(fact_9337_forall__pos__mono,axiom,
    ! [P: real > $o,E2: real] :
      ( ! [D6: real,E: real] :
          ( ( ord_less_real @ D6 @ E )
         => ( ( P @ D6 )
           => ( P @ E ) ) )
     => ( ! [N2: nat] :
            ( ( N2 != zero_zero_nat )
           => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( P @ E2 ) ) ) ) ).

% forall_pos_mono
thf(fact_9338_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_9339_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_9340_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_9341_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_9342_cmod__unit__one,axiom,
    ! [A2: real] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A2 ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A2 ) ) ) ) )
      = one_one_real ) ).

% cmod_unit_one
thf(fact_9343_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_9344_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_9345_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_9346_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_9347_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_9348_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M3: nat,N4: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_9349_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_9350_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_9351_Arg__ii,axiom,
    ( ( arg @ imaginary_unit )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% Arg_ii
thf(fact_9352_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_9353_drop__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% drop_bit_nonnegative_int_iff
thf(fact_9354_drop__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% drop_bit_negative_int_iff
thf(fact_9355_drop__bit__minus__one,axiom,
    ! [N: nat] :
      ( ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% drop_bit_minus_one
thf(fact_9356_csqrt__eq__0,axiom,
    ! [Z: complex] :
      ( ( ( csqrt @ Z )
        = zero_zero_complex )
      = ( Z = zero_zero_complex ) ) ).

% csqrt_eq_0
thf(fact_9357_csqrt__0,axiom,
    ( ( csqrt @ zero_zero_complex )
    = zero_zero_complex ) ).

% csqrt_0
thf(fact_9358_csqrt__eq__1,axiom,
    ! [Z: complex] :
      ( ( ( csqrt @ Z )
        = one_one_complex )
      = ( Z = one_one_complex ) ) ).

% csqrt_eq_1
thf(fact_9359_csqrt__1,axiom,
    ( ( csqrt @ one_one_complex )
    = one_one_complex ) ).

% csqrt_1
thf(fact_9360_norm__cis,axiom,
    ! [A2: real] :
      ( ( real_V1022390504157884413omplex @ ( cis @ A2 ) )
      = one_one_real ) ).

% norm_cis
thf(fact_9361_cis__zero,axiom,
    ( ( cis @ zero_zero_real )
    = one_one_complex ) ).

% cis_zero
thf(fact_9362_drop__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% drop_bit_Suc_minus_bit0
thf(fact_9363_drop__bit__of__Suc__0,axiom,
    ! [N: nat] :
      ( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).

% drop_bit_of_Suc_0
thf(fact_9364_cis__pi,axiom,
    ( ( cis @ pi )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% cis_pi
thf(fact_9365_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% drop_bit_numeral_minus_bit0
thf(fact_9366_drop__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).

% drop_bit_Suc_minus_bit1
thf(fact_9367_power2__csqrt,axiom,
    ! [Z: complex] :
      ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = Z ) ).

% power2_csqrt
thf(fact_9368_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).

% drop_bit_numeral_minus_bit1
thf(fact_9369_cis__pi__half,axiom,
    ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = imaginary_unit ) ).

% cis_pi_half
thf(fact_9370_cis__2pi,axiom,
    ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_complex ) ).

% cis_2pi
thf(fact_9371_sup__enat__def,axiom,
    sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).

% sup_enat_def
thf(fact_9372_drop__bit__int__code_I1_J,axiom,
    ! [I: int] :
      ( ( bit_se8568078237143864401it_int @ zero_zero_nat @ I )
      = I ) ).

% drop_bit_int_code(1)
thf(fact_9373_cis__Arg,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( cis @ ( arg @ Z ) )
        = ( sgn_sgn_complex @ Z ) ) ) ).

% cis_Arg
thf(fact_9374_cis__neq__zero,axiom,
    ! [A2: real] :
      ( ( cis @ A2 )
     != zero_zero_complex ) ).

% cis_neq_zero
thf(fact_9375_drop__bit__int__code_I2_J,axiom,
    ! [N: nat] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ zero_zero_int )
      = zero_zero_int ) ).

% drop_bit_int_code(2)
thf(fact_9376_DeMoivre,axiom,
    ! [A2: real,N: nat] :
      ( ( power_power_complex @ ( cis @ A2 ) @ N )
      = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A2 ) ) ) ).

% DeMoivre
thf(fact_9377_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_9378_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_9379_Arg__zero,axiom,
    ( ( arg @ zero_zero_complex )
    = zero_zero_real ) ).

% Arg_zero
thf(fact_9380_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_9381_drop__bit__int__def,axiom,
    ( bit_se8568078237143864401it_int
    = ( ^ [N4: nat,K3: int] : ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% drop_bit_int_def
thf(fact_9382_drop__bit__nat__def,axiom,
    ( bit_se8570568707652914677it_nat
    = ( ^ [N4: nat,M3: nat] : ( divide_divide_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% drop_bit_nat_def
thf(fact_9383_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_9384_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_9385_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_9386_shiftr__Suc__0,axiom,
    ! [N: nat] :
      ( ( bit_Sh2154871086232339855tr_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).

% shiftr_Suc_0
thf(fact_9387_sinh__real__zero__iff,axiom,
    ! [X: real] :
      ( ( ( sinh_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% sinh_real_zero_iff
thf(fact_9388_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_9389_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_9390_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_9391_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_9392_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_9393_atMost__0,axiom,
    ( ( set_ord_atMost_nat @ zero_zero_nat )
    = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).

% atMost_0
thf(fact_9394_sinh__less__cosh__real,axiom,
    ! [X: real] : ( ord_less_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).

% sinh_less_cosh_real
thf(fact_9395_sum__choose__upper,axiom,
    ! [M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ K3 @ M )
        @ ( set_ord_atMost_nat @ N ) )
      = ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).

% sum_choose_upper
thf(fact_9396_cosh__real__nonzero,axiom,
    ! [X: real] :
      ( ( cosh_real @ X )
     != zero_zero_real ) ).

% cosh_real_nonzero
thf(fact_9397_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ M ) ) ).

% choose_rising_sum(2)
thf(fact_9398_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).

% choose_rising_sum(1)
thf(fact_9399_sum__choose__lower,axiom,
    ! [R3: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R3 @ K3 ) @ K3 )
        @ ( set_ord_atMost_nat @ N ) )
      = ( binomial @ ( suc @ ( plus_plus_nat @ R3 @ N ) ) @ N ) ) ).

% sum_choose_lower
thf(fact_9400_sum__choose__diagonal,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3542108847815614940at_nat
          @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
          @ ( set_ord_atMost_nat @ M ) )
        = ( binomial @ ( suc @ N ) @ M ) ) ) ).

% sum_choose_diagonal
thf(fact_9401_vandermonde,axiom,
    ! [M: nat,N: nat,R3: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M @ K3 ) @ ( binomial @ N @ ( minus_minus_nat @ R3 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ R3 ) )
      = ( binomial @ ( plus_plus_nat @ M @ N ) @ R3 ) ) ).

% vandermonde
thf(fact_9402_cosh__real__pos,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X ) ) ).

% cosh_real_pos
thf(fact_9403_cosh__real__nonneg,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X ) ) ).

% cosh_real_nonneg
thf(fact_9404_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_9405_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_9406_atMost__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ K ) )
      = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).

% atMost_Suc
thf(fact_9407_cosh__real__ge__1,axiom,
    ! [X: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X ) ) ).

% cosh_real_ge_1
thf(fact_9408_choose__row__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat @ ( binomial @ N ) @ ( set_ord_atMost_nat @ N ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% choose_row_sum
thf(fact_9409_binomial,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A2 @ B2 ) @ N )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K3 ) ) @ ( power_power_nat @ A2 @ K3 ) ) @ ( power_power_nat @ B2 @ ( minus_minus_nat @ N @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% binomial
thf(fact_9410_polynomial__product__nat,axiom,
    ! [M: nat,A2: nat > nat,N: nat,B2: nat > nat,X: nat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ M @ I3 )
         => ( ( A2 @ I3 )
            = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N @ J2 )
           => ( ( B2 @ J2 )
              = zero_zero_nat ) )
       => ( ( times_times_nat
            @ ( groups3542108847815614940at_nat
              @ ^ [I4: nat] : ( times_times_nat @ ( A2 @ I4 ) @ ( power_power_nat @ X @ I4 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups3542108847815614940at_nat
              @ ^ [J3: nat] : ( times_times_nat @ ( B2 @ J3 ) @ ( power_power_nat @ X @ J3 ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( 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 @ N ) ) ) ) ) ) ).

% polynomial_product_nat
thf(fact_9411_choose__square__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( power_power_nat @ ( binomial @ N @ K3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).

% choose_square_sum
thf(fact_9412_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_9413_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_9414_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_9415_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_9416_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_9417_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_9418_choose__linear__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( times_times_nat @ I4 @ ( binomial @ N @ I4 ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( times_times_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% choose_linear_sum
thf(fact_9419_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q6: nat] : ( ord_less_nat @ Q6 @ N ) ) ) ) ).

% mask_eq_sum_exp_nat
thf(fact_9420_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_9421_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_9422_sum__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( groups7754918857620584856omplex
          @ ^ [X2: complex] : X2
          @ ( collect_complex
            @ ^ [Z4: complex] :
                ( ( power_power_complex @ Z4 @ N )
                = C ) ) )
        = zero_zero_complex ) ) ).

% sum_nth_roots
thf(fact_9423_sum__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( groups7754918857620584856omplex
          @ ^ [X2: complex] : X2
          @ ( collect_complex
            @ ^ [Z4: complex] :
                ( ( power_power_complex @ Z4 @ N )
                = one_one_complex ) ) )
        = zero_zero_complex ) ) ).

% sum_roots_unity
thf(fact_9424_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ? [T3: real] :
            ( ( ord_less_real @ zero_zero_real @ T3 )
            & ( ord_less_real @ T3 @ X )
            & ( ( cos_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_cos_expansion2
thf(fact_9425_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ? [T3: real] :
            ( ( ord_less_real @ X @ T3 )
            & ( ord_less_real @ T3 @ zero_zero_real )
            & ( ( cos_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
thf(fact_9426_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ? [T3: real] :
            ( ( ord_less_real @ zero_zero_real @ T3 )
            & ( ord_less_real @ T3 @ X )
            & ( ( sin_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_sin_expansion3
thf(fact_9427_lessThan__0,axiom,
    ( ( set_ord_lessThan_nat @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% lessThan_0
thf(fact_9428_sumr__cos__zero__one,axiom,
    ! [N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ zero_zero_real @ M3 ) )
        @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = one_one_real ) ).

% sumr_cos_zero_one
thf(fact_9429_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( ( set_ord_lessThan_nat @ N )
        = bot_bot_set_nat )
      = ( N = zero_zero_nat ) ) ).

% lessThan_empty_iff
thf(fact_9430_lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).

% lessThan_Suc
thf(fact_9431_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_9432_Maclaurin__lemma,axiom,
    ! [H2: real,F: real > real,J: nat > real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ? [B5: real] :
          ( ( F @ H2 )
          = ( plus_plus_real
            @ ( groups6591440286371151544t_real
              @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
              @ ( set_ord_lessThan_nat @ N ) )
            @ ( times_times_real @ B5 @ ( divide_divide_real @ ( power_power_real @ H2 @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).

% Maclaurin_lemma
thf(fact_9433_sum__split__even__odd,axiom,
    ! [F: nat > real,G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( F @ I4 ) @ ( G @ I4 ) )
        @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [I4: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ one_one_nat ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum_split_even_odd
thf(fact_9434_Maclaurin__exp__le,axiom,
    ! [X: real,N: nat] :
    ? [T3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
      & ( ( exp_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% Maclaurin_exp_le
thf(fact_9435_Maclaurin__sin__bound,axiom,
    ! [X: real,N: nat] :
      ( ord_less_eq_real
      @ ( abs_abs_real
        @ ( minus_minus_real @ ( sin_real @ X )
          @ ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) )
      @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X ) @ N ) ) ) ).

% Maclaurin_sin_bound
thf(fact_9436_sum__pos__lt__pair,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ! [D6: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D6 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D6 ) @ one_one_nat ) ) ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_pos_lt_pair
thf(fact_9437_Maclaurin__exp__lt,axiom,
    ! [X: real,N: nat] :
      ( ( X != zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ? [T3: real] :
            ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T3 ) )
            & ( ord_less_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
            & ( ( exp_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_exp_lt
thf(fact_9438_Maclaurin__sin__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T3: real] :
      ( ( sin_real @ X )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
          @ ( set_ord_lessThan_nat @ N ) )
        @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ).

% Maclaurin_sin_expansion
thf(fact_9439_Maclaurin__sin__expansion2,axiom,
    ! [X: real,N: nat] :
    ? [T3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
      & ( ( sin_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% Maclaurin_sin_expansion2
thf(fact_9440_Maclaurin__cos__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
      & ( ( cos_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% Maclaurin_cos_expansion
thf(fact_9441_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [T3: real] :
          ( ( ord_less_real @ zero_zero_real @ T3 )
          & ( ord_less_eq_real @ T3 @ X )
          & ( ( sin_real @ X )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
                @ ( set_ord_lessThan_nat @ N ) )
              @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).

% Maclaurin_sin_expansion4
thf(fact_9442_bij__betw__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( bij_betw_nat_complex
        @ ^ [K3: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K3 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        @ ( set_ord_lessThan_nat @ N )
        @ ( collect_complex
          @ ^ [Z4: complex] :
              ( ( power_power_complex @ Z4 @ N )
              = one_one_complex ) ) ) ) ).

% bij_betw_roots_unity
thf(fact_9443_ex__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M3: nat] :
            ( ( ord_less_eq_nat @ M3 @ N )
            & ( P @ M3 ) ) )
      = ( ? [X2: nat] :
            ( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
            & ( P @ X2 ) ) ) ) ).

% ex_nat_less
thf(fact_9444_all__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ M3 @ N )
           => ( P @ M3 ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
           => ( P @ X2 ) ) ) ) ).

% all_nat_less
thf(fact_9445_atMost__atLeast0,axiom,
    ( set_ord_atMost_nat
    = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).

% atMost_atLeast0
thf(fact_9446_atLeast0__atMost__Suc,axiom,
    ! [N: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% atLeast0_atMost_Suc
thf(fact_9447_atLeastAtMost__insertL,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
        = ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% atLeastAtMost_insertL
thf(fact_9448_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
        = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).

% atLeastAtMostSuc_conv
thf(fact_9449_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( set_or1269000886237332187st_nat @ M @ N )
        = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).

% Icc_eq_insert_lb_nat
thf(fact_9450_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_atMost_eq_remove0
thf(fact_9451_gauss__sum__nat,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_nat
thf(fact_9452_arith__series__nat,axiom,
    ! [A2: nat,D: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ A2 @ ( times_times_nat @ I4 @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) @ ( times_times_nat @ N @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series_nat
thf(fact_9453_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ ( set_or1269000886237332187st_nat @ M @ N ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Icc_nat
thf(fact_9454_bset_I1_J,axiom,
    ! [D3: int,B: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B )
                 => ( X3
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( minus_minus_int @ X3 @ D3 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B )
                   => ( X3
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( minus_minus_int @ X3 @ D3 ) ) ) )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B )
                   => ( X6
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
             => ( ( P @ ( minus_minus_int @ X6 @ D3 ) )
                & ( Q @ ( minus_minus_int @ X6 @ D3 ) ) ) ) ) ) ) ).

% bset(1)
thf(fact_9455_bset_I2_J,axiom,
    ! [D3: int,B: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B )
                 => ( X3
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( minus_minus_int @ X3 @ D3 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B )
                   => ( X3
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( minus_minus_int @ X3 @ D3 ) ) ) )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B )
                   => ( X6
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
             => ( ( P @ ( minus_minus_int @ X6 @ D3 ) )
                | ( Q @ ( minus_minus_int @ X6 @ D3 ) ) ) ) ) ) ) ).

% bset(2)
thf(fact_9456_aset_I1_J,axiom,
    ! [D3: int,A: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A )
                 => ( X3
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( plus_plus_int @ X3 @ D3 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A )
                   => ( X3
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( plus_plus_int @ X3 @ D3 ) ) ) )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A )
                   => ( X6
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
             => ( ( P @ ( plus_plus_int @ X6 @ D3 ) )
                & ( Q @ ( plus_plus_int @ X6 @ D3 ) ) ) ) ) ) ) ).

% aset(1)
thf(fact_9457_aset_I2_J,axiom,
    ! [D3: int,A: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A )
                 => ( X3
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( plus_plus_int @ X3 @ D3 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A )
                   => ( X3
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( plus_plus_int @ X3 @ D3 ) ) ) )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A )
                   => ( X6
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
             => ( ( P @ ( plus_plus_int @ X6 @ D3 ) )
                | ( Q @ ( plus_plus_int @ X6 @ D3 ) ) ) ) ) ) ) ).

% aset(2)
thf(fact_9458_aset_I10_J,axiom,
    ! [D: int,D3: int,A: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D3 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A )
                 => ( X6
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ T ) )
           => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X6 @ D3 ) @ T ) ) ) ) ) ).

% aset(10)
thf(fact_9459_aset_I9_J,axiom,
    ! [D: int,D3: int,A: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D3 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A )
                 => ( X6
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ T ) )
           => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X6 @ D3 ) @ T ) ) ) ) ) ).

% aset(9)
thf(fact_9460_bset_I10_J,axiom,
    ! [D: int,D3: int,B: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D3 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B )
                 => ( X6
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ T ) )
           => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X6 @ D3 ) @ T ) ) ) ) ) ).

% bset(10)
thf(fact_9461_bset_I9_J,axiom,
    ! [D: int,D3: int,B: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D3 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B )
                 => ( X6
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ T ) )
           => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X6 @ D3 ) @ T ) ) ) ) ) ).

% bset(9)
thf(fact_9462_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
     => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
        = ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M @ N ) ) ) ) ).

% atLeastAtMostPlus1_int_conv
thf(fact_9463_bset_I3_J,axiom,
    ! [D3: int,T: int,B: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B )
                   => ( X6
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X6 = T )
             => ( ( minus_minus_int @ X6 @ D3 )
                = T ) ) ) ) ) ).

% bset(3)
thf(fact_9464_bset_I4_J,axiom,
    ! [D3: int,T: int,B: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ T @ B )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B )
                   => ( X6
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X6 != T )
             => ( ( minus_minus_int @ X6 @ D3 )
               != T ) ) ) ) ) ).

% bset(4)
thf(fact_9465_bset_I5_J,axiom,
    ! [D3: int,B: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B )
                 => ( X6
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ X6 @ T )
           => ( ord_less_int @ ( minus_minus_int @ X6 @ D3 ) @ T ) ) ) ) ).

% bset(5)
thf(fact_9466_bset_I7_J,axiom,
    ! [D3: int,T: int,B: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ T @ B )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B )
                   => ( X6
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ T @ X6 )
             => ( ord_less_int @ T @ ( minus_minus_int @ X6 @ D3 ) ) ) ) ) ) ).

% bset(7)
thf(fact_9467_aset_I3_J,axiom,
    ! [D3: int,T: int,A: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A )
                   => ( X6
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X6 = T )
             => ( ( plus_plus_int @ X6 @ D3 )
                = T ) ) ) ) ) ).

% aset(3)
thf(fact_9468_aset_I4_J,axiom,
    ! [D3: int,T: int,A: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ T @ A )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A )
                   => ( X6
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X6 != T )
             => ( ( plus_plus_int @ X6 @ D3 )
               != T ) ) ) ) ) ).

% aset(4)
thf(fact_9469_aset_I5_J,axiom,
    ! [D3: int,T: int,A: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ T @ A )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A )
                   => ( X6
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ X6 @ T )
             => ( ord_less_int @ ( plus_plus_int @ X6 @ D3 ) @ T ) ) ) ) ) ).

% aset(5)
thf(fact_9470_aset_I7_J,axiom,
    ! [D3: int,A: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A )
                 => ( X6
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ T @ X6 )
           => ( ord_less_int @ T @ ( plus_plus_int @ X6 @ D3 ) ) ) ) ) ).

% aset(7)
thf(fact_9471_periodic__finite__ex,axiom,
    ! [D: int,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K2: int] :
            ( ( P @ X3 )
            = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ( ? [X8: int] : ( P @ X8 ) )
          = ( ? [X2: int] :
                ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                & ( P @ X2 ) ) ) ) ) ) ).

% periodic_finite_ex
thf(fact_9472_fact__eq__fact__times,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri1408675320244567234ct_nat @ M )
        = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
          @ ( groups708209901874060359at_nat
            @ ^ [X2: nat] : X2
            @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).

% fact_eq_fact_times
thf(fact_9473_simp__from__to,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I4: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I4 ) @ bot_bot_set_int @ ( insert_int @ I4 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ) ).

% simp_from_to
thf(fact_9474_bset_I6_J,axiom,
    ! [D3: int,B: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B )
                 => ( X6
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ X6 @ T )
           => ( ord_less_eq_int @ ( minus_minus_int @ X6 @ D3 ) @ T ) ) ) ) ).

% bset(6)
thf(fact_9475_bset_I8_J,axiom,
    ! [D3: int,T: int,B: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B )
                   => ( X6
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ T @ X6 )
             => ( ord_less_eq_int @ T @ ( minus_minus_int @ X6 @ D3 ) ) ) ) ) ) ).

% bset(8)
thf(fact_9476_aset_I6_J,axiom,
    ! [D3: int,T: int,A: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A )
                   => ( X6
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ X6 @ T )
             => ( ord_less_eq_int @ ( plus_plus_int @ X6 @ D3 ) @ T ) ) ) ) ) ).

% aset(6)
thf(fact_9477_aset_I8_J,axiom,
    ! [D3: int,A: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A )
                 => ( X6
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ T @ X6 )
           => ( ord_less_eq_int @ T @ ( plus_plus_int @ X6 @ D3 ) ) ) ) ) ).

% aset(8)
thf(fact_9478_cppi,axiom,
    ! [D3: int,P: int > $o,P5: int > $o,A: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z5 @ X3 )
           => ( ( P @ X3 )
              = ( P5 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                 => ! [Xb2: int] :
                      ( ( member_int @ Xb2 @ A )
                     => ( X3
                       != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
             => ( ( P @ X3 )
               => ( P @ ( plus_plus_int @ X3 @ D3 ) ) ) )
         => ( ! [X3: int,K2: int] :
                ( ( P5 @ X3 )
                = ( P5 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
           => ( ( ? [X8: int] : ( P @ X8 ) )
              = ( ? [X2: int] :
                    ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                    & ( P5 @ X2 ) )
                | ? [X2: int] :
                    ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                    & ? [Y2: int] :
                        ( ( member_int @ Y2 @ A )
                        & ( P @ ( minus_minus_int @ Y2 @ X2 ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_9479_cpmi,axiom,
    ! [D3: int,P: int > $o,P5: int > $o,B: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z5 )
           => ( ( P @ X3 )
              = ( P5 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                 => ! [Xb2: int] :
                      ( ( member_int @ Xb2 @ B )
                     => ( X3
                       != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
             => ( ( P @ X3 )
               => ( P @ ( minus_minus_int @ X3 @ D3 ) ) ) )
         => ( ! [X3: int,K2: int] :
                ( ( P5 @ X3 )
                = ( P5 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
           => ( ( ? [X8: int] : ( P @ X8 ) )
              = ( ? [X2: int] :
                    ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                    & ( P5 @ X2 ) )
                | ? [X2: int] :
                    ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                    & ? [Y2: int] :
                        ( ( member_int @ Y2 @ B )
                        & ( P @ ( plus_plus_int @ Y2 @ X2 ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_9480_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
        = ( groups708209901874060359at_nat
          @ ^ [X2: nat] : X2
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).

% fact_div_fact
thf(fact_9481_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_eq_int @ M @ N )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X2: int] : X2
          @ ( set_or1266510415728281911st_int @ M @ N ) )
        = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N @ ( plus_plus_int @ N @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% Sum_Icc_int
thf(fact_9482_bij__betw__nth__root__unity,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
          @ ( collect_complex
            @ ^ [Z4: complex] :
                ( ( power_power_complex @ Z4 @ N )
                = one_one_complex ) )
          @ ( collect_complex
            @ ^ [Z4: complex] :
                ( ( power_power_complex @ Z4 @ N )
                = C ) ) ) ) ) ).

% bij_betw_nth_root_unity
thf(fact_9483_Arg__def,axiom,
    ( arg
    = ( ^ [Z4: complex] :
          ( if_real @ ( Z4 = zero_zero_complex ) @ zero_zero_real
          @ ( fChoice_real
            @ ^ [A4: real] :
                ( ( ( sgn_sgn_complex @ Z4 )
                  = ( cis @ A4 ) )
                & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A4 )
                & ( ord_less_eq_real @ A4 @ pi ) ) ) ) ) ) ).

% Arg_def
thf(fact_9484_cis__multiple__2pi,axiom,
    ! [N: real] :
      ( ( member_real @ N @ ring_1_Ints_real )
     => ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
        = one_one_complex ) ) ).

% cis_multiple_2pi
thf(fact_9485_real__root__zero,axiom,
    ! [N: nat] :
      ( ( root @ N @ zero_zero_real )
      = zero_zero_real ) ).

% real_root_zero
thf(fact_9486_real__root__Suc__0,axiom,
    ! [X: real] :
      ( ( root @ ( suc @ zero_zero_nat ) @ X )
      = X ) ).

% real_root_Suc_0
thf(fact_9487_real__root__eq__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X )
          = ( root @ N @ Y ) )
        = ( X = Y ) ) ) ).

% real_root_eq_iff
thf(fact_9488_root__0,axiom,
    ! [X: real] :
      ( ( root @ zero_zero_nat @ X )
      = zero_zero_real ) ).

% root_0
thf(fact_9489_real__root__eq__0__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X )
          = zero_zero_real )
        = ( X = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
thf(fact_9490_real__root__less__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
        = ( ord_less_real @ X @ Y ) ) ) ).

% real_root_less_iff
thf(fact_9491_real__root__le__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% real_root_le_iff
thf(fact_9492_real__root__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ one_one_real )
        = one_one_real ) ) ).

% real_root_one
thf(fact_9493_real__root__eq__1__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X )
          = one_one_real )
        = ( X = one_one_real ) ) ) ).

% real_root_eq_1_iff
thf(fact_9494_real__root__gt__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y ) )
        = ( ord_less_real @ zero_zero_real @ Y ) ) ) ).

% real_root_gt_0_iff
thf(fact_9495_real__root__lt__0__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X ) @ zero_zero_real )
        = ( ord_less_real @ X @ zero_zero_real ) ) ) ).

% real_root_lt_0_iff
thf(fact_9496_real__root__le__0__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X ) @ zero_zero_real )
        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).

% real_root_le_0_iff
thf(fact_9497_real__root__ge__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y ) )
        = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).

% real_root_ge_0_iff
thf(fact_9498_real__root__gt__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y ) )
        = ( ord_less_real @ one_one_real @ Y ) ) ) ).

% real_root_gt_1_iff
thf(fact_9499_real__root__lt__1__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X ) @ one_one_real )
        = ( ord_less_real @ X @ one_one_real ) ) ) ).

% real_root_lt_1_iff
thf(fact_9500_real__root__le__1__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X ) @ one_one_real )
        = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).

% real_root_le_1_iff
thf(fact_9501_real__root__ge__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y ) )
        = ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).

% real_root_ge_1_iff
thf(fact_9502_real__root__pow__pos2,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( root @ N @ X ) @ N )
          = X ) ) ) ).

% real_root_pow_pos2
thf(fact_9503_real__root__divide,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( root @ N @ ( divide_divide_real @ X @ Y ) )
      = ( divide_divide_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ).

% real_root_divide
thf(fact_9504_real__root__minus,axiom,
    ! [N: nat,X: real] :
      ( ( root @ N @ ( uminus_uminus_real @ X ) )
      = ( uminus_uminus_real @ ( root @ N @ X ) ) ) ).

% real_root_minus
thf(fact_9505_real__root__commute,axiom,
    ! [M: nat,N: nat,X: real] :
      ( ( root @ M @ ( root @ N @ X ) )
      = ( root @ N @ ( root @ M @ X ) ) ) ).

% real_root_commute
thf(fact_9506_real__root__mult__exp,axiom,
    ! [M: nat,N: nat,X: real] :
      ( ( root @ ( times_times_nat @ M @ N ) @ X )
      = ( root @ M @ ( root @ N @ X ) ) ) ).

% real_root_mult_exp
thf(fact_9507_real__root__mult,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( root @ N @ ( times_times_real @ X @ Y ) )
      = ( times_times_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ).

% real_root_mult
thf(fact_9508_real__root__inverse,axiom,
    ! [N: nat,X: real] :
      ( ( root @ N @ ( inverse_inverse_real @ X ) )
      = ( inverse_inverse_real @ ( root @ N @ X ) ) ) ).

% real_root_inverse
thf(fact_9509_real__root__pos__pos__le,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ).

% real_root_pos_pos_le
thf(fact_9510_prod__int__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ J ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X2: int] : X2
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).

% prod_int_eq
thf(fact_9511_real__root__less__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ X @ Y )
       => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).

% real_root_less_mono
thf(fact_9512_real__root__le__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).

% real_root_le_mono
thf(fact_9513_real__root__power,axiom,
    ! [N: nat,X: real,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( power_power_real @ X @ K ) )
        = ( power_power_real @ ( root @ N @ X ) @ K ) ) ) ).

% real_root_power
thf(fact_9514_real__root__abs,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( abs_abs_real @ X ) )
        = ( abs_abs_real @ ( root @ N @ X ) ) ) ) ).

% real_root_abs
thf(fact_9515_sgn__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( sgn_sgn_real @ ( root @ N @ X ) )
        = ( sgn_sgn_real @ X ) ) ) ).

% sgn_root
thf(fact_9516_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X2: int] : X2
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_9517_real__root__gt__zero,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).

% real_root_gt_zero
thf(fact_9518_real__root__strict__decreasing,axiom,
    ! [N: nat,N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ N @ N3 )
       => ( ( ord_less_real @ one_one_real @ X )
         => ( ord_less_real @ ( root @ N3 @ X ) @ ( root @ N @ X ) ) ) ) ) ).

% real_root_strict_decreasing
thf(fact_9519_sqrt__def,axiom,
    ( sqrt
    = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% sqrt_def
thf(fact_9520_root__abs__power,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y @ N ) ) )
        = ( abs_abs_real @ Y ) ) ) ).

% root_abs_power
thf(fact_9521_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_9522_real__root__pos__pos,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).

% real_root_pos_pos
thf(fact_9523_real__root__strict__increasing,axiom,
    ! [N: nat,N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ N @ N3 )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ X @ one_one_real )
           => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N3 @ X ) ) ) ) ) ) ).

% real_root_strict_increasing
thf(fact_9524_real__root__decreasing,axiom,
    ! [N: nat,N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ( ord_less_eq_real @ one_one_real @ X )
         => ( ord_less_eq_real @ ( root @ N3 @ X ) @ ( root @ N @ X ) ) ) ) ) ).

% real_root_decreasing
thf(fact_9525_odd__real__root__pow,axiom,
    ! [N: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( root @ N @ X ) @ N )
        = X ) ) ).

% odd_real_root_pow
thf(fact_9526_odd__real__root__unique,axiom,
    ! [N: nat,Y: real,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ( power_power_real @ Y @ N )
          = X )
       => ( ( root @ N @ X )
          = Y ) ) ) ).

% odd_real_root_unique
thf(fact_9527_odd__real__root__power__cancel,axiom,
    ! [N: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( root @ N @ ( power_power_real @ X @ N ) )
        = X ) ) ).

% odd_real_root_power_cancel
thf(fact_9528_real__root__pow__pos,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( root @ N @ X ) @ N )
          = X ) ) ) ).

% real_root_pow_pos
thf(fact_9529_real__root__power__cancel,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( root @ N @ ( power_power_real @ X @ N ) )
          = X ) ) ) ).

% real_root_power_cancel
thf(fact_9530_real__root__pos__unique,axiom,
    ! [N: nat,Y: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( power_power_real @ Y @ N )
            = X )
         => ( ( root @ N @ X )
            = Y ) ) ) ) ).

% real_root_pos_unique
thf(fact_9531_real__root__increasing,axiom,
    ! [N: nat,N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ N3 )
       => ( ( ord_less_eq_real @ zero_zero_real @ X )
         => ( ( ord_less_eq_real @ X @ one_one_real )
           => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N3 @ X ) ) ) ) ) ) ).

% real_root_increasing
thf(fact_9532_sgn__power__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X ) ) @ N ) )
        = X ) ) ).

% sgn_power_root
thf(fact_9533_root__sgn__power,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) ) )
        = Y ) ) ).

% root_sgn_power
thf(fact_9534_log__root,axiom,
    ! [N: nat,A2: real,B2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A2 )
       => ( ( log @ B2 @ ( root @ N @ A2 ) )
          = ( divide_divide_real @ ( log @ B2 @ A2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_root
thf(fact_9535_log__base__root,axiom,
    ! [N: nat,B2: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ( log @ ( root @ N @ B2 ) @ X )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ X ) ) ) ) ) ).

% log_base_root
thf(fact_9536_ln__root,axiom,
    ! [N: nat,B2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ( ln_ln_real @ ( root @ N @ B2 ) )
          = ( divide_divide_real @ ( ln_ln_real @ B2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% ln_root
thf(fact_9537_split__root,axiom,
    ! [P: real > $o,N: nat,X: real] :
      ( ( P @ ( root @ N @ X ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ zero_zero_real ) )
        & ( ( ord_less_nat @ zero_zero_nat @ N )
         => ! [Y2: real] :
              ( ( ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N ) )
                = X )
             => ( P @ Y2 ) ) ) ) ) ).

% split_root
thf(fact_9538_sin__integer__2pi,axiom,
    ! [N: real] :
      ( ( member_real @ N @ ring_1_Ints_real )
     => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
        = zero_zero_real ) ) ).

% sin_integer_2pi
thf(fact_9539_cos__integer__2pi,axiom,
    ! [N: real] :
      ( ( member_real @ N @ ring_1_Ints_real )
     => ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
        = one_one_real ) ) ).

% cos_integer_2pi
thf(fact_9540_root__powr__inverse,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( root @ N @ X )
          = ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).

% root_powr_inverse
thf(fact_9541_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_9542_concat__bit__Suc,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
      = ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).

% concat_bit_Suc
thf(fact_9543_concat__bit__0,axiom,
    ! [K: int,L: int] :
      ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
      = L ) ).

% concat_bit_0
thf(fact_9544_concat__bit__of__zero__2,axiom,
    ! [N: nat,K: int] :
      ( ( bit_concat_bit @ N @ K @ zero_zero_int )
      = ( bit_se2923211474154528505it_int @ N @ K ) ) ).

% concat_bit_of_zero_2
thf(fact_9545_concat__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L ) )
      = ( ord_less_eq_int @ zero_zero_int @ L ) ) ).

% concat_bit_nonnegative_iff
thf(fact_9546_concat__bit__negative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L ) @ zero_zero_int )
      = ( ord_less_int @ L @ zero_zero_int ) ) ).

% concat_bit_negative_iff
thf(fact_9547_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N )
      = ( ( ( ord_less_nat @ N @ M )
          & ( bit_se1146084159140164899it_int @ K @ N ) )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% bit_concat_bit_iff
thf(fact_9548_norm__assertion__simps_I2_J,axiom,
    ! [A2: assn] :
      ( ( times_times_assn @ A2 @ one_one_assn )
      = A2 ) ).

% norm_assertion_simps(2)
thf(fact_9549_norm__assertion__simps_I1_J,axiom,
    ! [A2: assn] :
      ( ( times_times_assn @ one_one_assn @ A2 )
      = A2 ) ).

% norm_assertion_simps(1)
thf(fact_9550_norm__assertion__simps_I5_J,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ bot_bot_assn @ X )
      = X ) ).

% norm_assertion_simps(5)
thf(fact_9551_norm__assertion__simps_I6_J,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ bot_bot_assn )
      = X ) ).

% norm_assertion_simps(6)
thf(fact_9552_xor__num_Ocases,axiom,
    ! [X: product_prod_num_num] :
      ( ( X
       != ( product_Pair_num_num @ one @ one ) )
     => ( ! [N2: num] :
            ( X
           != ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) )
       => ( ! [N2: num] :
              ( X
             != ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) )
         => ( ! [M4: num] :
                ( X
               != ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) )
           => ( ! [M4: num,N2: num] :
                  ( X
                 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) )
             => ( ! [M4: num,N2: num] :
                    ( X
                   != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) )
               => ( ! [M4: num] :
                      ( X
                     != ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) )
                 => ( ! [M4: num,N2: num] :
                        ( X
                       != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) )
                   => ~ ! [M4: num,N2: num] :
                          ( X
                         != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.cases
thf(fact_9553_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_9554_eucl__rel__int__by0,axiom,
    ! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).

% eucl_rel_int_by0
thf(fact_9555_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q3: int] :
      ( ( L != zero_zero_int )
     => ( ( K
          = ( times_times_int @ Q3 @ L ) )
       => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ zero_zero_int ) ) ) ) ).

% eucl_rel_int_dividesI
thf(fact_9556_divmod__int__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M3: num,N4: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N4 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N4 ) ) ) ) ) ).

% divmod_int_def
thf(fact_9557_divmod_H__nat__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M3: num,N4: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N4 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N4 ) ) ) ) ) ).

% divmod'_nat_def
thf(fact_9558_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_9559_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q3: int,R3: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R3 ) )
      = ( ( K
          = ( plus_plus_int @ ( times_times_int @ L @ Q3 ) @ R3 ) )
        & ( ( ord_less_int @ zero_zero_int @ L )
         => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
            & ( ord_less_int @ R3 @ L ) ) )
        & ( ~ ( ord_less_int @ zero_zero_int @ L )
         => ( ( ( ord_less_int @ L @ zero_zero_int )
             => ( ( ord_less_int @ L @ R3 )
                & ( ord_less_eq_int @ R3 @ zero_zero_int ) ) )
            & ( ~ ( ord_less_int @ L @ zero_zero_int )
             => ( Q3 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_9560_eucl__rel__int__remainderI,axiom,
    ! [R3: int,L: int,K: int,Q3: int] :
      ( ( ( sgn_sgn_int @ R3 )
        = ( sgn_sgn_int @ L ) )
     => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ L ) )
       => ( ( K
            = ( plus_plus_int @ ( times_times_int @ Q3 @ L ) @ R3 ) )
         => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R3 ) ) ) ) ) ).

% eucl_rel_int_remainderI
thf(fact_9561_eucl__rel__int_Osimps,axiom,
    ( eucl_rel_int
    = ( ^ [A1: int,A22: int,A32: product_prod_int_int] :
          ( ? [K3: int] :
              ( ( A1 = K3 )
              & ( A22 = zero_zero_int )
              & ( A32
                = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
          | ? [L2: int,K3: int,Q6: int] :
              ( ( A1 = K3 )
              & ( A22 = L2 )
              & ( A32
                = ( product_Pair_int_int @ Q6 @ zero_zero_int ) )
              & ( L2 != zero_zero_int )
              & ( K3
                = ( times_times_int @ Q6 @ L2 ) ) )
          | ? [R5: int,L2: int,K3: int,Q6: int] :
              ( ( A1 = K3 )
              & ( A22 = L2 )
              & ( A32
                = ( product_Pair_int_int @ Q6 @ R5 ) )
              & ( ( sgn_sgn_int @ R5 )
                = ( sgn_sgn_int @ L2 ) )
              & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L2 ) )
              & ( K3
                = ( plus_plus_int @ ( times_times_int @ Q6 @ L2 ) @ R5 ) ) ) ) ) ) ).

% eucl_rel_int.simps
thf(fact_9562_eucl__rel__int_Ocases,axiom,
    ! [A12: int,A23: int,A33: product_prod_int_int] :
      ( ( eucl_rel_int @ A12 @ A23 @ A33 )
     => ( ( ( A23 = zero_zero_int )
         => ( A33
           != ( product_Pair_int_int @ zero_zero_int @ A12 ) ) )
       => ( ! [Q5: int] :
              ( ( A33
                = ( product_Pair_int_int @ Q5 @ zero_zero_int ) )
             => ( ( A23 != zero_zero_int )
               => ( A12
                 != ( times_times_int @ Q5 @ A23 ) ) ) )
         => ~ ! [R2: int,Q5: int] :
                ( ( A33
                  = ( 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_9563_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_9564_minus__one__div__numeral,axiom,
    ! [N: num] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).

% minus_one_div_numeral
thf(fact_9565_one__div__minus__numeral,axiom,
    ! [N: num] :
      ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).

% one_div_minus_numeral
thf(fact_9566_Divides_Oadjust__div__eq,axiom,
    ! [Q3: int,R3: int] :
      ( ( adjust_div @ ( product_Pair_int_int @ Q3 @ R3 ) )
      = ( plus_plus_int @ Q3 @ ( zero_n2684676970156552555ol_int @ ( R3 != zero_zero_int ) ) ) ) ).

% Divides.adjust_div_eq
thf(fact_9567_numeral__div__minus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).

% numeral_div_minus_numeral
thf(fact_9568_minus__numeral__div__numeral,axiom,
    ! [M: num,N: num] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).

% minus_numeral_div_numeral
thf(fact_9569_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_9570_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_9571_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
     => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K @ L )
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
        & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K @ L )
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.psimps
thf(fact_9572_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
thf(fact_9573_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
thf(fact_9574_zero__integer_Orsp,axiom,
    zero_zero_int = zero_zero_int ).

% zero_integer.rsp
thf(fact_9575_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
thf(fact_9576_uminus__integer__code_I1_J,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% uminus_integer_code(1)
thf(fact_9577_times__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(1)
thf(fact_9578_times__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(2)
thf(fact_9579_less__eq__integer__code_I1_J,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).

% less_eq_integer_code(1)
thf(fact_9580_less__integer__code_I1_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).

% less_integer_code(1)
thf(fact_9581_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_9582_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,L3: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L3 ) )
           => ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ L3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P @ K2 @ L3 ) ) )
       => ( P @ A0 @ A12 ) ) ) ).

% and_int.pinduct
thf(fact_9583_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L )
      = L ) ).

% plus_integer_code(2)
thf(fact_9584_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( plus_p5714425477246183910nteger @ K @ zero_z3403309356797280102nteger )
      = K ) ).

% plus_integer_code(1)
thf(fact_9585_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
      = K ) ).

% minus_integer_code(1)
thf(fact_9586_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L )
      = ( uminus1351360451143612070nteger @ L ) ) ).

% minus_integer_code(2)
thf(fact_9587_upto_Opinduct,axiom,
    ! [A0: int,A12: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
     => ( ! [I3: int,J2: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J2 ) )
           => ( ( ( ord_less_eq_int @ I3 @ J2 )
               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) )
             => ( P @ I3 @ J2 ) ) )
       => ( P @ A0 @ A12 ) ) ) ).

% upto.pinduct
thf(fact_9588_bitNOT__integer__code,axiom,
    ( bit_ri7632146776885996613nteger
    = ( ^ [I4: code_integer] : ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ I4 ) @ one_one_Code_integer ) ) ) ).

% bitNOT_integer_code
thf(fact_9589_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_9590_zero__integer__def,axiom,
    ( zero_z3403309356797280102nteger
    = ( code_integer_of_int @ zero_zero_int ) ) ).

% zero_integer_def
thf(fact_9591_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_9592_one__integer__def,axiom,
    ( one_one_Code_integer
    = ( code_integer_of_int @ one_one_int ) ) ).

% one_integer_def
thf(fact_9593_lsb__integer__code,axiom,
    ( least_7544222001954398261nteger
    = ( ^ [X2: code_integer] : ( bit_se9216721137139052372nteger @ X2 @ zero_zero_nat ) ) ) ).

% lsb_integer_code
thf(fact_9594_shiftr__integer__conv__div__pow2,axiom,
    ( bit_se3928097537394005634nteger
    = ( ^ [N4: nat,X2: code_integer] : ( divide6298287555418463151nteger @ X2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% shiftr_integer_conv_div_pow2
thf(fact_9595_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_9596_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_9597_pred__numeral__simps_I2_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit0 @ K ) )
      = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).

% pred_numeral_simps(2)
thf(fact_9598_semiring__norm_I26_J,axiom,
    ( ( bitM @ one )
    = one ) ).

% semiring_norm(26)
thf(fact_9599_semiring__norm_I28_J,axiom,
    ! [N: num] :
      ( ( bitM @ ( bit1 @ N ) )
      = ( bit1 @ ( bit0 @ N ) ) ) ).

% semiring_norm(28)
thf(fact_9600_semiring__norm_I27_J,axiom,
    ! [N: num] :
      ( ( bitM @ ( bit0 @ N ) )
      = ( bit1 @ ( bitM @ N ) ) ) ).

% semiring_norm(27)
thf(fact_9601_inc__BitM__eq,axiom,
    ! [N: num] :
      ( ( inc @ ( bitM @ N ) )
      = ( bit0 @ N ) ) ).

% inc_BitM_eq
thf(fact_9602_BitM__inc__eq,axiom,
    ! [N: num] :
      ( ( bitM @ ( inc @ N ) )
      = ( bit1 @ N ) ) ).

% BitM_inc_eq
thf(fact_9603_eval__nat__numeral_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N ) )
      = ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).

% eval_nat_numeral(2)
thf(fact_9604_BitM__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ ( bitM @ N ) @ one )
      = ( bit0 @ N ) ) ).

% BitM_plus_one
thf(fact_9605_one__plus__BitM,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bitM @ N ) )
      = ( bit0 @ N ) ) ).

% one_plus_BitM
thf(fact_9606_push__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% push_bit_nonnegative_int_iff
thf(fact_9607_push__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% push_bit_negative_int_iff
thf(fact_9608_concat__bit__of__zero__1,axiom,
    ! [N: nat,L: int] :
      ( ( bit_concat_bit @ N @ zero_zero_int @ L )
      = ( bit_se545348938243370406it_int @ N @ L ) ) ).

% concat_bit_of_zero_1
thf(fact_9609_push__bit__of__Suc__0,axiom,
    ! [N: nat] :
      ( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% push_bit_of_Suc_0
thf(fact_9610_push__bit__int__code_I1_J,axiom,
    ! [I: int] :
      ( ( bit_se545348938243370406it_int @ zero_zero_nat @ I )
      = I ) ).

% push_bit_int_code(1)
thf(fact_9611_flip__bit__nat__def,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [M3: nat,N4: nat] : ( bit_se6528837805403552850or_nat @ N4 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).

% flip_bit_nat_def
thf(fact_9612_set__bit__nat__def,axiom,
    ( bit_se7882103937844011126it_nat
    = ( ^ [M3: nat,N4: nat] : ( bit_se1412395901928357646or_nat @ N4 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).

% set_bit_nat_def
thf(fact_9613_Bit__integer__code_I1_J,axiom,
    ! [I: code_integer] :
      ( ( bits_Bit_integer @ I @ $false )
      = ( bit_se7788150548672797655nteger @ one_one_nat @ I ) ) ).

% Bit_integer_code(1)
thf(fact_9614_Bit__Operations_Oset__bit__int__def,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N4: nat,K3: int] : ( bit_se1409905431419307370or_int @ K3 @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ).

% Bit_Operations.set_bit_int_def
thf(fact_9615_flip__bit__int__def,axiom,
    ( bit_se2159334234014336723it_int
    = ( ^ [N4: nat,K3: int] : ( bit_se6526347334894502574or_int @ K3 @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ).

% flip_bit_int_def
thf(fact_9616_unset__bit__int__def,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N4: nat,K3: int] : ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ) ).

% unset_bit_int_def
thf(fact_9617_push__bit__int__def,axiom,
    ( bit_se545348938243370406it_int
    = ( ^ [N4: nat,K3: int] : ( times_times_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% push_bit_int_def
thf(fact_9618_Bit__integer__code_I2_J,axiom,
    ! [I: code_integer] :
      ( ( bits_Bit_integer @ I @ $true )
      = ( plus_p5714425477246183910nteger @ ( bit_se7788150548672797655nteger @ one_one_nat @ I ) @ one_one_Code_integer ) ) ).

% Bit_integer_code(2)
thf(fact_9619_push__bit__nat__def,axiom,
    ( bit_se547839408752420682it_nat
    = ( ^ [N4: nat,M3: nat] : ( times_times_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% push_bit_nat_def
thf(fact_9620_push__bit__minus__one,axiom,
    ! [N: nat] :
      ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% push_bit_minus_one
thf(fact_9621_shiftl__integer__conv__mult__pow2,axiom,
    ( bit_se7788150548672797655nteger
    = ( ^ [N4: nat,X2: code_integer] : ( times_3573771949741848930nteger @ X2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% shiftl_integer_conv_mult_pow2
thf(fact_9622_or__minus__numerals_I5_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).

% or_minus_numerals(5)
thf(fact_9623_or__minus__numerals_I1_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).

% or_minus_numerals(1)
thf(fact_9624_or__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).

% or_minus_numerals(8)
thf(fact_9625_or__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).

% or_minus_numerals(4)
thf(fact_9626_or__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).

% or_minus_numerals(3)
thf(fact_9627_or__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).

% or_minus_numerals(7)
thf(fact_9628_or__not__num__neg_Osimps_I1_J,axiom,
    ( ( bit_or_not_num_neg @ one @ one )
    = one ) ).

% or_not_num_neg.simps(1)
thf(fact_9629_or__not__num__neg_Osimps_I4_J,axiom,
    ! [N: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one )
      = ( bit0 @ one ) ) ).

% or_not_num_neg.simps(4)
thf(fact_9630_or__not__num__neg_Osimps_I6_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
      = ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).

% or_not_num_neg.simps(6)
thf(fact_9631_or__not__num__neg_Osimps_I7_J,axiom,
    ! [N: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one )
      = one ) ).

% or_not_num_neg.simps(7)
thf(fact_9632_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_9633_or__not__num__neg_Osimps_I5_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).

% or_not_num_neg.simps(5)
thf(fact_9634_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_9635_or__not__num__neg_Osimps_I8_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).

% or_not_num_neg.simps(8)
thf(fact_9636_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 ) ) ) )
           => ( ( ? [N2: num] :
                    ( X
                    = ( bit0 @ N2 ) )
               => ( ( Xa = one )
                 => ( Y
                   != ( bit0 @ one ) ) ) )
             => ( ! [N2: num] :
                    ( ( X
                      = ( bit0 @ N2 ) )
                   => ! [M4: num] :
                        ( ( Xa
                          = ( bit0 @ M4 ) )
                       => ( Y
                         != ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) )
               => ( ! [N2: num] :
                      ( ( X
                        = ( bit0 @ N2 ) )
                     => ! [M4: num] :
                          ( ( Xa
                            = ( bit1 @ M4 ) )
                         => ( Y
                           != ( bit0 @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) )
                 => ( ( ? [N2: num] :
                          ( X
                          = ( bit1 @ N2 ) )
                     => ( ( Xa = one )
                       => ( Y != one ) ) )
                   => ( ! [N2: num] :
                          ( ( X
                            = ( bit1 @ N2 ) )
                         => ! [M4: num] :
                              ( ( Xa
                                = ( bit0 @ M4 ) )
                             => ( Y
                               != ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) )
                     => ~ ! [N2: num] :
                            ( ( X
                              = ( bit1 @ N2 ) )
                           => ! [M4: num] :
                                ( ( Xa
                                  = ( bit1 @ M4 ) )
                               => ( Y
                                 != ( bitM @ ( bit_or_not_num_neg @ N2 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
thf(fact_9637_int__numeral__or__not__num__neg,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).

% int_numeral_or_not_num_neg
thf(fact_9638_int__numeral__not__or__num__neg,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).

% int_numeral_not_or_num_neg
thf(fact_9639_numeral__or__not__num__eq,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) )
      = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% numeral_or_not_num_eq
thf(fact_9640_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L2: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q6: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q6 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q6 ) @ R5 ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_9641_divmod__step__int__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L2: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q6: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q6 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q6 ) @ R5 ) ) ) ) ) ).

% divmod_step_int_def
thf(fact_9642_divmod__step__integer__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L2: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q6: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q6 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q6 ) @ R5 ) ) ) ) ) ).

% divmod_step_integer_def
thf(fact_9643_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M3: nat,N4: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N4 = zero_zero_nat )
            | ( ord_less_nat @ M3 @ N4 ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M3 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q6: nat] : ( product_Pair_nat_nat @ ( suc @ Q6 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M3 @ N4 ) @ N4 ) ) ) ) ) ).

% divmod_nat_if
thf(fact_9644_Divides_Oadjust__div__def,axiom,
    ( adjust_div
    = ( produc8211389475949308722nt_int
      @ ^ [Q6: int,R5: int] : ( plus_plus_int @ Q6 @ ( zero_n2684676970156552555ol_int @ ( R5 != zero_zero_int ) ) ) ) ) ).

% Divides.adjust_div_def
thf(fact_9645_int__ge__less__than__def,axiom,
    ( int_ge_less_than
    = ( ^ [D4: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z7: int,Z4: int] :
                ( ( ord_less_eq_int @ D4 @ Z7 )
                & ( ord_less_int @ Z7 @ Z4 ) ) ) ) ) ) ).

% int_ge_less_than_def
thf(fact_9646_int__ge__less__than2__def,axiom,
    ( int_ge_less_than2
    = ( ^ [D4: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z7: int,Z4: int] :
                ( ( ord_less_eq_int @ D4 @ Z4 )
                & ( ord_less_int @ Z7 @ Z4 ) ) ) ) ) ) ).

% int_ge_less_than2_def
thf(fact_9647_dup__1,axiom,
    ( ( code_dup @ one_one_Code_integer )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% dup_1
thf(fact_9648_rat__inverse__code,axiom,
    ! [P4: rat] :
      ( ( quotient_of @ ( inverse_inverse_rat @ P4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A4: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( A4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A4 ) @ B4 ) @ ( abs_abs_int @ A4 ) ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_inverse_code
thf(fact_9649_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_9650_quotient__of__number_I3_J,axiom,
    ! [K: num] :
      ( ( quotient_of @ ( numeral_numeral_rat @ K ) )
      = ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).

% quotient_of_number(3)
thf(fact_9651_rat__one__code,axiom,
    ( ( quotient_of @ one_one_rat )
    = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).

% rat_one_code
thf(fact_9652_rat__zero__code,axiom,
    ( ( quotient_of @ zero_zero_rat )
    = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% rat_zero_code
thf(fact_9653_quotient__of__number_I5_J,axiom,
    ! [K: num] :
      ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
      = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).

% quotient_of_number(5)
thf(fact_9654_quotient__of__number_I4_J,axiom,
    ( ( quotient_of @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( product_Pair_int_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ) ) ).

% quotient_of_number(4)
thf(fact_9655_Code__Numeral_Odup__code_I1_J,axiom,
    ( ( code_dup @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% Code_Numeral.dup_code(1)
thf(fact_9656_quotient__of__denom__pos,axiom,
    ! [R3: rat,P4: int,Q3: int] :
      ( ( ( quotient_of @ R3 )
        = ( product_Pair_int_int @ P4 @ Q3 ) )
     => ( ord_less_int @ zero_zero_int @ Q3 ) ) ).

% quotient_of_denom_pos
thf(fact_9657_rat__uminus__code,axiom,
    ! [P4: rat] :
      ( ( quotient_of @ ( uminus_uminus_rat @ P4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A4: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A4 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_uminus_code
thf(fact_9658_rat__less__code,axiom,
    ( ord_less_rat
    = ( ^ [P6: rat,Q6: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A4: int,C5: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B4: int,D4: int] : ( ord_less_int @ ( times_times_int @ A4 @ D4 ) @ ( times_times_int @ C5 @ B4 ) )
              @ ( quotient_of @ Q6 ) )
          @ ( quotient_of @ P6 ) ) ) ) ).

% rat_less_code
thf(fact_9659_rat__floor__code,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [P6: rat] : ( produc8211389475949308722nt_int @ divide_divide_int @ ( quotient_of @ P6 ) ) ) ) ).

% rat_floor_code
thf(fact_9660_rat__abs__code,axiom,
    ! [P4: rat] :
      ( ( quotient_of @ ( abs_abs_rat @ P4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A4: int] : ( product_Pair_int_int @ ( abs_abs_int @ A4 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_abs_code
thf(fact_9661_rat__less__eq__code,axiom,
    ( ord_less_eq_rat
    = ( ^ [P6: rat,Q6: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A4: int,C5: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B4: int,D4: int] : ( ord_less_eq_int @ ( times_times_int @ A4 @ D4 ) @ ( times_times_int @ C5 @ B4 ) )
              @ ( quotient_of @ Q6 ) )
          @ ( quotient_of @ P6 ) ) ) ) ).

% rat_less_eq_code
thf(fact_9662_bin__rest__integer__code,axiom,
    ( bits_b2549910563261871055nteger
    = ( ^ [I4: code_integer] : ( divide6298287555418463151nteger @ I4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% bin_rest_integer_code
thf(fact_9663_quotient__of__int,axiom,
    ! [A2: int] :
      ( ( quotient_of @ ( of_int @ A2 ) )
      = ( product_Pair_int_int @ A2 @ one_one_int ) ) ).

% quotient_of_int
thf(fact_9664_bitXOR__integer__unfold,axiom,
    ( bit_se3222712562003087583nteger
    = ( ^ [X2: code_integer,Y2: code_integer] :
          ( if_Code_integer @ ( X2 = zero_z3403309356797280102nteger ) @ Y2
          @ ( if_Code_integer
            @ ( X2
              = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
            @ ( bit_ri7632146776885996613nteger @ Y2 )
            @ ( bits_Bit_integer @ ( bit_se3222712562003087583nteger @ ( bits_b2549910563261871055nteger @ X2 ) @ ( bits_b2549910563261871055nteger @ Y2 ) )
              @ ( ( ~ ( bits_b8758750999018896077nteger @ X2 ) )
                = ( bits_b8758750999018896077nteger @ Y2 ) ) ) ) ) ) ) ).

% bitXOR_integer_unfold
thf(fact_9665_rat__plus__code,axiom,
    ! [P4: rat,Q3: rat] :
      ( ( quotient_of @ ( plus_plus_rat @ P4 @ Q3 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A4: int,C5: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A4 @ D4 ) @ ( times_times_int @ B4 @ C5 ) ) @ ( times_times_int @ C5 @ D4 ) ) )
            @ ( quotient_of @ Q3 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_plus_code
thf(fact_9666_normalize__denom__zero,axiom,
    ! [P4: int] :
      ( ( normalize @ ( product_Pair_int_int @ P4 @ zero_zero_int ) )
      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% normalize_denom_zero
thf(fact_9667_normalize__negative,axiom,
    ! [Q3: int,P4: int] :
      ( ( ord_less_int @ Q3 @ zero_zero_int )
     => ( ( normalize @ ( product_Pair_int_int @ P4 @ Q3 ) )
        = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P4 ) @ ( uminus_uminus_int @ Q3 ) ) ) ) ) ).

% normalize_negative
thf(fact_9668_bitval__bin__last__integer,axiom,
    ! [I: code_integer] :
      ( ( zero_n356916108424825756nteger @ ( bits_b8758750999018896077nteger @ I ) )
      = ( bit_se3949692690581998587nteger @ I @ one_one_Code_integer ) ) ).

% bitval_bin_last_integer
thf(fact_9669_bin__last__integer__code,axiom,
    ( bits_b8758750999018896077nteger
    = ( ^ [I4: code_integer] :
          ( ( bit_se3949692690581998587nteger @ I4 @ one_one_Code_integer )
         != zero_z3403309356797280102nteger ) ) ) ).

% bin_last_integer_code
thf(fact_9670_normalize__denom__pos,axiom,
    ! [R3: product_prod_int_int,P4: int,Q3: int] :
      ( ( ( normalize @ R3 )
        = ( product_Pair_int_int @ P4 @ Q3 ) )
     => ( ord_less_int @ zero_zero_int @ Q3 ) ) ).

% normalize_denom_pos
thf(fact_9671_normalize__crossproduct,axiom,
    ! [Q3: int,S2: int,P4: int,R3: int] :
      ( ( Q3 != zero_zero_int )
     => ( ( S2 != zero_zero_int )
       => ( ( ( normalize @ ( product_Pair_int_int @ P4 @ Q3 ) )
            = ( normalize @ ( product_Pair_int_int @ R3 @ S2 ) ) )
         => ( ( times_times_int @ P4 @ S2 )
            = ( times_times_int @ R3 @ Q3 ) ) ) ) ) ).

% normalize_crossproduct
thf(fact_9672_bin__last__integer__nbe,axiom,
    ( bits_b8758750999018896077nteger
    = ( ^ [I4: code_integer] :
          ( ( modulo364778990260209775nteger @ I4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
         != zero_z3403309356797280102nteger ) ) ) ).

% bin_last_integer_nbe
thf(fact_9673_rat__divide__code,axiom,
    ! [P4: rat,Q3: rat] :
      ( ( quotient_of @ ( divide_divide_rat @ P4 @ Q3 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A4: int,C5: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A4 @ D4 ) @ ( times_times_int @ C5 @ B4 ) ) )
            @ ( quotient_of @ Q3 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_divide_code
thf(fact_9674_rat__times__code,axiom,
    ! [P4: rat,Q3: rat] :
      ( ( quotient_of @ ( times_times_rat @ P4 @ Q3 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A4: int,C5: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A4 @ B4 ) @ ( times_times_int @ C5 @ D4 ) ) )
            @ ( quotient_of @ Q3 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_times_code
thf(fact_9675_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_9676_rat__minus__code,axiom,
    ! [P4: rat,Q3: rat] :
      ( ( quotient_of @ ( minus_minus_rat @ P4 @ Q3 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A4: int,C5: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A4 @ D4 ) @ ( times_times_int @ B4 @ C5 ) ) @ ( times_times_int @ C5 @ D4 ) ) )
            @ ( quotient_of @ Q3 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_minus_code
thf(fact_9677_bitOR__integer__unfold,axiom,
    ( bit_se1080825931792720795nteger
    = ( ^ [X2: code_integer,Y2: code_integer] :
          ( if_Code_integer @ ( X2 = zero_z3403309356797280102nteger ) @ Y2
          @ ( if_Code_integer
            @ ( X2
              = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
            @ ( uminus1351360451143612070nteger @ one_one_Code_integer )
            @ ( bits_Bit_integer @ ( bit_se1080825931792720795nteger @ ( bits_b2549910563261871055nteger @ X2 ) @ ( bits_b2549910563261871055nteger @ Y2 ) )
              @ ( ( bits_b8758750999018896077nteger @ X2 )
                | ( bits_b8758750999018896077nteger @ Y2 ) ) ) ) ) ) ) ).

% bitOR_integer_unfold
thf(fact_9678_bitAND__integer__unfold,axiom,
    ( bit_se3949692690581998587nteger
    = ( ^ [X2: code_integer,Y2: code_integer] :
          ( if_Code_integer @ ( X2 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger
          @ ( if_Code_integer
            @ ( X2
              = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
            @ Y2
            @ ( bits_Bit_integer @ ( bit_se3949692690581998587nteger @ ( bits_b2549910563261871055nteger @ X2 ) @ ( bits_b2549910563261871055nteger @ Y2 ) )
              @ ( ( bits_b8758750999018896077nteger @ X2 )
                & ( bits_b8758750999018896077nteger @ Y2 ) ) ) ) ) ) ) ).

% bitAND_integer_unfold
thf(fact_9679_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_9680_Frct__code__post_I2_J,axiom,
    ! [A2: int] :
      ( ( frct @ ( product_Pair_int_int @ A2 @ zero_zero_int ) )
      = zero_zero_rat ) ).

% Frct_code_post(2)
thf(fact_9681_Frct__code__post_I1_J,axiom,
    ! [A2: int] :
      ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A2 ) )
      = zero_zero_rat ) ).

% Frct_code_post(1)
thf(fact_9682_Frct__code__post_I3_J,axiom,
    ( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
    = one_one_rat ) ).

% Frct_code_post(3)
thf(fact_9683_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] :
      ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L ) ) )
      = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L ) ) ) ).

% Frct_code_post(6)
thf(fact_9684_Frct__code__post_I4_J,axiom,
    ! [K: num] :
      ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
      = ( numeral_numeral_rat @ K ) ) ).

% Frct_code_post(4)
thf(fact_9685_set__bit__integer__conv__masks,axiom,
    ( generi2397576812484419408nteger
    = ( ^ [X2: code_integer,I4: nat,B4: $o] : ( if_Code_integer @ B4 @ ( bit_se1080825931792720795nteger @ X2 @ ( bit_se7788150548672797655nteger @ I4 @ one_one_Code_integer ) ) @ ( bit_se3949692690581998587nteger @ X2 @ ( bit_ri7632146776885996613nteger @ ( bit_se7788150548672797655nteger @ I4 @ one_one_Code_integer ) ) ) ) ) ) ).

% set_bit_integer_conv_masks
thf(fact_9686_Uint32__code,axiom,
    ( uint322
    = ( ^ [I4: code_integer] : ( if_uint32 @ ( bit_se9216721137139052372nteger @ ( bit_se3949692690581998587nteger @ I4 @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( uint32_signed @ ( minus_8373710615458151222nteger @ ( bit_se3949692690581998587nteger @ I4 @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( uint32_signed @ ( bit_se3949692690581998587nteger @ I4 @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% Uint32_code
thf(fact_9687_Uint32__signed__def,axiom,
    ( uint32_signed
    = ( ^ [I4: code_integer] :
          ( if_uint32
          @ ( ( ord_le6747313008572928689nteger @ I4 @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
            | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ I4 ) )
          @ ( undefi2040150642751712519uint32 @ uint322 @ I4 )
          @ ( uint322 @ I4 ) ) ) ) ).

% Uint32_signed_def
thf(fact_9688_int__set__bit__True__conv__OR,axiom,
    ! [I: int,N: nat] :
      ( ( generi8991105624351003935it_int @ I @ N @ $true )
      = ( bit_se1409905431419307370or_int @ I @ ( bit_se545348938243370406it_int @ N @ one_one_int ) ) ) ).

% int_set_bit_True_conv_OR
thf(fact_9689_int__set__bit__False__conv__NAND,axiom,
    ! [I: int,N: nat] :
      ( ( generi8991105624351003935it_int @ I @ N @ $false )
      = ( bit_se725231765392027082nd_int @ I @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N @ one_one_int ) ) ) ) ).

% int_set_bit_False_conv_NAND
thf(fact_9690_int__set__bit__conv__ops,axiom,
    ( generi8991105624351003935it_int
    = ( ^ [I4: int,N4: nat,B4: $o] : ( if_int @ B4 @ ( bit_se1409905431419307370or_int @ I4 @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) @ ( bit_se725231765392027082nd_int @ I4 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ) ) ).

% int_set_bit_conv_ops
thf(fact_9691_set__encode__def,axiom,
    ( nat_set_encode
    = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% set_encode_def
thf(fact_9692_set__encode__empty,axiom,
    ( ( nat_set_encode @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% set_encode_empty
thf(fact_9693_set__encode__insert,axiom,
    ! [A: set_nat,N: nat] :
      ( ( finite_finite_nat @ A )
     => ( ~ ( member_nat @ N @ A )
       => ( ( nat_set_encode @ ( insert_nat @ N @ A ) )
          = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A ) ) ) ) ) ).

% set_encode_insert
thf(fact_9694_Cauchy__iff2,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X8: nat > real] :
        ! [J3: nat] :
        ? [M8: nat] :
        ! [M3: nat] :
          ( ( ord_less_eq_nat @ M8 @ M3 )
         => ! [N4: nat] :
              ( ( ord_less_eq_nat @ M8 @ N4 )
             => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X8 @ M3 ) @ ( X8 @ N4 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).

% Cauchy_iff2
thf(fact_9695_sdiv__int__div__0,axiom,
    ! [X: int] :
      ( ( signed6714573509424544716de_int @ X @ zero_zero_int )
      = zero_zero_int ) ).

% sdiv_int_div_0
thf(fact_9696_sdiv__int__0__div,axiom,
    ! [X: int] :
      ( ( signed6714573509424544716de_int @ zero_zero_int @ X )
      = zero_zero_int ) ).

% sdiv_int_0_div
thf(fact_9697_int__sdiv__simps_I2_J,axiom,
    ! [A2: int] :
      ( ( signed6714573509424544716de_int @ A2 @ zero_zero_int )
      = zero_zero_int ) ).

% int_sdiv_simps(2)
thf(fact_9698_int__sdiv__simps_I1_J,axiom,
    ! [A2: int] :
      ( ( signed6714573509424544716de_int @ A2 @ one_one_int )
      = A2 ) ).

% int_sdiv_simps(1)
thf(fact_9699_int__sdiv__same__is__1,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( ( signed6714573509424544716de_int @ A2 @ B2 )
          = A2 )
        = ( B2 = one_one_int ) ) ) ).

% int_sdiv_same_is_1
thf(fact_9700_int__sdiv__simps_I3_J,axiom,
    ! [A2: int] :
      ( ( signed6714573509424544716de_int @ A2 @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ A2 ) ) ).

% int_sdiv_simps(3)
thf(fact_9701_sdiv__int__numeral__numeral,axiom,
    ! [M: num,N: num] :
      ( ( signed6714573509424544716de_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ).

% sdiv_int_numeral_numeral
thf(fact_9702_int__sdiv__negated__is__minus1,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( ( signed6714573509424544716de_int @ A2 @ B2 )
          = ( uminus_uminus_int @ A2 ) )
        = ( B2
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% int_sdiv_negated_is_minus1
thf(fact_9703_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_9704_bounded__nat__set__is__finite,axiom,
    ! [N3: set_nat,N: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ N3 )
         => ( ord_less_nat @ X3 @ N ) )
     => ( finite_finite_nat @ N3 ) ) ).

% bounded_nat_set_is_finite
thf(fact_9705_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N11: set_nat] :
        ? [M3: nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ N11 )
         => ( ord_less_nat @ X2 @ M3 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_9706_finite__less__ub,axiom,
    ! [F: nat > nat,U2: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F @ N2 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ U2 ) ) ) ) ).

% finite_less_ub
thf(fact_9707_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N3: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
     => ( finite_finite_nat @ N3 ) ) ).

% subset_eq_atLeast0_atMost_finite
thf(fact_9708_set__encode__inf,axiom,
    ! [A: set_nat] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( nat_set_encode @ A )
        = zero_zero_nat ) ) ).

% set_encode_inf
thf(fact_9709_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [D4: nat] : ( dvd_dvd_nat @ D4 @ M ) ) ) ) ).

% finite_divisors_nat
thf(fact_9710_sgn__sdiv__eq__sgn__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( ( signed6714573509424544716de_int @ A2 @ B2 )
       != zero_zero_int )
     => ( ( sgn_sgn_int @ ( signed6714573509424544716de_int @ A2 @ B2 ) )
        = ( sgn_sgn_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).

% sgn_sdiv_eq_sgn_mult
thf(fact_9711_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_9712_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_9713_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N4: nat] : ( ord_less_nat @ N4 @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_9714_finite__interval__int1,axiom,
    ! [A2: int,B2: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I4: int] :
            ( ( ord_less_eq_int @ A2 @ I4 )
            & ( ord_less_eq_int @ I4 @ B2 ) ) ) ) ).

% finite_interval_int1
thf(fact_9715_finite__interval__int4,axiom,
    ! [A2: int,B2: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I4: int] :
            ( ( ord_less_int @ A2 @ I4 )
            & ( ord_less_int @ I4 @ B2 ) ) ) ) ).

% finite_interval_int4
thf(fact_9716_finite__interval__int3,axiom,
    ! [A2: int,B2: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I4: int] :
            ( ( ord_less_int @ A2 @ I4 )
            & ( ord_less_eq_int @ I4 @ B2 ) ) ) ) ).

% finite_interval_int3
thf(fact_9717_finite__interval__int2,axiom,
    ! [A2: int,B2: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I4: int] :
            ( ( ord_less_eq_int @ A2 @ I4 )
            & ( ord_less_int @ I4 @ B2 ) ) ) ) ).

% finite_interval_int2
thf(fact_9718_finite__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z4: complex] :
              ( ( power_power_complex @ Z4 @ N )
              = C ) ) ) ) ).

% finite_nth_roots
thf(fact_9719_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero_int )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [D4: int] : ( dvd_dvd_int @ D4 @ I ) ) ) ) ).

% finite_divisors_int
thf(fact_9720_int__of__nat__def,axiom,
    code_T6385005292777649522of_nat = semiri1314217659103216013at_int ).

% int_of_nat_def
thf(fact_9721_infinite__int__iff__unbounded,axiom,
    ! [S: set_int] :
      ( ( ~ ( finite_finite_int @ S ) )
      = ( ! [M3: int] :
          ? [N4: int] :
            ( ( ord_less_int @ M3 @ ( abs_abs_int @ N4 ) )
            & ( member_int @ N4 @ S ) ) ) ) ).

% infinite_int_iff_unbounded
thf(fact_9722_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ ( set_or4665077453230672383an_nat @ M @ N ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Ico_nat
thf(fact_9723_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_9724_atLeastLessThan__singleton,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
      = ( insert_nat @ M @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_9725_atLeastLessThan0,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
      = bot_bot_set_nat ) ).

% atLeastLessThan0
thf(fact_9726_lessThan__atLeast0,axiom,
    ( set_ord_lessThan_nat
    = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).

% lessThan_atLeast0
thf(fact_9727_all__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M3: nat] :
            ( ( ord_less_nat @ M3 @ N )
           => ( P @ M3 ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
           => ( P @ X2 ) ) ) ) ).

% all_nat_less_eq
thf(fact_9728_ex__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M3: nat] :
            ( ( ord_less_nat @ M3 @ N )
            & ( P @ M3 ) ) )
      = ( ? [X2: nat] :
            ( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
            & ( P @ X2 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_9729_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).

% atLeast0_lessThan_Suc
thf(fact_9730_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
        = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% atLeastLessThan_add_Un
thf(fact_9731_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N3: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
     => ( finite_finite_nat @ N3 ) ) ).

% subset_eq_atLeast0_lessThan_finite
thf(fact_9732_atLeastLessThanSuc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
          = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
thf(fact_9733_prod__Suc__fact,axiom,
    ! [N: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
      = ( semiri1408675320244567234ct_nat @ N ) ) ).

% prod_Suc_fact
thf(fact_9734_prod__Suc__Suc__fact,axiom,
    ! [N: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( semiri1408675320244567234ct_nat @ N ) ) ).

% prod_Suc_Suc_fact
thf(fact_9735_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_9736_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_lessThan_eq_remove0
thf(fact_9737_infinite__nat__iff__unbounded,axiom,
    ! [S: set_nat] :
      ( ( ~ ( finite_finite_nat @ S ) )
      = ( ! [M3: nat] :
          ? [N4: nat] :
            ( ( ord_less_nat @ M3 @ N4 )
            & ( member_nat @ N4 @ S ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_9738_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_9739_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A2: nat > nat,B2: nat > nat] :
      ( ! [I3: nat,J2: nat] :
          ( ( ord_less_eq_nat @ I3 @ J2 )
         => ( ( ord_less_nat @ J2 @ N )
           => ( ord_less_eq_nat @ ( A2 @ I3 ) @ ( A2 @ J2 ) ) ) )
     => ( ! [I3: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I3 @ J2 )
           => ( ( ord_less_nat @ J2 @ N )
             => ( ord_less_eq_nat @ ( B2 @ J2 ) @ ( B2 @ I3 ) ) ) )
       => ( ord_less_eq_nat
          @ ( times_times_nat @ N
            @ ( groups3542108847815614940at_nat
              @ ^ [I4: nat] : ( times_times_nat @ ( A2 @ I4 ) @ ( B2 @ I4 ) )
              @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
          @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).

% Chebyshev_sum_upper_nat
thf(fact_9740_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_9741_smod__int__mod__0,axiom,
    ! [X: int] :
      ( ( signed6292675348222524329lo_int @ X @ zero_zero_int )
      = X ) ).

% smod_int_mod_0
thf(fact_9742_smod__int__0__mod,axiom,
    ! [X: int] :
      ( ( signed6292675348222524329lo_int @ zero_zero_int @ X )
      = zero_zero_int ) ).

% smod_int_0_mod
thf(fact_9743_smod__int__numeral__numeral,axiom,
    ! [M: num,N: num] :
      ( ( signed6292675348222524329lo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ).

% smod_int_numeral_numeral
thf(fact_9744_finite__atLeastZeroLessThan__integer,axiom,
    ! [U2: code_integer] : ( finite6017078050557962740nteger @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ U2 ) ) ).

% finite_atLeastZeroLessThan_integer
thf(fact_9745_finite__atLeastZeroLessThan__int,axiom,
    ! [U2: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U2 ) ) ).

% finite_atLeastZeroLessThan_int
thf(fact_9746_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U2: code_integer] :
      ( ( set_or8404916559141939852nteger @ L @ ( plus_p5714425477246183910nteger @ U2 @ one_one_Code_integer ) )
      = ( set_or189985376899183464nteger @ L @ U2 ) ) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
thf(fact_9747_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U2: int] :
      ( ( set_or4662586982721622107an_int @ L @ ( plus_plus_int @ U2 @ one_one_int ) )
      = ( set_or1266510415728281911st_int @ L @ U2 ) ) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_9748_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_9749_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_9750_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_9751_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_9752_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_9753_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_9754_smod__mod__positive,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ( signed6292675348222524329lo_int @ A2 @ B2 )
          = ( modulo_modulo_int @ A2 @ B2 ) ) ) ) ).

% smod_mod_positive
thf(fact_9755_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_9756_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_9757_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_9758_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_9759_len__num0,axiom,
    ( type_l4264026598287037464l_num0
    = ( ^ [Uu2: itself_Numeral_num0] : zero_zero_nat ) ) ).

% len_num0
thf(fact_9760_len__of__finite__1__def,axiom,
    ( type_l31302759751748491nite_1
    = ( ^ [X2: itself_finite_1] : one_one_nat ) ) ).

% len_of_finite_1_def
thf(fact_9761_len__num1,axiom,
    ( type_l4264026598287037465l_num1
    = ( ^ [Uu2: itself_Numeral_num1] : one_one_nat ) ) ).

% len_num1
thf(fact_9762_len__of__finite__2__def,axiom,
    ( type_l31302759751748492nite_2
    = ( ^ [X2: itself_finite_2] : ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% len_of_finite_2_def
thf(fact_9763_len__of__finite__3__def,axiom,
    ( type_l31302759751748493nite_3
    = ( ^ [X2: itself_finite_3] : ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% len_of_finite_3_def
thf(fact_9764_bij__betw__Suc,axiom,
    ! [M7: set_nat,N3: set_nat] :
      ( ( bij_betw_nat_nat @ suc @ M7 @ N3 )
      = ( ( image_nat_nat @ suc @ M7 )
        = N3 ) ) ).

% bij_betw_Suc
thf(fact_9765_zero__notin__Suc__image,axiom,
    ! [A: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A ) ) ).

% zero_notin_Suc_image
thf(fact_9766_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_9767_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_9768_image__Suc__lessThan,axiom,
    ! [N: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).

% image_Suc_lessThan
thf(fact_9769_image__Suc__atMost,axiom,
    ! [N: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).

% image_Suc_atMost
thf(fact_9770_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).

% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_9771_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).

% atLeast0_atMost_Suc_eq_insert_0
thf(fact_9772_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% lessThan_Suc_eq_insert_0
thf(fact_9773_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% atMost_Suc_eq_insert_0
thf(fact_9774_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U2: int] :
      ( ( image_int_int
        @ ^ [X2: int] : ( plus_plus_int @ X2 @ L )
        @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U2 @ L ) ) )
      = ( set_or4662586982721622107an_int @ L @ U2 ) ) ).

% image_add_int_atLeastLessThan
thf(fact_9775_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U2: code_integer] :
      ( ( image_4470545334726330049nteger
        @ ^ [X2: code_integer] : ( plus_p5714425477246183910nteger @ X2 @ L )
        @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ U2 @ L ) ) )
      = ( set_or8404916559141939852nteger @ L @ U2 ) ) ).

% image_add_integer_atLeastLessThan
thf(fact_9776_image__atLeastZeroLessThan__int,axiom,
    ! [U2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ U2 )
     => ( ( set_or4662586982721622107an_int @ zero_zero_int @ U2 )
        = ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U2 ) ) ) ) ) ).

% image_atLeastZeroLessThan_int
thf(fact_9777_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y: nat,X: nat] :
      ( ( ( ord_less_nat @ C @ Y )
       => ( ( image_nat_nat
            @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
            @ ( set_or4665077453230672383an_nat @ X @ Y ) )
          = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
      & ( ~ ( ord_less_nat @ C @ Y )
       => ( ( ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_9778_min__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( suc @ ( ord_min_nat @ M @ N ) ) ) ).

% min_Suc_Suc
thf(fact_9779_min__0L,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% min_0L
thf(fact_9780_min__0R,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ N @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_9781_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_9782_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_9783_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_9784_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_9785_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_9786_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_9787_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_9788_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_9789_min__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ord_min_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( suc @ ( ord_min_nat @ N @ ( pred_numeral @ K ) ) ) ) ).

% min_Suc_numeral
thf(fact_9790_min__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N ) ) ) ).

% min_numeral_Suc
thf(fact_9791_min__diff,axiom,
    ! [M: nat,I: nat,N: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
      = ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).

% min_diff
thf(fact_9792_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q3: nat] :
      ( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q3 )
      = ( ord_min_nat @ ( times_times_nat @ M @ Q3 ) @ ( times_times_nat @ N @ Q3 ) ) ) ).

% nat_mult_min_left
thf(fact_9793_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q3: nat] :
      ( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q3 ) )
      = ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q3 ) ) ) ).

% nat_mult_min_right
thf(fact_9794_mod__mod__power,axiom,
    ! [K: nat,M: nat,N: 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 ) ) @ N ) )
      = ( modulo_modulo_nat @ K @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( ord_min_nat @ M @ N ) ) ) ) ).

% mod_mod_power
thf(fact_9795_min__enat__simps_I3_J,axiom,
    ! [Q3: extended_enat] :
      ( ( ord_mi8085742599997312461d_enat @ zero_z5237406670263579293d_enat @ Q3 )
      = zero_z5237406670263579293d_enat ) ).

% min_enat_simps(3)
thf(fact_9796_min__enat__simps_I2_J,axiom,
    ! [Q3: extended_enat] :
      ( ( ord_mi8085742599997312461d_enat @ Q3 @ zero_z5237406670263579293d_enat )
      = zero_z5237406670263579293d_enat ) ).

% min_enat_simps(2)
thf(fact_9797_assn__basic__inequalities_I1_J,axiom,
    top_top_assn != one_one_assn ).

% assn_basic_inequalities(1)
thf(fact_9798_assn__basic__inequalities_I5_J,axiom,
    top_top_assn != bot_bot_assn ).

% assn_basic_inequalities(5)
thf(fact_9799_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_9800_infinite__UNIV__int,axiom,
    ~ ( finite_finite_int @ top_top_set_int ) ).

% infinite_UNIV_int
thf(fact_9801_int__in__range__abs,axiom,
    ! [N: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N ) @ ( image_int_int @ abs_abs_int @ top_top_set_int ) ) ).

% int_in_range_abs
thf(fact_9802_range__mod,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( image_nat_nat
          @ ^ [M3: nat] : ( modulo_modulo_nat @ M3 @ N )
          @ top_top_set_nat )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).

% range_mod
thf(fact_9803_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_9804_root__def,axiom,
    ( root
    = ( ^ [N4: nat,X2: real] :
          ( if_real @ ( N4 = 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 ) @ N4 ) )
            @ X2 ) ) ) ) ).

% root_def
thf(fact_9805_inj__on__diff__nat,axiom,
    ! [N3: set_nat,K: nat] :
      ( ! [N2: nat] :
          ( ( member_nat @ N2 @ N3 )
         => ( ord_less_eq_nat @ K @ N2 ) )
     => ( inj_on_nat_nat
        @ ^ [N4: nat] : ( minus_minus_nat @ N4 @ K )
        @ N3 ) ) ).

% inj_on_diff_nat
thf(fact_9806_inj__Suc,axiom,
    ! [N3: set_nat] : ( inj_on_nat_nat @ suc @ N3 ) ).

% inj_Suc
thf(fact_9807_inj__sgn__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( inj_on_real_real
        @ ^ [Y2: real] : ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N ) )
        @ top_top_set_real ) ) ).

% inj_sgn_power
thf(fact_9808_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I4: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_9809_DERIV__even__real__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( ord_less_real @ X @ zero_zero_real )
         => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_even_real_root
thf(fact_9810_DERIV__real__root__generic,axiom,
    ! [N: nat,X: real,D3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( X != zero_zero_real )
       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
           => ( ( ord_less_real @ zero_zero_real @ X )
             => ( D3
                = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
             => ( ( ord_less_real @ X @ zero_zero_real )
               => ( D3
                  = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
           => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
               => ( D3
                  = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
             => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D3 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).

% DERIV_real_root_generic
thf(fact_9811_has__real__derivative__neg__dec__right,axiom,
    ! [F: real > real,L: real,X: real,S: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D6: real] :
            ( ( ord_less_real @ zero_zero_real @ D6 )
            & ! [H3: real] :
                ( ( ord_less_real @ zero_zero_real @ H3 )
               => ( ( member_real @ ( plus_plus_real @ X @ H3 ) @ S )
                 => ( ( ord_less_real @ H3 @ D6 )
                   => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H3 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
thf(fact_9812_has__real__derivative__pos__inc__right,axiom,
    ! [F: real > real,L: real,X: real,S: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D6: real] :
            ( ( ord_less_real @ zero_zero_real @ D6 )
            & ! [H3: real] :
                ( ( ord_less_real @ zero_zero_real @ H3 )
               => ( ( member_real @ ( plus_plus_real @ X @ H3 ) @ S )
                 => ( ( ord_less_real @ H3 @ D6 )
                   => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H3 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
thf(fact_9813_has__real__derivative__pos__inc__left,axiom,
    ! [F: real > real,L: real,X: real,S: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D6: real] :
            ( ( ord_less_real @ zero_zero_real @ D6 )
            & ! [H3: real] :
                ( ( ord_less_real @ zero_zero_real @ H3 )
               => ( ( member_real @ ( minus_minus_real @ X @ H3 ) @ S )
                 => ( ( ord_less_real @ H3 @ D6 )
                   => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H3 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
thf(fact_9814_has__real__derivative__neg__dec__left,axiom,
    ! [F: real > real,L: real,X: real,S: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D6: real] :
            ( ( ord_less_real @ zero_zero_real @ D6 )
            & ! [H3: real] :
                ( ( ord_less_real @ zero_zero_real @ H3 )
               => ( ( member_real @ ( minus_minus_real @ X @ H3 ) @ S )
                 => ( ( ord_less_real @ H3 @ D6 )
                   => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H3 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
thf(fact_9815_MVT2,axiom,
    ! [A2: real,B2: real,F: real > real,F3: real > real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A2 @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B2 )
             => ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
       => ? [Z3: real] :
            ( ( ord_less_real @ A2 @ Z3 )
            & ( ord_less_real @ Z3 @ B2 )
            & ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A2 ) )
              = ( times_times_real @ ( minus_minus_real @ B2 @ A2 ) @ ( F3 @ Z3 ) ) ) ) ) ) ).

% MVT2
thf(fact_9816_DERIV__neg__imp__decreasing,axiom,
    ! [A2: real,B2: real,F: real > real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A2 @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B2 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
       => ( ord_less_real @ ( F @ B2 ) @ ( F @ A2 ) ) ) ) ).

% DERIV_neg_imp_decreasing
thf(fact_9817_DERIV__pos__imp__increasing,axiom,
    ! [A2: real,B2: real,F: real > real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A2 @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B2 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
       => ( ord_less_real @ ( F @ A2 ) @ ( F @ B2 ) ) ) ) ).

% DERIV_pos_imp_increasing
thf(fact_9818_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A2: real,B2: real,F: real > real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A2 @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B2 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ zero_zero_real @ Y5 ) ) ) )
       => ( ord_less_eq_real @ ( F @ A2 ) @ ( F @ B2 ) ) ) ) ).

% DERIV_nonneg_imp_nondecreasing
thf(fact_9819_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A2: real,B2: real,F: real > real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A2 @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B2 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ Y5 @ zero_zero_real ) ) ) )
       => ( ord_less_eq_real @ ( F @ B2 ) @ ( F @ A2 ) ) ) ) ).

% DERIV_nonpos_imp_nonincreasing
thf(fact_9820_DERIV__isconst__all,axiom,
    ! [F: real > real,X: real,Y: real] :
      ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( ( F @ X )
        = ( F @ Y ) ) ) ).

% DERIV_isconst_all
thf(fact_9821_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_9822_DERIV__mirror,axiom,
    ! [F: real > real,Y: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ Y @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ X ) @ top_top_set_real ) )
      = ( has_fi5821293074295781190e_real
        @ ^ [X2: real] : ( F @ ( uminus_uminus_real @ X2 ) )
        @ ( uminus_uminus_real @ Y )
        @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_mirror
thf(fact_9823_DERIV__pos__inc__right,axiom,
    ! [F: real > real,L: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D6: real] :
            ( ( ord_less_real @ zero_zero_real @ D6 )
            & ! [H3: real] :
                ( ( ord_less_real @ zero_zero_real @ H3 )
               => ( ( ord_less_real @ H3 @ D6 )
                 => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H3 ) ) ) ) ) ) ) ) ).

% DERIV_pos_inc_right
thf(fact_9824_DERIV__neg__dec__right,axiom,
    ! [F: real > real,L: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D6: real] :
            ( ( ord_less_real @ zero_zero_real @ D6 )
            & ! [H3: real] :
                ( ( ord_less_real @ zero_zero_real @ H3 )
               => ( ( ord_less_real @ H3 @ D6 )
                 => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H3 ) ) @ ( F @ X ) ) ) ) ) ) ) ).

% DERIV_neg_dec_right
thf(fact_9825_DERIV__local__const,axiom,
    ! [F: real > real,L: real,X: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y4: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
             => ( ( F @ X )
                = ( F @ Y4 ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
thf(fact_9826_DERIV__pos__inc__left,axiom,
    ! [F: real > real,L: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D6: real] :
            ( ( ord_less_real @ zero_zero_real @ D6 )
            & ! [H3: real] :
                ( ( ord_less_real @ zero_zero_real @ H3 )
               => ( ( ord_less_real @ H3 @ D6 )
                 => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H3 ) ) @ ( F @ X ) ) ) ) ) ) ) ).

% DERIV_pos_inc_left
thf(fact_9827_DERIV__neg__dec__left,axiom,
    ! [F: real > real,L: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D6: real] :
            ( ( ord_less_real @ zero_zero_real @ D6 )
            & ! [H3: real] :
                ( ( ord_less_real @ zero_zero_real @ H3 )
               => ( ( ord_less_real @ H3 @ D6 )
                 => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H3 ) ) ) ) ) ) ) ) ).

% DERIV_neg_dec_left
thf(fact_9828_deriv__nonneg__imp__mono,axiom,
    ! [A2: real,B2: real,G: real > real,G2: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A2 @ B2 ) )
         => ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A2 @ B2 ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X3 ) ) )
       => ( ( ord_less_eq_real @ A2 @ B2 )
         => ( ord_less_eq_real @ ( G @ A2 ) @ ( G @ B2 ) ) ) ) ) ).

% deriv_nonneg_imp_mono
thf(fact_9829_DERIV__const__average,axiom,
    ! [A2: real,B2: real,V: real > real,K: real] :
      ( ( A2 != B2 )
     => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X3 @ 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_9830_DERIV__local__min,axiom,
    ! [F: real > real,L: real,X: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y4: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
thf(fact_9831_DERIV__local__max,axiom,
    ! [F: real > real,L: real,X: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y4: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
thf(fact_9832_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_9833_DERIV__pow,axiom,
    ! [N: nat,X: real,S2: set_real] :
      ( has_fi5821293074295781190e_real
      @ ^ [X2: real] : ( power_power_real @ X2 @ N )
      @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
      @ ( topolo2177554685111907308n_real @ X @ S2 ) ) ).

% DERIV_pow
thf(fact_9834_DERIV__fun__pow,axiom,
    ! [G: real > real,M: real,X: real,N: nat] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( has_fi5821293074295781190e_real
        @ ^ [X2: real] : ( power_power_real @ ( G @ X2 ) @ N )
        @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G @ X ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M )
        @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_fun_pow
thf(fact_9835_has__real__derivative__powr,axiom,
    ! [Z: real,R3: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( has_fi5821293074295781190e_real
        @ ^ [Z4: real] : ( powr_real @ Z4 @ 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_9836_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
          @ ^ [X2: real] : ( powr_real @ ( G @ X2 ) @ 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_9837_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_9838_DERIV__powr,axiom,
    ! [G: real > real,M: real,X: real,F: 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 @ F @ R3 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
         => ( has_fi5821293074295781190e_real
            @ ^ [X2: real] : ( powr_real @ ( G @ X2 ) @ ( F @ X2 ) )
            @ ( times_times_real @ ( powr_real @ ( G @ X ) @ ( F @ X ) ) @ ( plus_plus_real @ ( times_times_real @ R3 @ ( ln_ln_real @ ( G @ X ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X ) ) @ ( G @ X ) ) ) )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_powr
thf(fact_9839_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_9840_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_9841_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_9842_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D3: real] :
      ( ( X != zero_zero_real )
     => ( ( ( ord_less_real @ zero_zero_real @ X )
         => ( D3
            = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( ( ord_less_real @ X @ zero_zero_real )
           => ( D3
              = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
         => ( has_fi5821293074295781190e_real @ sqrt @ D3 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_real_sqrt_generic
thf(fact_9843_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_9844_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_9845_DERIV__real__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_real_root
thf(fact_9846_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_9847_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_9848_Maclaurin__all__le,axiom,
    ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
       => ? [T3: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
            & ( ( F @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_all_le
thf(fact_9849_Maclaurin__all__le__objl,axiom,
    ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
      ( ( ( ( Diff @ zero_zero_nat )
          = F )
        & ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
     => ? [T3: real] :
          ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
          & ( ( F @ X )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
                @ ( set_ord_lessThan_nat @ N ) )
              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).

% Maclaurin_all_le_objl
thf(fact_9850_DERIV__odd__real__root,axiom,
    ! [N: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( X != zero_zero_real )
       => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_odd_real_root
thf(fact_9851_Maclaurin,axiom,
    ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M4: nat,T3: real] :
                ( ( ( ord_less_nat @ M4 @ N )
                  & ( ord_less_eq_real @ zero_zero_real @ T3 )
                  & ( ord_less_eq_real @ T3 @ H2 ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
           => ? [T3: real] :
                ( ( ord_less_real @ zero_zero_real @ T3 )
                & ( ord_less_real @ T3 @ H2 )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin
thf(fact_9852_Maclaurin2,axiom,
    ! [H2: real,Diff: nat > real > real,F: real > real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T3: real] :
              ( ( ( ord_less_nat @ M4 @ N )
                & ( ord_less_eq_real @ zero_zero_real @ T3 )
                & ( ord_less_eq_real @ T3 @ H2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
         => ? [T3: real] :
              ( ( ord_less_real @ zero_zero_real @ T3 )
              & ( ord_less_eq_real @ T3 @ H2 )
              & ( ( F @ H2 )
                = ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
                    @ ( set_ord_lessThan_nat @ N ) )
                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ).

% Maclaurin2
thf(fact_9853_Maclaurin__minus,axiom,
    ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ H2 @ zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M4: nat,T3: real] :
                ( ( ( ord_less_nat @ M4 @ N )
                  & ( ord_less_eq_real @ H2 @ T3 )
                  & ( ord_less_eq_real @ T3 @ zero_zero_real ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
           => ? [T3: real] :
                ( ( ord_less_real @ H2 @ T3 )
                & ( ord_less_real @ T3 @ zero_zero_real )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin_minus
thf(fact_9854_Maclaurin__all__lt,axiom,
    ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( X != zero_zero_real )
         => ( ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
           => ? [T3: real] :
                ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T3 ) )
                & ( ord_less_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
                & ( ( F @ X )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin_all_lt
thf(fact_9855_Maclaurin__bi__le,axiom,
    ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M4: nat,T3: real] :
            ( ( ( ord_less_nat @ M4 @ N )
              & ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) ) )
           => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
       => ? [T3: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
            & ( ( F @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_bi_le
thf(fact_9856_Taylor__down,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A2: real,B2: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T3: real] :
              ( ( ( ord_less_nat @ M4 @ N )
                & ( ord_less_eq_real @ A2 @ T3 )
                & ( ord_less_eq_real @ T3 @ B2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
         => ( ( ord_less_real @ A2 @ C )
           => ( ( ord_less_eq_real @ C @ B2 )
             => ? [T3: real] :
                  ( ( ord_less_real @ A2 @ T3 )
                  & ( ord_less_real @ T3 @ C )
                  & ( ( F @ A2 )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ A2 @ C ) @ M3 ) )
                        @ ( set_ord_lessThan_nat @ N ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A2 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).

% Taylor_down
thf(fact_9857_Taylor__up,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A2: real,B2: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T3: real] :
              ( ( ( ord_less_nat @ M4 @ N )
                & ( ord_less_eq_real @ A2 @ T3 )
                & ( ord_less_eq_real @ T3 @ B2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A2 @ C )
           => ( ( ord_less_real @ C @ B2 )
             => ? [T3: real] :
                  ( ( ord_less_real @ C @ T3 )
                  & ( ord_less_real @ T3 @ B2 )
                  & ( ( F @ B2 )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ B2 @ C ) @ M3 ) )
                        @ ( set_ord_lessThan_nat @ N ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B2 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).

% Taylor_up
thf(fact_9858_Taylor,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A2: real,B2: real,C: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T3: real] :
              ( ( ( ord_less_nat @ M4 @ N )
                & ( ord_less_eq_real @ A2 @ T3 )
                & ( ord_less_eq_real @ T3 @ B2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A2 @ C )
           => ( ( ord_less_eq_real @ C @ B2 )
             => ( ( ord_less_eq_real @ A2 @ X )
               => ( ( ord_less_eq_real @ X @ B2 )
                 => ( ( X != C )
                   => ? [T3: real] :
                        ( ( ( ord_less_real @ X @ C )
                         => ( ( ord_less_real @ X @ T3 )
                            & ( ord_less_real @ T3 @ C ) ) )
                        & ( ~ ( ord_less_real @ X @ C )
                         => ( ( ord_less_real @ C @ T3 )
                            & ( ord_less_real @ T3 @ X ) ) )
                        & ( ( F @ X )
                          = ( plus_plus_real
                            @ ( groups6591440286371151544t_real
                              @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ M3 ) )
                              @ ( set_ord_lessThan_nat @ N ) )
                            @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
thf(fact_9859_Maclaurin__lemma2,axiom,
    ! [N: nat,H2: real,Diff: nat > real > real,K: nat,B: real] :
      ( ! [M4: nat,T3: real] :
          ( ( ( ord_less_nat @ M4 @ N )
            & ( ord_less_eq_real @ zero_zero_real @ T3 )
            & ( ord_less_eq_real @ T3 @ H2 ) )
         => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
     => ( ( N
          = ( suc @ K ) )
       => ! [M2: nat,T4: real] :
            ( ( ( ord_less_nat @ M2 @ N )
              & ( ord_less_eq_real @ zero_zero_real @ T4 )
              & ( ord_less_eq_real @ T4 @ H2 ) )
           => ( has_fi5821293074295781190e_real
              @ ^ [U: real] :
                  ( minus_minus_real @ ( Diff @ M2 @ U )
                  @ ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [P6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M2 @ P6 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ U @ P6 ) )
                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M2 ) ) )
                    @ ( times_times_real @ B @ ( divide_divide_real @ ( power_power_real @ U @ ( minus_minus_nat @ N @ M2 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) )
              @ ( minus_minus_real @ ( Diff @ ( suc @ M2 ) @ T4 )
                @ ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [P6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M2 ) @ P6 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ T4 @ P6 ) )
                    @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) )
                  @ ( times_times_real @ B @ ( divide_divide_real @ ( power_power_real @ T4 @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) ) ) ) )
              @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) ) ) ) ).

% Maclaurin_lemma2
thf(fact_9860_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( has_fi5821293074295781190e_real
        @ ^ [X9: real] :
            ( suminf_real
            @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
        @ ( suminf_real
          @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_arctan_series
thf(fact_9861_DERIV__power__series_H,axiom,
    ! [R: real,F: nat > real,X0: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
         => ( summable_real
            @ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( F @ N4 ) @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ ( power_power_real @ X3 @ N4 ) ) ) )
     => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
       => ( ( ord_less_real @ zero_zero_real @ R )
         => ( has_fi5821293074295781190e_real
            @ ^ [X2: real] :
                ( suminf_real
                @ ^ [N4: nat] : ( times_times_real @ ( F @ N4 ) @ ( power_power_real @ X2 @ ( suc @ N4 ) ) ) )
            @ ( suminf_real
              @ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( F @ N4 ) @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ ( power_power_real @ X0 @ N4 ) ) )
            @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).

% DERIV_power_series'
thf(fact_9862_tanh__real__bounds,axiom,
    ! [X: real] : ( member_real @ ( tanh_real @ X ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) ).

% tanh_real_bounds
thf(fact_9863_DERIV__isconst3,axiom,
    ! [A2: real,B2: real,X: real,Y: real,F: real > real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( member_real @ X @ ( set_or1633881224788618240n_real @ A2 @ B2 ) )
       => ( ( member_real @ Y @ ( set_or1633881224788618240n_real @ A2 @ B2 ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A2 @ B2 ) )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
           => ( ( F @ X )
              = ( F @ Y ) ) ) ) ) ) ).

% DERIV_isconst3
thf(fact_9864_DERIV__series_H,axiom,
    ! [F: real > nat > real,F3: real > nat > real,X0: real,A2: real,B2: real,L4: nat > real] :
      ( ! [N2: nat] :
          ( has_fi5821293074295781190e_real
          @ ^ [X2: real] : ( F @ X2 @ N2 )
          @ ( F3 @ X0 @ N2 )
          @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A2 @ B2 ) )
           => ( summable_real @ ( F @ X3 ) ) )
       => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A2 @ B2 ) )
         => ( ( summable_real @ ( F3 @ X0 ) )
           => ( ( summable_real @ L4 )
             => ( ! [N2: nat,X3: real,Y4: real] :
                    ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A2 @ B2 ) )
                   => ( ( member_real @ Y4 @ ( set_or1633881224788618240n_real @ A2 @ B2 ) )
                     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X3 @ N2 ) @ ( F @ Y4 @ N2 ) ) ) @ ( times_times_real @ ( L4 @ N2 ) @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y4 ) ) ) ) ) )
               => ( has_fi5821293074295781190e_real
                  @ ^ [X2: real] : ( suminf_real @ ( F @ X2 ) )
                  @ ( suminf_real @ ( F3 @ X0 ) )
                  @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_series'
thf(fact_9865_LIM__fun__less__zero,axiom,
    ! [F: real > real,L: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [R2: real] :
            ( ( ord_less_real @ zero_zero_real @ R2 )
            & ! [X6: real] :
                ( ( ( X6 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X6 ) ) @ R2 ) )
               => ( ord_less_real @ ( F @ X6 ) @ zero_zero_real ) ) ) ) ) ).

% LIM_fun_less_zero
thf(fact_9866_LIM__fun__not__zero,axiom,
    ! [F: real > real,L: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( L != zero_zero_real )
       => ? [R2: real] :
            ( ( ord_less_real @ zero_zero_real @ R2 )
            & ! [X6: real] :
                ( ( ( X6 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X6 ) ) @ R2 ) )
               => ( ( F @ X6 )
                 != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
thf(fact_9867_LIM__fun__gt__zero,axiom,
    ! [F: real > real,L: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [R2: real] :
            ( ( ord_less_real @ zero_zero_real @ R2 )
            & ! [X6: real] :
                ( ( ( X6 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X6 ) ) @ R2 ) )
               => ( ord_less_real @ zero_zero_real @ ( F @ X6 ) ) ) ) ) ) ).

% LIM_fun_gt_zero
thf(fact_9868_isCont__real__sqrt,axiom,
    ! [X: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ sqrt ) ).

% isCont_real_sqrt
thf(fact_9869_isCont__real__root,axiom,
    ! [X: real,N: nat] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ( root @ N ) ) ).

% isCont_real_root
thf(fact_9870_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U2: int] :
      ( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L @ one_one_int ) @ U2 )
      = ( set_or5832277885323065728an_int @ L @ U2 ) ) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_9871_isCont__inverse__function2,axiom,
    ! [A2: real,X: real,B2: real,G: real > real,F: real > real] :
      ( ( ord_less_real @ A2 @ X )
     => ( ( ord_less_real @ X @ B2 )
       => ( ! [Z3: real] :
              ( ( ord_less_eq_real @ A2 @ Z3 )
             => ( ( ord_less_eq_real @ Z3 @ B2 )
               => ( ( G @ ( F @ Z3 ) )
                  = Z3 ) ) )
         => ( ! [Z3: real] :
                ( ( ord_less_eq_real @ A2 @ Z3 )
               => ( ( ord_less_eq_real @ Z3 @ B2 )
                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ) ).

% isCont_inverse_function2
thf(fact_9872_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_9873_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U2: code_integer] :
      ( ( set_or8404916559141939852nteger @ ( plus_p5714425477246183910nteger @ L @ one_one_Code_integer ) @ U2 )
      = ( set_or4266950643985792945nteger @ L @ U2 ) ) ).

% atLeastPlusOneLessThan_greaterThanLessThan_integer
thf(fact_9874_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_9875_LIM__cos__div__sin,axiom,
    ( filterlim_real_real
    @ ^ [X2: real] : ( divide_divide_real @ ( cos_real @ X2 ) @ ( sin_real @ X2 ) )
    @ ( 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_9876_DERIV__inverse__function,axiom,
    ! [F: real > real,D3: real,G: real > real,X: real,A2: real,B2: real] :
      ( ( has_fi5821293074295781190e_real @ F @ D3 @ ( topolo2177554685111907308n_real @ ( G @ X ) @ top_top_set_real ) )
     => ( ( D3 != zero_zero_real )
       => ( ( ord_less_real @ A2 @ X )
         => ( ( ord_less_real @ X @ B2 )
           => ( ! [Y4: real] :
                  ( ( ord_less_real @ A2 @ Y4 )
                 => ( ( ord_less_real @ Y4 @ B2 )
                   => ( ( F @ ( G @ Y4 ) )
                      = Y4 ) ) )
             => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ G )
               => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D3 ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_inverse_function
thf(fact_9877_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_9878_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_9879_LIM__less__bound,axiom,
    ! [B2: real,X: real,F: real > real] :
      ( ( ord_less_real @ B2 @ X )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ B2 @ X ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) ) ) ) ).

% LIM_less_bound
thf(fact_9880_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_9881_isCont__inverse__function,axiom,
    ! [D: real,X: real,G: real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ D )
     => ( ! [Z3: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z3 @ X ) ) @ D )
           => ( ( G @ ( F @ Z3 ) )
              = Z3 ) )
       => ( ! [Z3: real] :
              ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z3 @ X ) ) @ D )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) )
         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ).

% isCont_inverse_function
thf(fact_9882_GMVT_H,axiom,
    ! [A2: real,B2: real,F: real > real,G: real > real,G2: real > real,F3: real > real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ! [Z3: real] :
            ( ( ord_less_eq_real @ A2 @ Z3 )
           => ( ( ord_less_eq_real @ Z3 @ B2 )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) ) )
       => ( ! [Z3: real] :
              ( ( ord_less_eq_real @ A2 @ Z3 )
             => ( ( ord_less_eq_real @ Z3 @ B2 )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ G ) ) )
         => ( ! [Z3: real] :
                ( ( ord_less_real @ A2 @ Z3 )
               => ( ( ord_less_real @ Z3 @ B2 )
                 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z3 ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) )
           => ( ! [Z3: real] :
                  ( ( ord_less_real @ A2 @ Z3 )
                 => ( ( ord_less_real @ Z3 @ B2 )
                   => ( has_fi5821293074295781190e_real @ F @ ( F3 @ Z3 ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) )
             => ? [C4: real] :
                  ( ( ord_less_real @ A2 @ C4 )
                  & ( ord_less_real @ C4 @ B2 )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F @ B2 ) @ ( F @ A2 ) ) @ ( G2 @ C4 ) )
                    = ( times_times_real @ ( minus_minus_real @ ( G @ B2 ) @ ( G @ A2 ) ) @ ( F3 @ C4 ) ) ) ) ) ) ) ) ) ).

% GMVT'
thf(fact_9883_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
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N10 ) @ one_one_nat ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N10 ) ) ) ) ) ) ) ) ).

% summable_Leibniz(3)
thf(fact_9884_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
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N10 ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N10 ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% summable_Leibniz(2)
thf(fact_9885_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
thf(fact_9886_filterlim__Suc,axiom,
    filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).

% filterlim_Suc
thf(fact_9887_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( filterlim_nat_nat
        @ ^ [X2: nat] : ( times_times_nat @ X2 @ C )
        @ at_top_nat
        @ at_top_nat ) ) ).

% mult_nat_right_at_top
thf(fact_9888_mult__nat__left__at__top,axiom,
    ! [C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).

% mult_nat_left_at_top
thf(fact_9889_LIMSEQ__root,axiom,
    ( filterlim_nat_real
    @ ^ [N4: nat] : ( root @ N4 @ ( semiri5074537144036343181t_real @ N4 ) )
    @ ( topolo2815343760600316023s_real @ one_one_real )
    @ at_top_nat ) ).

% LIMSEQ_root
thf(fact_9890_nested__sequence__unique,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ! [N2: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N2 ) ) @ ( G @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G @ N2 ) )
         => ( ( filterlim_nat_real
              @ ^ [N4: nat] : ( minus_minus_real @ ( F @ N4 ) @ ( G @ N4 ) )
              @ ( topolo2815343760600316023s_real @ zero_zero_real )
              @ at_top_nat )
           => ? [L3: real] :
                ( ! [N10: nat] : ( ord_less_eq_real @ ( F @ N10 ) @ L3 )
                & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L3 ) @ at_top_nat )
                & ! [N10: nat] : ( ord_less_eq_real @ L3 @ ( G @ N10 ) )
                & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L3 ) @ at_top_nat ) ) ) ) ) ) ).

% nested_sequence_unique
thf(fact_9891_LIMSEQ__inverse__zero,axiom,
    ! [X5: nat > real] :
      ( ! [R2: real] :
        ? [N8: nat] :
        ! [N2: nat] :
          ( ( ord_less_eq_nat @ N8 @ N2 )
         => ( ord_less_real @ R2 @ ( X5 @ N2 ) ) )
     => ( filterlim_nat_real
        @ ^ [N4: nat] : ( inverse_inverse_real @ ( X5 @ N4 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_zero
thf(fact_9892_lim__inverse__n_H,axiom,
    ( filterlim_nat_real
    @ ^ [N4: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N4 ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% lim_inverse_n'
thf(fact_9893_LIMSEQ__inverse__real__of__nat,axiom,
    ( filterlim_nat_real
    @ ^ [N4: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat
thf(fact_9894_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( filterlim_nat_real
        @ ^ [N4: nat] : ( root @ N4 @ C )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat ) ) ).

% LIMSEQ_root_const
thf(fact_9895_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R3: real] :
      ( filterlim_nat_real
      @ ^ [N4: nat] : ( plus_plus_real @ R3 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) )
      @ ( topolo2815343760600316023s_real @ R3 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add
thf(fact_9896_increasing__LIMSEQ,axiom,
    ! [F: nat > real,L: real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ L )
       => ( ! [E: real] :
              ( ( ord_less_real @ zero_zero_real @ E )
             => ? [N10: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N10 ) @ E ) ) )
         => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).

% increasing_LIMSEQ
thf(fact_9897_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_9898_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,A2: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( filterlim_nat_real
        @ ^ [N4: nat] : ( divide_divide_real @ A2 @ ( power_power_real @ X @ N4 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_divide_realpow_zero
thf(fact_9899_LIMSEQ__abs__realpow__zero,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
     => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).

% LIMSEQ_abs_realpow_zero
thf(fact_9900_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
     => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).

% LIMSEQ_abs_realpow_zero2
thf(fact_9901_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( filterlim_nat_real
        @ ^ [N4: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N4 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_realpow_zero
thf(fact_9902_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R3: real] :
      ( filterlim_nat_real
      @ ^ [N4: nat] : ( plus_plus_real @ R3 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R3 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_9903_tendsto__exp__limit__sequentially,axiom,
    ! [X: real] :
      ( filterlim_nat_real
      @ ^ [N4: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N4 ) ) ) @ N4 )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
      @ at_top_nat ) ).

% tendsto_exp_limit_sequentially
thf(fact_9904_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R3: real] :
      ( filterlim_nat_real
      @ ^ [N4: nat] : ( times_times_real @ R3 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R3 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_9905_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
          @ ^ [N4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( A2 @ N4 ) ) ) ) ) ).

% summable_Leibniz(1)
thf(fact_9906_summable,axiom,
    ! [A2: nat > real] :
      ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N2 ) ) @ ( A2 @ N2 ) )
         => ( summable_real
            @ ^ [N4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( A2 @ N4 ) ) ) ) ) ) ).

% summable
thf(fact_9907_cos__diff__limit__1,axiom,
    ! [Theta: nat > real,Theta2: real] :
      ( ( filterlim_nat_real
        @ ^ [J3: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J3 ) @ Theta2 ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ~ ! [K2: nat > int] :
            ~ ( filterlim_nat_real
              @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
              @ ( topolo2815343760600316023s_real @ Theta2 )
              @ at_top_nat ) ) ).

% cos_diff_limit_1
thf(fact_9908_cos__limit__1,axiom,
    ! [Theta: nat > real] :
      ( ( filterlim_nat_real
        @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ? [K2: nat > int] :
          ( filterlim_nat_real
          @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
          @ ( topolo2815343760600316023s_real @ zero_zero_real )
          @ at_top_nat ) ) ).

% cos_limit_1
thf(fact_9909_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
          @ ^ [N4: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(4)
thf(fact_9910_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( filterlim_nat_real
        @ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% zeroseq_arctan_series
thf(fact_9911_summable__Leibniz_H_I3_J,axiom,
    ! [A2: nat > real] :
      ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N2 ) ) @ ( A2 @ N2 ) )
         => ( filterlim_nat_real
            @ ^ [N4: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(3)
thf(fact_9912_summable__Leibniz_H_I2_J,axiom,
    ! [A2: nat > real,N: nat] :
      ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N2 ) ) @ ( A2 @ N2 ) )
         => ( ord_less_eq_real
            @ ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
            @ ( suminf_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) ) ) ) ) ) ) ).

% summable_Leibniz'(2)
thf(fact_9913_sums__alternating__upper__lower,axiom,
    ! [A2: nat > real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N2 ) ) @ ( A2 @ N2 ) )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N2 ) )
       => ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
         => ? [L3: real] :
              ( ! [N10: nat] :
                  ( ord_less_eq_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N10 ) ) )
                  @ L3 )
              & ( filterlim_nat_real
                @ ^ [N4: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
                @ ( topolo2815343760600316023s_real @ L3 )
                @ at_top_nat )
              & ! [N10: nat] :
                  ( ord_less_eq_real @ L3
                  @ ( groups6591440286371151544t_real
                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N10 ) @ one_one_nat ) ) ) )
              & ( filterlim_nat_real
                @ ^ [N4: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
                @ ( topolo2815343760600316023s_real @ L3 )
                @ at_top_nat ) ) ) ) ) ).

% sums_alternating_upper_lower
thf(fact_9914_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
          @ ^ [N4: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(5)
thf(fact_9915_summable__Leibniz_H_I4_J,axiom,
    ! [A2: nat > real,N: nat] :
      ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N2 ) ) @ ( A2 @ N2 ) )
         => ( ord_less_eq_real
            @ ( suminf_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) ) )
            @ ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).

% summable_Leibniz'(4)
thf(fact_9916_summable__Leibniz_H_I5_J,axiom,
    ! [A2: nat > real] :
      ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N2 ) ) @ ( A2 @ N2 ) )
         => ( filterlim_nat_real
            @ ^ [N4: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(5)
thf(fact_9917_filterlim__int__sequentially,axiom,
    filterlim_nat_int @ semiri1314217659103216013at_int @ at_top_int @ at_top_nat ).

% filterlim_int_sequentially
thf(fact_9918_exp__at__top,axiom,
    filterlim_real_real @ exp_real @ at_top_real @ at_top_real ).

% exp_at_top
thf(fact_9919_sqrt__at__top,axiom,
    filterlim_real_real @ sqrt @ at_top_real @ at_top_real ).

% sqrt_at_top
thf(fact_9920_ln__at__top,axiom,
    filterlim_real_real @ ln_ln_real @ at_top_real @ at_top_real ).

% ln_at_top
thf(fact_9921_filterlim__real__sequentially,axiom,
    filterlim_nat_real @ semiri5074537144036343181t_real @ at_top_real @ at_top_nat ).

% filterlim_real_sequentially
thf(fact_9922_tanh__real__at__top,axiom,
    filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ one_one_real ) @ at_top_real ).

% tanh_real_at_top
thf(fact_9923_artanh__real__at__left__1,axiom,
    filterlim_real_real @ artanh_real @ at_top_real @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5984915006950818249n_real @ one_one_real ) ) ).

% artanh_real_at_left_1
thf(fact_9924_ln__x__over__x__tendsto__0,axiom,
    ( filterlim_real_real
    @ ^ [X2: real] : ( divide_divide_real @ ( ln_ln_real @ X2 ) @ X2 )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_real ) ).

% ln_x_over_x_tendsto_0
thf(fact_9925_tendsto__power__div__exp__0,axiom,
    ! [K: nat] :
      ( filterlim_real_real
      @ ^ [X2: real] : ( divide_divide_real @ ( power_power_real @ X2 @ K ) @ ( exp_real @ X2 ) )
      @ ( topolo2815343760600316023s_real @ zero_zero_real )
      @ at_top_real ) ).

% tendsto_power_div_exp_0
thf(fact_9926_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_9927_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_9928_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_9929_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B2: real,F: real > real,Flim: real] :
      ( ! [X3: real] :
          ( ( ord_less_eq_real @ B2 @ X3 )
         => ? [Y5: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              & ( ord_less_real @ Y5 @ zero_zero_real ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
       => ( ord_less_real @ Flim @ ( F @ B2 ) ) ) ) ).

% DERIV_neg_imp_decreasing_at_top
thf(fact_9930_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F: real > real,F4: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
         => ( filterlim_real_real
            @ ^ [X2: real] : ( power_power_real @ ( F @ X2 ) @ N )
            @ at_top_real
            @ F4 ) ) ) ) ).

% filterlim_pow_at_bot_even
thf(fact_9931_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_9932_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_9933_artanh__real__at__right__1,axiom,
    filterlim_real_real @ artanh_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ one_one_real ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ one_one_real ) ) ) ).

% artanh_real_at_right_1
thf(fact_9934_filterlim__uminus__at__bot__at__top,axiom,
    filterlim_real_real @ uminus_uminus_real @ at_bot_real @ at_top_real ).

% filterlim_uminus_at_bot_at_top
thf(fact_9935_filterlim__uminus__at__top__at__bot,axiom,
    filterlim_real_real @ uminus_uminus_real @ at_top_real @ at_bot_real ).

% filterlim_uminus_at_top_at_bot
thf(fact_9936_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_9937_exp__at__bot,axiom,
    filterlim_real_real @ exp_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_bot_real ).

% exp_at_bot
thf(fact_9938_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_9939_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_9940_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_9941_tanh__real__at__bot,axiom,
    filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ one_one_real ) ) @ at_bot_real ).

% tanh_real_at_bot
thf(fact_9942_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_9943_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_9944_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B2: real,F: real > real,Flim: real] :
      ( ! [X3: real] :
          ( ( ord_less_eq_real @ X3 @ B2 )
         => ? [Y5: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              & ( ord_less_real @ zero_zero_real @ Y5 ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
       => ( ord_less_real @ Flim @ ( F @ B2 ) ) ) ) ).

% DERIV_pos_imp_increasing_at_bot
thf(fact_9945_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F: real > real,F4: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
       => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
         => ( filterlim_real_real
            @ ^ [X2: real] : ( power_power_real @ ( F @ X2 ) @ N )
            @ at_bot_real
            @ F4 ) ) ) ) ).

% filterlim_pow_at_bot_odd
thf(fact_9946_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_9947_lhopital__left__at__top,axiom,
    ! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
     => ( ( eventually_real
          @ ^ [X2: real] :
              ( ( G2 @ X2 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ).

% lhopital_left_at_top
thf(fact_9948_lhopital__right__at__top,axiom,
    ! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
     => ( ( eventually_real
          @ ^ [X2: real] :
              ( ( G2 @ X2 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ).

% lhopital_right_at_top
thf(fact_9949_greaterThan__0,axiom,
    ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
    = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).

% greaterThan_0
thf(fact_9950_eventually__at__right__to__0,axiom,
    ! [P: real > $o,A2: real] :
      ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
      = ( eventually_real
        @ ^ [X2: real] : ( P @ ( plus_plus_real @ X2 @ A2 ) )
        @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% eventually_at_right_to_0
thf(fact_9951_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_9952_eventually__at__left__to__right,axiom,
    ! [P: real > $o,A2: real] :
      ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
      = ( eventually_real
        @ ^ [X2: real] : ( P @ ( uminus_uminus_real @ X2 ) )
        @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ A2 ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ A2 ) ) ) ) ) ).

% eventually_at_left_to_right
thf(fact_9953_eventually__at__right__real,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( eventually_real
        @ ^ [X2: real] : ( member_real @ X2 @ ( set_or1633881224788618240n_real @ A2 @ B2 ) )
        @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) ) ) ).

% eventually_at_right_real
thf(fact_9954_eventually__at__left__real,axiom,
    ! [B2: real,A2: real] :
      ( ( ord_less_real @ B2 @ A2 )
     => ( eventually_real
        @ ^ [X2: real] : ( member_real @ X2 @ ( set_or1633881224788618240n_real @ B2 @ A2 ) )
        @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) ) ) ).

% eventually_at_left_real
thf(fact_9955_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
        @ ^ [X2: real] : ( P @ ( inverse_inverse_real @ X2 ) )
        @ at_top_real ) ) ).

% eventually_at_right_to_top
thf(fact_9956_eventually__at__top__to__right,axiom,
    ! [P: real > $o] :
      ( ( eventually_real @ P @ at_top_real )
      = ( eventually_real
        @ ^ [X2: real] : ( P @ ( inverse_inverse_real @ X2 ) )
        @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% eventually_at_top_to_right
thf(fact_9957_lhopital__at__top__at__top,axiom,
    ! [F: real > real,A2: real,G: real > real,F3: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F @ 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
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ) ) ).

% lhopital_at_top_at_top
thf(fact_9958_lhopital,axiom,
    ! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X2: real] :
                ( ( G @ X2 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X2: real] :
                  ( ( G2 @ X2 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
           => ( ( eventually_real
                @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
             => ( ( eventually_real
                  @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
               => ( ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                    @ F4
                    @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                 => ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                    @ F4
                    @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ) ).

% lhopital
thf(fact_9959_lhopital__right__at__top__at__top,axiom,
    ! [F: real > real,A2: real,G: real > real,F3: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F @ 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
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) ) ) ) ) ) ) ).

% lhopital_right_at_top_at_top
thf(fact_9960_lhopital__at__top__at__bot,axiom,
    ! [F: real > real,A2: real,G: real > real,F3: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F @ 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
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ) ) ).

% lhopital_at_top_at_bot
thf(fact_9961_lhopital__left__at__top__at__top,axiom,
    ! [F: real > real,A2: real,G: real > real,F3: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F @ 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
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) ) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
thf(fact_9962_lhopital__at__top,axiom,
    ! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( eventually_real
          @ ^ [X2: real] :
              ( ( G2 @ X2 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).

% lhopital_at_top
thf(fact_9963_lhospital__at__top__at__top,axiom,
    ! [G: real > real,G2: real > real,F: real > real,F3: real > real,X: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
     => ( ( eventually_real
          @ ^ [X2: real] :
              ( ( G2 @ X2 )
             != zero_zero_real )
          @ at_top_real )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ at_top_real )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ at_top_real )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ at_top_real )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ at_top_real ) ) ) ) ) ) ).

% lhospital_at_top_at_top
thf(fact_9964_lhopital__right,axiom,
    ! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] :
                ( ( G @ X2 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] :
                  ( ( G2 @ X2 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
           => ( ( eventually_real
                @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
             => ( ( eventually_real
                  @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                    @ F4
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
                 => ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                    @ F4
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ) ) ).

% lhopital_right
thf(fact_9965_lhopital__right__0,axiom,
    ! [F0: real > real,G0: real > real,G2: real > real,F3: real > real,F4: 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
            @ ^ [X2: real] :
                ( ( G0 @ X2 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] :
                  ( ( G2 @ X2 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
           => ( ( eventually_real
                @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F0 @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
             => ( ( eventually_real
                  @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G0 @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                    @ F4
                    @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
                 => ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F0 @ X2 ) @ ( G0 @ X2 ) )
                    @ F4
                    @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ) ) ).

% lhopital_right_0
thf(fact_9966_lhopital__left,axiom,
    ! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] :
                ( ( G @ X2 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] :
                  ( ( G2 @ X2 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
           => ( ( eventually_real
                @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
             => ( ( eventually_real
                  @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                    @ F4
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
                 => ( filterlim_real_real
                    @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                    @ F4
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ) ) ).

% lhopital_left
thf(fact_9967_lhopital__right__at__top__at__bot,axiom,
    ! [F: real > real,A2: real,G: real > real,F3: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F @ 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
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) ) ) ) ) ) ) ).

% lhopital_right_at_top_at_bot
thf(fact_9968_lhopital__left__at__top__at__bot,axiom,
    ! [F: real > real,A2: real,G: real > real,F3: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F @ 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
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) ) ) ) ) ) ) ).

% lhopital_left_at_top_at_bot
thf(fact_9969_lhopital__right__0__at__top,axiom,
    ! [G: real > real,G2: real > real,F: real > real,F3: real > real,X: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
     => ( ( eventually_real
          @ ^ [X2: real] :
              ( ( G2 @ X2 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
       => ( ( eventually_real
            @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
         => ( ( eventually_real
              @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
           => ( ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
             => ( filterlim_real_real
                @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ).

% lhopital_right_0_at_top
thf(fact_9970_eventually__sequentially__Suc,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat
        @ ^ [I4: nat] : ( P @ ( suc @ I4 ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_Suc
thf(fact_9971_eventually__sequentially__seg,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat
        @ ^ [N4: nat] : ( P @ ( plus_plus_nat @ N4 @ K ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_seg
thf(fact_9972_le__sequentially,axiom,
    ! [F4: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ F4 @ at_top_nat )
      = ( ! [N11: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N11 ) @ F4 ) ) ) ).

% le_sequentially
thf(fact_9973_sequentially__offset,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat @ P @ at_top_nat )
     => ( eventually_nat
        @ ^ [I4: nat] : ( P @ ( plus_plus_nat @ I4 @ K ) )
        @ at_top_nat ) ) ).

% sequentially_offset
thf(fact_9974_eventually__False__sequentially,axiom,
    ~ ( eventually_nat
      @ ^ [N4: nat] : $false
      @ at_top_nat ) ).

% eventually_False_sequentially
thf(fact_9975_eventually__sequentially,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat @ P @ at_top_nat )
      = ( ? [N11: nat] :
          ! [N4: nat] :
            ( ( ord_less_eq_nat @ N11 @ N4 )
           => ( P @ N4 ) ) ) ) ).

% eventually_sequentially
thf(fact_9976_eventually__sequentiallyI,axiom,
    ! [C: nat,P: nat > $o] :
      ( ! [X3: nat] :
          ( ( ord_less_eq_nat @ C @ X3 )
         => ( P @ X3 ) )
     => ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentiallyI
thf(fact_9977_filterlim__real__at__infinity__sequentially,axiom,
    filterlim_nat_real @ semiri5074537144036343181t_real @ at_infinity_real @ at_top_nat ).

% filterlim_real_at_infinity_sequentially
thf(fact_9978_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_9979_GreatestI__nat,axiom,
    ! [P: nat > $o,K: nat,B2: nat] :
      ( ( P @ K )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B2 ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_nat
thf(fact_9980_Greatest__le__nat,axiom,
    ! [P: nat > $o,K: nat,B2: nat] :
      ( ( P @ K )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B2 ) )
       => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).

% Greatest_le_nat
thf(fact_9981_GreatestI__ex__nat,axiom,
    ! [P: nat > $o,B2: nat] :
      ( ? [X_12: nat] : ( P @ X_12 )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B2 ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_9982_GMVT,axiom,
    ! [A2: real,B2: real,F: real > real,G: real > real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ! [X3: real] :
            ( ( ( ord_less_eq_real @ A2 @ X3 )
              & ( ord_less_eq_real @ X3 @ B2 ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F ) )
       => ( ! [X3: real] :
              ( ( ( ord_less_real @ A2 @ X3 )
                & ( ord_less_real @ X3 @ B2 ) )
             => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
         => ( ! [X3: real] :
                ( ( ( ord_less_eq_real @ A2 @ X3 )
                  & ( ord_less_eq_real @ X3 @ B2 ) )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ G ) )
           => ( ! [X3: real] :
                  ( ( ( ord_less_real @ A2 @ X3 )
                    & ( ord_less_real @ X3 @ B2 ) )
                 => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X3 @ 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 @ F @ 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 @ ( F @ B2 ) @ ( F @ A2 ) ) @ G_c )
                    = ( times_times_real @ ( minus_minus_real @ ( G @ B2 ) @ ( G @ A2 ) ) @ F_c ) ) ) ) ) ) ) ) ).

% GMVT
thf(fact_9983_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U2: int] :
      ( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L @ one_one_int ) @ U2 )
      = ( set_or6656581121297822940st_int @ L @ U2 ) ) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_9984_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U2: code_integer] :
      ( ( set_or189985376899183464nteger @ ( plus_p5714425477246183910nteger @ L @ one_one_Code_integer ) @ U2 )
      = ( set_or2715278749043346189nteger @ L @ U2 ) ) ).

% atLeastPlusOneAtMost_greaterThanAtMost_integer
thf(fact_9985_MVT,axiom,
    ! [A2: real,B2: real,F: real > real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ F )
       => ( ! [X3: real] :
              ( ( ord_less_real @ A2 @ X3 )
             => ( ( ord_less_real @ X3 @ B2 )
               => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
         => ? [L3: real,Z3: real] :
              ( ( ord_less_real @ A2 @ Z3 )
              & ( ord_less_real @ Z3 @ B2 )
              & ( has_fi5821293074295781190e_real @ F @ L3 @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) )
              & ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A2 ) )
                = ( times_times_real @ ( minus_minus_real @ B2 @ A2 ) @ L3 ) ) ) ) ) ) ).

% MVT
thf(fact_9986_atLeast__0,axiom,
    ( ( set_ord_atLeast_nat @ zero_zero_nat )
    = top_top_set_nat ) ).

% atLeast_0
thf(fact_9987_continuous__on__arcosh,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_set_real @ A @ ( set_ord_atLeast_real @ one_one_real ) )
     => ( topolo5044208981011980120l_real @ A @ arcosh_real ) ) ).

% continuous_on_arcosh
thf(fact_9988_continuous__on__arsinh_H,axiom,
    ! [A: set_real,F: real > real] :
      ( ( topolo5044208981011980120l_real @ A @ F )
     => ( topolo5044208981011980120l_real @ A
        @ ^ [X2: real] : ( arsinh_real @ ( F @ X2 ) ) ) ) ).

% continuous_on_arsinh'
thf(fact_9989_continuous__on__arcosh_H,axiom,
    ! [A: set_real,F: real > real] :
      ( ( topolo5044208981011980120l_real @ A @ F )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A )
           => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
       => ( topolo5044208981011980120l_real @ A
          @ ^ [X2: real] : ( arcosh_real @ ( F @ X2 ) ) ) ) ) ).

% continuous_on_arcosh'
thf(fact_9990_continuous__on__arccos_H,axiom,
    topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arccos ).

% continuous_on_arccos'
thf(fact_9991_continuous__on__arcsin_H,axiom,
    topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arcsin ).

% continuous_on_arcsin'
thf(fact_9992_continuous__on__artanh,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_set_real @ A @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
     => ( topolo5044208981011980120l_real @ A @ artanh_real ) ) ).

% continuous_on_artanh
thf(fact_9993_continuous__on__artanh_H,axiom,
    ! [A: set_real,F: real > real] :
      ( ( topolo5044208981011980120l_real @ A @ F )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A )
           => ( member_real @ ( F @ X3 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) )
       => ( topolo5044208981011980120l_real @ A
          @ ^ [X2: real] : ( artanh_real @ ( F @ X2 ) ) ) ) ) ).

% continuous_on_artanh'
thf(fact_9994_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_9995_Rolle__deriv,axiom,
    ! [A2: real,B2: real,F: real > real,F3: real > real > real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ( F @ A2 )
          = ( F @ B2 ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ F )
         => ( ! [X3: real] :
                ( ( ord_less_real @ A2 @ X3 )
               => ( ( ord_less_real @ X3 @ B2 )
                 => ( has_de1759254742604945161l_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
           => ? [Z3: real] :
                ( ( ord_less_real @ A2 @ Z3 )
                & ( ord_less_real @ Z3 @ B2 )
                & ( ( F3 @ Z3 )
                  = ( ^ [V2: real] : zero_zero_real ) ) ) ) ) ) ) ).

% Rolle_deriv
thf(fact_9996_mvt,axiom,
    ! [A2: real,B2: real,F: real > real,F3: real > real > real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ F )
       => ( ! [X3: real] :
              ( ( ord_less_real @ A2 @ X3 )
             => ( ( ord_less_real @ X3 @ B2 )
               => ( has_de1759254742604945161l_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
         => ~ ! [Xi: real] :
                ( ( ord_less_real @ A2 @ Xi )
               => ( ( ord_less_real @ Xi @ B2 )
                 => ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A2 ) )
                   != ( F3 @ Xi @ ( minus_minus_real @ B2 @ A2 ) ) ) ) ) ) ) ) ).

% mvt
thf(fact_9997_DERIV__pos__imp__increasing__open,axiom,
    ! [A2: real,B2: real,F: real > real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ! [X3: real] :
            ( ( ord_less_real @ A2 @ X3 )
           => ( ( ord_less_real @ X3 @ B2 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ F )
         => ( ord_less_real @ ( F @ A2 ) @ ( F @ B2 ) ) ) ) ) ).

% DERIV_pos_imp_increasing_open
thf(fact_9998_DERIV__neg__imp__decreasing__open,axiom,
    ! [A2: real,B2: real,F: real > real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ! [X3: real] :
            ( ( ord_less_real @ A2 @ X3 )
           => ( ( ord_less_real @ X3 @ B2 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ F )
         => ( ord_less_real @ ( F @ B2 ) @ ( F @ A2 ) ) ) ) ) ).

% DERIV_neg_imp_decreasing_open
thf(fact_9999_DERIV__isconst__end,axiom,
    ! [A2: real,B2: real,F: real > real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ F )
       => ( ! [X3: real] :
              ( ( ord_less_real @ A2 @ X3 )
             => ( ( ord_less_real @ X3 @ B2 )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
         => ( ( F @ B2 )
            = ( F @ A2 ) ) ) ) ) ).

% DERIV_isconst_end
thf(fact_10000_DERIV__isconst2,axiom,
    ! [A2: real,B2: real,F: real > real,X: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ F )
       => ( ! [X3: real] :
              ( ( ord_less_real @ A2 @ X3 )
             => ( ( ord_less_real @ X3 @ B2 )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
         => ( ( ord_less_eq_real @ A2 @ X )
           => ( ( ord_less_eq_real @ X @ B2 )
             => ( ( F @ X )
                = ( F @ A2 ) ) ) ) ) ) ) ).

% DERIV_isconst2
thf(fact_10001_Rolle,axiom,
    ! [A2: real,B2: real,F: real > real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ( F @ A2 )
          = ( F @ B2 ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ F )
         => ( ! [X3: real] :
                ( ( ord_less_real @ A2 @ X3 )
               => ( ( ord_less_real @ X3 @ B2 )
                 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
           => ? [Z3: real] :
                ( ( ord_less_real @ A2 @ Z3 )
                & ( ord_less_real @ Z3 @ B2 )
                & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) ) ) ) ) ).

% Rolle
thf(fact_10002_int__of__integer__code,axiom,
    ( code_int_of_integer
    = ( ^ [K3: code_integer] :
          ( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
          @ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
            @ ( produc1553301316500091796er_int
              @ ^ [L2: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ one_one_int ) )
              @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% int_of_integer_code
thf(fact_10003_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_10004_int__of__integer__of__nat,axiom,
    ! [N: nat] :
      ( ( code_int_of_integer @ ( semiri4939895301339042750nteger @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% int_of_integer_of_nat
thf(fact_10005_complex__Re__of__nat,axiom,
    ! [N: nat] :
      ( ( re @ ( semiri8010041392384452111omplex @ N ) )
      = ( semiri5074537144036343181t_real @ N ) ) ).

% complex_Re_of_nat
thf(fact_10006_zero__integer_Orep__eq,axiom,
    ( ( code_int_of_integer @ zero_z3403309356797280102nteger )
    = zero_zero_int ) ).

% zero_integer.rep_eq
thf(fact_10007_int__of__integer__numeral,axiom,
    ! [K: num] :
      ( ( code_int_of_integer @ ( numera6620942414471956472nteger @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% int_of_integer_numeral
thf(fact_10008_one__integer_Orep__eq,axiom,
    ( ( code_int_of_integer @ one_one_Code_integer )
    = one_one_int ) ).

% one_integer.rep_eq
thf(fact_10009_complex__Re__numeral,axiom,
    ! [V: num] :
      ( ( re @ ( numera6690914467698888265omplex @ V ) )
      = ( numeral_numeral_real @ V ) ) ).

% complex_Re_numeral
thf(fact_10010_Re__divide__of__nat,axiom,
    ! [Z: complex,N: nat] :
      ( ( re @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N ) ) )
      = ( divide_divide_real @ ( re @ Z ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% Re_divide_of_nat
thf(fact_10011_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_10012_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_10013_summable__Re,axiom,
    ! [F: nat > complex] :
      ( ( summable_complex @ F )
     => ( summable_real
        @ ^ [X2: nat] : ( re @ ( F @ X2 ) ) ) ) ).

% summable_Re
thf(fact_10014_zero__complex_Osimps_I1_J,axiom,
    ( ( re @ zero_zero_complex )
    = zero_zero_real ) ).

% zero_complex.simps(1)
thf(fact_10015_one__complex_Osimps_I1_J,axiom,
    ( ( re @ one_one_complex )
    = one_one_real ) ).

% one_complex.simps(1)
thf(fact_10016_imaginary__unit_Osimps_I1_J,axiom,
    ( ( re @ imaginary_unit )
    = zero_zero_real ) ).

% imaginary_unit.simps(1)
thf(fact_10017_sums__Re,axiom,
    ! [X5: nat > complex,A2: complex] :
      ( ( sums_complex @ X5 @ A2 )
     => ( sums_real
        @ ^ [N4: nat] : ( re @ ( X5 @ N4 ) )
        @ ( re @ A2 ) ) ) ).

% sums_Re
thf(fact_10018_Cauchy__Re,axiom,
    ! [X5: nat > complex] :
      ( ( topolo6517432010174082258omplex @ X5 )
     => ( topolo4055970368930404560y_real
        @ ^ [N4: nat] : ( re @ ( X5 @ N4 ) ) ) ) ).

% Cauchy_Re
thf(fact_10019_less__integer_Orep__eq,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [X2: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).

% less_integer.rep_eq
thf(fact_10020_integer__less__iff,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [K3: code_integer,L2: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).

% integer_less_iff
thf(fact_10021_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_10022_Re__csqrt,axiom,
    ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).

% Re_csqrt
thf(fact_10023_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_10024_bin__last__integer_Orep__eq,axiom,
    ( bits_b8758750999018896077nteger
    = ( ^ [X2: code_integer] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ X2 ) ) ) ) ).

% bin_last_integer.rep_eq
thf(fact_10025_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_10026_cos__n__Re__cis__pow__n,axiom,
    ! [N: nat,A2: real] :
      ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A2 ) )
      = ( re @ ( power_power_complex @ ( cis @ A2 ) @ N ) ) ) ).

% cos_n_Re_cis_pow_n
thf(fact_10027_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_10028_num__of__integer__code,axiom,
    ( code_num_of_integer
    = ( ^ [K3: code_integer] :
          ( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
          @ ( produc7336495610019696514er_num
            @ ^ [L2: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ one ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% num_of_integer_code
thf(fact_10029_csqrt_Ocode,axiom,
    ( csqrt
    = ( ^ [Z4: complex] :
          ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z4 ) @ ( re @ Z4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          @ ( times_times_real
            @ ( if_real
              @ ( ( im @ Z4 )
                = zero_zero_real )
              @ one_one_real
              @ ( sgn_sgn_real @ ( im @ Z4 ) ) )
            @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z4 ) @ ( re @ Z4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% csqrt.code
thf(fact_10030_complex__Im__fact,axiom,
    ! [N: nat] :
      ( ( im @ ( semiri5044797733671781792omplex @ N ) )
      = zero_zero_real ) ).

% complex_Im_fact
thf(fact_10031_complex__Im__of__int,axiom,
    ! [Z: int] :
      ( ( im @ ( ring_17405671764205052669omplex @ Z ) )
      = zero_zero_real ) ).

% complex_Im_of_int
thf(fact_10032_Im__complex__of__real,axiom,
    ! [Z: real] :
      ( ( im @ ( real_V4546457046886955230omplex @ Z ) )
      = zero_zero_real ) ).

% Im_complex_of_real
thf(fact_10033_Im__power__real,axiom,
    ! [X: complex,N: nat] :
      ( ( ( im @ X )
        = zero_zero_real )
     => ( ( im @ ( power_power_complex @ X @ N ) )
        = zero_zero_real ) ) ).

% Im_power_real
thf(fact_10034_complex__Im__numeral,axiom,
    ! [V: num] :
      ( ( im @ ( numera6690914467698888265omplex @ V ) )
      = zero_zero_real ) ).

% complex_Im_numeral
thf(fact_10035_complex__Im__of__nat,axiom,
    ! [N: nat] :
      ( ( im @ ( semiri8010041392384452111omplex @ N ) )
      = zero_zero_real ) ).

% complex_Im_of_nat
thf(fact_10036_Re__power__real,axiom,
    ! [X: complex,N: nat] :
      ( ( ( im @ X )
        = zero_zero_real )
     => ( ( re @ ( power_power_complex @ X @ N ) )
        = ( power_power_real @ ( re @ X ) @ N ) ) ) ).

% Re_power_real
thf(fact_10037_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_10038_Im__divide__of__nat,axiom,
    ! [Z: complex,N: nat] :
      ( ( im @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N ) ) )
      = ( divide_divide_real @ ( im @ Z ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% Im_divide_of_nat
thf(fact_10039_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_10040_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_10041_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_10042_complex__is__Int__iff,axiom,
    ! [Z: complex] :
      ( ( member_complex @ Z @ ring_1_Ints_complex )
      = ( ( ( im @ Z )
          = zero_zero_real )
        & ? [I4: int] :
            ( ( re @ Z )
            = ( ring_1_of_int_real @ I4 ) ) ) ) ).

% complex_is_Int_iff
thf(fact_10043_sums__complex__iff,axiom,
    ( sums_complex
    = ( ^ [F2: nat > complex,X2: complex] :
          ( ( sums_real
            @ ^ [Y2: nat] : ( re @ ( F2 @ Y2 ) )
            @ ( re @ X2 ) )
          & ( sums_real
            @ ^ [Y2: nat] : ( im @ ( F2 @ Y2 ) )
            @ ( im @ X2 ) ) ) ) ) ).

% sums_complex_iff
thf(fact_10044_sums__Im,axiom,
    ! [X5: nat > complex,A2: complex] :
      ( ( sums_complex @ X5 @ A2 )
     => ( sums_real
        @ ^ [N4: nat] : ( im @ ( X5 @ N4 ) )
        @ ( im @ A2 ) ) ) ).

% sums_Im
thf(fact_10045_Cauchy__Im,axiom,
    ! [X5: nat > complex] :
      ( ( topolo6517432010174082258omplex @ X5 )
     => ( topolo4055970368930404560y_real
        @ ^ [N4: nat] : ( im @ ( X5 @ N4 ) ) ) ) ).

% Cauchy_Im
thf(fact_10046_zero__complex_Osimps_I2_J,axiom,
    ( ( im @ zero_zero_complex )
    = zero_zero_real ) ).

% zero_complex.simps(2)
thf(fact_10047_imaginary__unit_Osimps_I2_J,axiom,
    ( ( im @ imaginary_unit )
    = one_one_real ) ).

% imaginary_unit.simps(2)
thf(fact_10048_one__complex_Osimps_I2_J,axiom,
    ( ( im @ one_one_complex )
    = zero_zero_real ) ).

% one_complex.simps(2)
thf(fact_10049_summable__Im,axiom,
    ! [F: nat > complex] :
      ( ( summable_complex @ F )
     => ( summable_real
        @ ^ [X2: nat] : ( im @ ( F @ X2 ) ) ) ) ).

% summable_Im
thf(fact_10050_summable__complex__iff,axiom,
    ( summable_complex
    = ( ^ [F2: nat > complex] :
          ( ( summable_real
            @ ^ [X2: nat] : ( re @ ( F2 @ X2 ) ) )
          & ( summable_real
            @ ^ [X2: nat] : ( im @ ( F2 @ X2 ) ) ) ) ) ) ).

% summable_complex_iff
thf(fact_10051_Im__eq__0,axiom,
    ! [Z: complex] :
      ( ( ( abs_abs_real @ ( re @ Z ) )
        = ( real_V1022390504157884413omplex @ Z ) )
     => ( ( im @ Z )
        = zero_zero_real ) ) ).

% Im_eq_0
thf(fact_10052_cmod__eq__Im,axiom,
    ! [Z: complex] :
      ( ( ( re @ Z )
        = zero_zero_real )
     => ( ( real_V1022390504157884413omplex @ Z )
        = ( abs_abs_real @ ( im @ Z ) ) ) ) ).

% cmod_eq_Im
thf(fact_10053_cmod__eq__Re,axiom,
    ! [Z: complex] :
      ( ( ( im @ Z )
        = zero_zero_real )
     => ( ( real_V1022390504157884413omplex @ Z )
        = ( abs_abs_real @ ( re @ Z ) ) ) ) ).

% cmod_eq_Re
thf(fact_10054_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_10055_sin__n__Im__cis__pow__n,axiom,
    ! [N: nat,A2: real] :
      ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A2 ) )
      = ( im @ ( power_power_complex @ ( cis @ A2 ) @ N ) ) ) ).

% sin_n_Im_cis_pow_n
thf(fact_10056_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_10057_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_10058_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_10059_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_10060_norm__complex__def,axiom,
    ( real_V1022390504157884413omplex
    = ( ^ [Z4: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% norm_complex_def
thf(fact_10061_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_10062_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_10063_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_10064_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_10065_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_10066_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_10067_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_10068_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_10069_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_10070_inverse__complex_Ocode,axiom,
    ( invers8013647133539491842omplex
    = ( ^ [X2: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X2 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X2 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% inverse_complex.code
thf(fact_10071_Complex__divide,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [X2: complex,Y2: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X2 ) @ ( re @ Y2 ) ) @ ( times_times_real @ ( im @ X2 ) @ ( 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 @ X2 ) @ ( re @ Y2 ) ) @ ( times_times_real @ ( re @ X2 ) @ ( 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_10072_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_10073_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_10074_nat__of__integer__code,axiom,
    ( code_nat_of_integer
    = ( ^ [K3: code_integer] :
          ( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
          @ ( produc1555791787009142072er_nat
            @ ^ [L2: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ one_one_nat ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% nat_of_integer_code
thf(fact_10075_nat__of__integer__numeral,axiom,
    ! [N: num] :
      ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( numeral_numeral_nat @ N ) ) ).

% nat_of_integer_numeral
thf(fact_10076_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_10077_nat__of__integer__1,axiom,
    ( ( code_nat_of_integer @ one_one_Code_integer )
    = one_one_nat ) ).

% nat_of_integer_1
thf(fact_10078_of__nat__of__integer,axiom,
    ! [K: code_integer] :
      ( ( semiri4939895301339042750nteger @ ( code_nat_of_integer @ K ) )
      = ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ K ) ) ).

% of_nat_of_integer
thf(fact_10079_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_10080_real__eq__imaginary__iff,axiom,
    ! [Y: complex,X: complex] :
      ( ( member_complex @ Y @ real_V2521375963428798218omplex )
     => ( ( member_complex @ X @ real_V2521375963428798218omplex )
       => ( ( X
            = ( times_times_complex @ imaginary_unit @ Y ) )
          = ( ( X = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% real_eq_imaginary_iff
thf(fact_10081_imaginary__eq__real__iff,axiom,
    ! [Y: complex,X: complex] :
      ( ( member_complex @ Y @ real_V2521375963428798218omplex )
     => ( ( member_complex @ X @ real_V2521375963428798218omplex )
       => ( ( ( times_times_complex @ imaginary_unit @ Y )
            = X )
          = ( ( X = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% imaginary_eq_real_iff
thf(fact_10082_complex__is__Real__iff,axiom,
    ! [Z: complex] :
      ( ( member_complex @ Z @ real_V2521375963428798218omplex )
      = ( ( im @ Z )
        = zero_zero_real ) ) ).

% complex_is_Real_iff
thf(fact_10083_Complex__in__Reals,axiom,
    ! [X: real] : ( member_complex @ ( complex2 @ X @ zero_zero_real ) @ real_V2521375963428798218omplex ) ).

% Complex_in_Reals
thf(fact_10084_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_10085_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_10086_image__atLeastZeroLessThan__integer,axiom,
    ! [U2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U2 )
     => ( ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ U2 )
        = ( image_1215581382706833972nteger @ semiri4939895301339042750nteger @ ( set_ord_lessThan_nat @ ( code_nat_of_integer @ U2 ) ) ) ) ) ).

% image_atLeastZeroLessThan_integer
thf(fact_10087_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_10088_integer__set__bit__code,axiom,
    ( bits_integer_set_bit
    = ( ^ [X2: code_integer,N4: code_integer,B4: $o] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ N4 @ zero_z3403309356797280102nteger ) @ ( undefi1878487536576149250nteger @ X2 @ N4 @ B4 ) @ ( if_Code_integer @ B4 @ ( bit_se1080825931792720795nteger @ X2 @ ( bit_se7788150548672797655nteger @ ( code_nat_of_integer @ N4 ) @ one_one_Code_integer ) ) @ ( bit_se3949692690581998587nteger @ X2 @ ( bit_ri7632146776885996613nteger @ ( bit_se7788150548672797655nteger @ ( code_nat_of_integer @ N4 ) @ one_one_Code_integer ) ) ) ) ) ) ) ).

% integer_set_bit_code
thf(fact_10089_uint32__shiftr__def,axiom,
    ( uint32_shiftr
    = ( ^ [X2: uint32,N4: code_integer] :
          ( if_uint32
          @ ( ( ord_le6747313008572928689nteger @ N4 @ zero_z3403309356797280102nteger )
            | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N4 ) )
          @ ( undefi8952517107220742160uint32 @ bit_se3964402333458159761uint32 @ X2 @ N4 )
          @ ( bit_se3964402333458159761uint32 @ ( code_nat_of_integer @ N4 ) @ X2 ) ) ) ) ).

% uint32_shiftr_def
thf(fact_10090_integer__set__bit__def,axiom,
    ( bits_integer_set_bit
    = ( ^ [X2: code_integer,N4: code_integer,B4: $o] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ N4 @ zero_z3403309356797280102nteger ) @ ( undefi1878487536576149250nteger @ X2 @ N4 @ B4 ) @ ( generi2397576812484419408nteger @ X2 @ ( code_nat_of_integer @ N4 ) @ B4 ) ) ) ) ).

% integer_set_bit_def
thf(fact_10091_uint32__shiftl__def,axiom,
    ( uint32_shiftl
    = ( ^ [X2: uint32,N4: code_integer] :
          ( if_uint32
          @ ( ( ord_le6747313008572928689nteger @ N4 @ zero_z3403309356797280102nteger )
            | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N4 ) )
          @ ( undefi8952517107220742160uint32 @ bit_se5742574853984576102uint32 @ X2 @ N4 )
          @ ( bit_se5742574853984576102uint32 @ ( code_nat_of_integer @ N4 ) @ X2 ) ) ) ) ).

% uint32_shiftl_def
thf(fact_10092_uint32__set__bit__def,axiom,
    ( uint32_set_bit
    = ( ^ [X2: uint32,N4: code_integer,B4: $o] :
          ( if_uint32
          @ ( ( ord_le6747313008572928689nteger @ N4 @ zero_z3403309356797280102nteger )
            | ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ N4 ) )
          @ ( undefi8537048349889504752uint32 @ generi1993664874377053279uint32 @ X2 @ N4 @ B4 )
          @ ( generi1993664874377053279uint32 @ X2 @ ( code_nat_of_integer @ N4 ) @ B4 ) ) ) ) ).

% uint32_set_bit_def
thf(fact_10093_uint32__test__bit__def,axiom,
    ( uint32_test_bit
    = ( ^ [X2: uint32,N4: code_integer] :
          ( ( ( ( ord_le6747313008572928689nteger @ N4 @ zero_z3403309356797280102nteger )
              | ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ N4 ) )
           => ( undefi6981832269580975664eger_o @ bit_se5367290876889521763uint32 @ X2 @ N4 ) )
          & ( ~ ( ( ord_le6747313008572928689nteger @ N4 @ zero_z3403309356797280102nteger )
                | ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ N4 ) )
           => ( bit_se5367290876889521763uint32 @ X2 @ ( code_nat_of_integer @ N4 ) ) ) ) ) ) ).

% uint32_test_bit_def
thf(fact_10094_uint32__shiftr__code,axiom,
    ! [N: code_integer,W: uint32] :
      ( ( ( ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger )
          | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N ) )
       => ( ( rep_uint322 @ ( uint32_shiftr @ W @ N ) )
          = ( rep_uint322 @ ( undefi8952517107220742160uint32 @ bit_se3964402333458159761uint32 @ W @ N ) ) ) )
      & ( ~ ( ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger )
            | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N ) )
       => ( ( rep_uint322 @ ( uint32_shiftr @ W @ N ) )
          = ( bit_se5176125413884933531l_num1 @ ( code_nat_of_integer @ N ) @ ( rep_uint322 @ W ) ) ) ) ) ).

% uint32_shiftr_code
thf(fact_10095_uint32_Oless__iff__word__of,axiom,
    ( ord_less_uint32
    = ( ^ [P6: uint32,Q6: uint32] : ( ord_le750835935415966154l_num1 @ ( rep_uint322 @ P6 ) @ ( rep_uint322 @ Q6 ) ) ) ) ).

% uint32.less_iff_word_of
thf(fact_10096_less__uint32_Orep__eq,axiom,
    ( ord_less_uint32
    = ( ^ [X2: uint32,Xa4: uint32] : ( ord_le750835935415966154l_num1 @ ( rep_uint322 @ X2 ) @ ( rep_uint322 @ Xa4 ) ) ) ) ).

% less_uint32.rep_eq
thf(fact_10097_zero__uint32_Orep__eq,axiom,
    ( ( rep_uint322 @ zero_zero_uint32 )
    = zero_z3563351764282998399l_num1 ) ).

% zero_uint32.rep_eq
thf(fact_10098_one__uint32_Orep__eq,axiom,
    ( ( rep_uint322 @ one_one_uint32 )
    = one_on7727431528512463931l_num1 ) ).

% one_uint32.rep_eq
thf(fact_10099_uint32_Oeven__iff__word__of,axiom,
    ! [P4: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ P4 )
      = ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( rep_uint322 @ P4 ) ) ) ).

% uint32.even_iff_word_of
thf(fact_10100_uint32__test__bit__code,axiom,
    ( uint32_test_bit
    = ( ^ [W2: uint32,N4: code_integer] :
          ( ( ( ( ord_le6747313008572928689nteger @ N4 @ zero_z3403309356797280102nteger )
              | ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ N4 ) )
           => ( undefi6981832269580975664eger_o @ bit_se5367290876889521763uint32 @ W2 @ N4 ) )
          & ( ~ ( ( ord_le6747313008572928689nteger @ N4 @ zero_z3403309356797280102nteger )
                | ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ N4 ) )
           => ( bit_se6859397288646540909l_num1 @ ( rep_uint322 @ W2 ) @ ( code_nat_of_integer @ N4 ) ) ) ) ) ) ).

% uint32_test_bit_code
thf(fact_10101_uint32__set__bit__code,axiom,
    ! [N: code_integer,W: uint32,B2: $o] :
      ( ( ( ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger )
          | ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ N ) )
       => ( ( rep_uint322 @ ( uint32_set_bit @ W @ N @ B2 ) )
          = ( rep_uint322 @ ( undefi8537048349889504752uint32 @ generi1993664874377053279uint32 @ W @ N @ B2 ) ) ) )
      & ( ~ ( ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger )
            | ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ N ) )
       => ( ( rep_uint322 @ ( uint32_set_bit @ W @ N @ B2 ) )
          = ( generi5268133209446125161l_num1 @ ( rep_uint322 @ W ) @ ( code_nat_of_integer @ N ) @ B2 ) ) ) ) ).

% uint32_set_bit_code
thf(fact_10102_uint32__shiftl__code,axiom,
    ! [N: code_integer,W: uint32] :
      ( ( ( ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger )
          | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N ) )
       => ( ( rep_uint322 @ ( uint32_shiftl @ W @ N ) )
          = ( rep_uint322 @ ( undefi8952517107220742160uint32 @ bit_se5742574853984576102uint32 @ W @ N ) ) ) )
      & ( ~ ( ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger )
            | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N ) )
       => ( ( rep_uint322 @ ( uint32_shiftl @ W @ N ) )
          = ( bit_se837345729053750000l_num1 @ ( code_nat_of_integer @ N ) @ ( rep_uint322 @ W ) ) ) ) ) ).

% uint32_shiftl_code
thf(fact_10103_Uint32__signed__code,axiom,
    ! [I: code_integer] :
      ( ( ( ( ord_le6747313008572928689nteger @ I @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
          | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ I ) )
       => ( ( rep_uint322 @ ( uint32_signed @ I ) )
          = ( rep_uint322 @ ( undefi2040150642751712519uint32 @ uint322 @ I ) ) ) )
      & ( ~ ( ( ord_le6747313008572928689nteger @ I @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
            | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ I ) )
       => ( ( rep_uint322 @ ( uint32_signed @ I ) )
          = ( ring_17408606157368542149l_num1 @ ( code_I935103866777955880mbolic @ I ) ) ) ) ) ).

% Uint32_signed_code
thf(fact_10104_integer__shiftl__def,axiom,
    ( bits_integer_shiftl
    = ( ^ [X2: code_integer,N4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ N4 @ zero_z3403309356797280102nteger ) @ ( undefi8133104259855420269nteger @ X2 @ N4 ) @ ( bit_se7788150548672797655nteger @ ( code_nat_of_integer @ N4 ) @ X2 ) ) ) ) ).

% integer_shiftl_def
thf(fact_10105_one__uint32_Orsp,axiom,
    one_on7727431528512463931l_num1 = one_on7727431528512463931l_num1 ).

% one_uint32.rsp
thf(fact_10106_zero__uint32_Orsp,axiom,
    zero_z3563351764282998399l_num1 = zero_z3563351764282998399l_num1 ).

% zero_uint32.rsp
thf(fact_10107_integer__shiftl__code_I2_J,axiom,
    ! [X: code_integer] :
      ( ( bits_integer_shiftl @ X @ zero_z3403309356797280102nteger )
      = X ) ).

% integer_shiftl_code(2)
thf(fact_10108_int__of__integer__symbolic__aux__code_I1_J,axiom,
    ( ( code_I935103866777955880mbolic @ zero_z3403309356797280102nteger )
    = zero_zero_int ) ).

% int_of_integer_symbolic_aux_code(1)
thf(fact_10109_uint32_Oset__bits__aux__code,axiom,
    ( set_bits_aux_uint32
    = ( ^ [F2: nat > $o,N4: nat,W2: uint32] : ( if_uint32 @ ( N4 = zero_zero_nat ) @ W2 @ ( set_bits_aux_uint32 @ F2 @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( bit_se2966626333419230250uint32 @ ( bit_se5742574853984576102uint32 @ one_one_nat @ W2 ) @ ( if_uint32 @ ( F2 @ ( minus_minus_nat @ N4 @ one_one_nat ) ) @ one_one_uint32 @ zero_zero_uint32 ) ) ) ) ) ) ).

% uint32.set_bits_aux_code
thf(fact_10110_shiftr__uint32__code,axiom,
    ( bit_se3964402333458159761uint32
    = ( ^ [N4: nat,X2: uint32] : ( if_uint32 @ ( ord_less_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) @ ( uint32_shiftr @ X2 @ ( code_integer_of_nat @ N4 ) ) @ zero_zero_uint32 ) ) ) ).

% shiftr_uint32_code
thf(fact_10111_shiftl__uint32__code,axiom,
    ( bit_se5742574853984576102uint32
    = ( ^ [N4: nat,X2: uint32] : ( if_uint32 @ ( ord_less_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) @ ( uint32_shiftl @ X2 @ ( code_integer_of_nat @ N4 ) ) @ zero_zero_uint32 ) ) ) ).

% shiftl_uint32_code
thf(fact_10112_integer__of__nat_Orep__eq,axiom,
    ! [X: nat] :
      ( ( code_int_of_integer @ ( code_integer_of_nat @ X ) )
      = ( semiri1314217659103216013at_int @ X ) ) ).

% integer_of_nat.rep_eq
thf(fact_10113_int__of__integer__integer__of__nat,axiom,
    ! [N: nat] :
      ( ( code_int_of_integer @ ( code_integer_of_nat @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% int_of_integer_integer_of_nat
thf(fact_10114_integer__of__nat_Oabs__eq,axiom,
    ( code_integer_of_nat
    = ( ^ [X2: nat] : ( code_integer_of_int @ ( semiri1314217659103216013at_int @ X2 ) ) ) ) ).

% integer_of_nat.abs_eq
thf(fact_10115_integer__of__nat__less__0__conv,axiom,
    ! [N: nat] :
      ~ ( ord_le6747313008572928689nteger @ ( code_integer_of_nat @ N ) @ zero_z3403309356797280102nteger ) ).

% integer_of_nat_less_0_conv
thf(fact_10116_integer__of__nat__numeral,axiom,
    ! [N: num] :
      ( ( code_integer_of_nat @ ( numeral_numeral_nat @ N ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% integer_of_nat_numeral
thf(fact_10117_integer__of__nat__0,axiom,
    ( ( code_integer_of_nat @ zero_zero_nat )
    = zero_z3403309356797280102nteger ) ).

% integer_of_nat_0
thf(fact_10118_integer__of__nat__1,axiom,
    ( ( code_integer_of_nat @ one_one_nat )
    = one_one_Code_integer ) ).

% integer_of_nat_1
thf(fact_10119_test__bit__uint32__code,axiom,
    ( bit_se5367290876889521763uint32
    = ( ^ [X2: uint32,N4: nat] :
          ( ( ord_less_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) )
          & ( uint32_test_bit @ X2 @ ( code_integer_of_nat @ N4 ) ) ) ) ) ).

% test_bit_uint32_code
thf(fact_10120_set__bit__uint32__code,axiom,
    ( generi1993664874377053279uint32
    = ( ^ [X2: uint32,N4: nat,B4: $o] : ( if_uint32 @ ( ord_less_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) @ ( uint32_set_bit @ X2 @ ( code_integer_of_nat @ N4 ) @ B4 ) @ X2 ) ) ) ).

% set_bit_uint32_code
thf(fact_10121_uint32__divmod__code,axiom,
    ( uint32_divmod
    = ( ^ [X2: uint32,Y2: uint32] : ( if_Pro1135515155860407935uint32 @ ( ord_less_eq_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ Y2 ) @ ( if_Pro1135515155860407935uint32 @ ( ord_less_uint32 @ X2 @ Y2 ) @ ( produc1400373151660368625uint32 @ zero_zero_uint32 @ X2 ) @ ( produc1400373151660368625uint32 @ one_one_uint32 @ ( minus_minus_uint32 @ X2 @ Y2 ) ) ) @ ( if_Pro1135515155860407935uint32 @ ( Y2 = zero_zero_uint32 ) @ ( produc1400373151660368625uint32 @ ( div0_uint32 @ X2 ) @ ( mod0_uint32 @ X2 ) ) @ ( if_Pro1135515155860407935uint32 @ ( ord_less_eq_uint32 @ Y2 @ ( minus_minus_uint32 @ X2 @ ( times_times_uint32 @ ( bit_se5742574853984576102uint32 @ one_one_nat @ ( uint32_sdiv @ ( bit_se3964402333458159761uint32 @ one_one_nat @ X2 ) @ Y2 ) ) @ Y2 ) ) ) @ ( produc1400373151660368625uint32 @ ( plus_plus_uint32 @ ( bit_se5742574853984576102uint32 @ one_one_nat @ ( uint32_sdiv @ ( bit_se3964402333458159761uint32 @ one_one_nat @ X2 ) @ Y2 ) ) @ one_one_uint32 ) @ ( minus_minus_uint32 @ ( minus_minus_uint32 @ X2 @ ( times_times_uint32 @ ( bit_se5742574853984576102uint32 @ one_one_nat @ ( uint32_sdiv @ ( bit_se3964402333458159761uint32 @ one_one_nat @ X2 ) @ Y2 ) ) @ Y2 ) ) @ Y2 ) ) @ ( produc1400373151660368625uint32 @ ( bit_se5742574853984576102uint32 @ one_one_nat @ ( uint32_sdiv @ ( bit_se3964402333458159761uint32 @ one_one_nat @ X2 ) @ Y2 ) ) @ ( minus_minus_uint32 @ X2 @ ( times_times_uint32 @ ( bit_se5742574853984576102uint32 @ one_one_nat @ ( uint32_sdiv @ ( bit_se3964402333458159761uint32 @ one_one_nat @ X2 ) @ Y2 ) ) @ Y2 ) ) ) ) ) ) ) ) ).

% uint32_divmod_code
thf(fact_10122_sshiftr__uint32__code,axiom,
    ( signed489701013188660438uint32
    = ( ^ [N4: nat,X2: uint32] : ( if_uint32 @ ( ord_less_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) @ ( uint32_sshiftr @ X2 @ ( code_integer_of_nat @ N4 ) ) @ ( if_uint32 @ ( bit_se5367290876889521763uint32 @ X2 @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ zero_zero_uint32 ) ) ) ) ).

% sshiftr_uint32_code
thf(fact_10123_uint32__divmod__def,axiom,
    ( uint32_divmod
    = ( ^ [X2: uint32,Y2: uint32] : ( if_Pro1135515155860407935uint32 @ ( Y2 = zero_zero_uint32 ) @ ( produc1400373151660368625uint32 @ ( undefi332904144742839227uint32 @ divide_divide_uint32 @ X2 @ zero_zero_uint32 ) @ ( undefi332904144742839227uint32 @ modulo_modulo_uint32 @ X2 @ zero_zero_uint32 ) ) @ ( produc1400373151660368625uint32 @ ( divide_divide_uint32 @ X2 @ Y2 ) @ ( modulo_modulo_uint32 @ X2 @ Y2 ) ) ) ) ) ).

% uint32_divmod_def
thf(fact_10124_mod0__uint32__def,axiom,
    ( mod0_uint32
    = ( ^ [X2: uint32] : ( undefi332904144742839227uint32 @ modulo_modulo_uint32 @ X2 @ zero_zero_uint32 ) ) ) ).

% mod0_uint32_def
thf(fact_10125_div0__uint32__def,axiom,
    ( div0_uint32
    = ( ^ [X2: uint32] : ( undefi332904144742839227uint32 @ divide_divide_uint32 @ X2 @ zero_zero_uint32 ) ) ) ).

% div0_uint32_def
thf(fact_10126_uint32__sdiv__code,axiom,
    ! [Y: uint32,X: uint32] :
      ( ( ( Y = zero_zero_uint32 )
       => ( ( rep_uint322 @ ( uint32_sdiv @ X @ Y ) )
          = ( rep_uint322 @ ( undefi332904144742839227uint32 @ divide_divide_uint32 @ X @ zero_zero_uint32 ) ) ) )
      & ( ( Y != zero_zero_uint32 )
       => ( ( rep_uint322 @ ( uint32_sdiv @ X @ Y ) )
          = ( signed6753297604338940182l_num1 @ ( rep_uint322 @ X ) @ ( rep_uint322 @ Y ) ) ) ) ) ).

% uint32_sdiv_code
thf(fact_10127_uint32__sshiftr__def,axiom,
    ( uint32_sshiftr
    = ( ^ [X2: uint32,N4: code_integer] :
          ( if_uint32
          @ ( ( ord_le6747313008572928689nteger @ N4 @ zero_z3403309356797280102nteger )
            | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N4 ) )
          @ ( undefi7330133036835070352uint32 @ signed489701013188660438uint32 @ N4 @ X2 )
          @ ( signed489701013188660438uint32 @ ( code_nat_of_integer @ N4 ) @ X2 ) ) ) ) ).

% uint32_sshiftr_def
thf(fact_10128_uint32__sshiftr__code,axiom,
    ! [N: code_integer,W: uint32] :
      ( ( ( ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger )
          | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N ) )
       => ( ( rep_uint322 @ ( uint32_sshiftr @ W @ N ) )
          = ( rep_uint322 @ ( undefi7330133036835070352uint32 @ signed489701013188660438uint32 @ N @ W ) ) ) )
      & ( ~ ( ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger )
            | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N ) )
       => ( ( rep_uint322 @ ( uint32_sshiftr @ W @ N ) )
          = ( signed5000768011106662067l_num1 @ ( code_nat_of_integer @ N ) @ ( rep_uint322 @ W ) ) ) ) ) ).

% uint32_sshiftr_code
thf(fact_10129_integer__shiftr__def,axiom,
    ( bits_integer_shiftr
    = ( ^ [X2: code_integer,N4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ N4 @ zero_z3403309356797280102nteger ) @ ( undefi8133104259855420269nteger @ X2 @ N4 ) @ ( bit_se3928097537394005634nteger @ ( code_nat_of_integer @ N4 ) @ X2 ) ) ) ) ).

% integer_shiftr_def
thf(fact_10130_uint32_Oset__bits__code,axiom,
    ( bit_bi705532357378895591uint32
    = ( ^ [P3: nat > $o] : ( set_bits_aux_uint32 @ P3 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ zero_zero_uint32 ) ) ) ).

% uint32.set_bits_code
thf(fact_10131_integer__shiftr__code_I2_J,axiom,
    ! [X: code_integer] :
      ( ( bits_integer_shiftr @ X @ zero_z3403309356797280102nteger )
      = X ) ).

% integer_shiftr_code(2)
thf(fact_10132_mod__uint32__code,axiom,
    ( modulo_modulo_uint32
    = ( ^ [X2: uint32,Y2: uint32] : ( if_uint32 @ ( Y2 = zero_zero_uint32 ) @ X2 @ ( uint32_mod @ X2 @ Y2 ) ) ) ) ).

% mod_uint32_code
thf(fact_10133_int__set__bits__K__False,axiom,
    ( ( bit_bi6516823479961619367ts_int
      @ ^ [Uu: nat] : $false )
    = zero_zero_int ) ).

% int_set_bits_K_False
thf(fact_10134_int__set__bits__K__True,axiom,
    ( ( bit_bi6516823479961619367ts_int
      @ ^ [Uu: nat] : $true )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% int_set_bits_K_True
thf(fact_10135_div__uint32__code,axiom,
    ( divide_divide_uint32
    = ( ^ [X2: uint32,Y2: uint32] : ( if_uint32 @ ( Y2 = zero_zero_uint32 ) @ zero_zero_uint32 @ ( uint32_div @ X2 @ Y2 ) ) ) ) ).

% div_uint32_code
thf(fact_10136_bin__last__set__bits,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int @ F )
     => ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_bi6516823479961619367ts_int @ F ) ) )
        = ( F @ zero_zero_nat ) ) ) ).

% bin_last_set_bits
thf(fact_10137_wf__set__bits__int__Suc,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int
        @ ^ [N4: nat] : ( F @ ( suc @ N4 ) ) )
      = ( bit_wf_set_bits_int @ F ) ) ).

% wf_set_bits_int_Suc
thf(fact_10138_wf__set__bits__int__const,axiom,
    ! [B2: $o] :
      ( bit_wf_set_bits_int
      @ ^ [Uu: nat] : B2 ) ).

% wf_set_bits_int_const
thf(fact_10139_int__set__bits__unfold__BIT,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int @ F )
     => ( ( bit_bi6516823479961619367ts_int @ F )
        = ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( F @ zero_zero_nat ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_bi6516823479961619367ts_int @ ( comp_nat_o_nat @ F @ suc ) ) ) ) ) ) ).

% int_set_bits_unfold_BIT
thf(fact_10140_msb__uint32__code,axiom,
    ( most_s9063628576841037300uint32
    = ( ^ [X2: uint32] : ( uint32_test_bit @ X2 @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ).

% msb_uint32_code
thf(fact_10141_bin__rest__set__bits,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int @ F )
     => ( ( divide_divide_int @ ( bit_bi6516823479961619367ts_int @ F ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( bit_bi6516823479961619367ts_int @ ( comp_nat_o_nat @ F @ suc ) ) ) ) ).

% bin_rest_set_bits
thf(fact_10142_uint32__msb__test__bit,axiom,
    ( most_s9063628576841037300uint32
    = ( ^ [X2: uint32] : ( bit_se5367290876889521763uint32 @ X2 @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ).

% uint32_msb_test_bit
thf(fact_10143_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_10144_complex__cnj__zero,axiom,
    ( ( cnj @ zero_zero_complex )
    = zero_zero_complex ) ).

% complex_cnj_zero
thf(fact_10145_complex__cnj__zero__iff,axiom,
    ! [Z: complex] :
      ( ( ( cnj @ Z )
        = zero_zero_complex )
      = ( Z = zero_zero_complex ) ) ).

% complex_cnj_zero_iff
thf(fact_10146_complex__cnj__one__iff,axiom,
    ! [Z: complex] :
      ( ( ( cnj @ Z )
        = one_one_complex )
      = ( Z = one_one_complex ) ) ).

% complex_cnj_one_iff
thf(fact_10147_complex__cnj__one,axiom,
    ( ( cnj @ one_one_complex )
    = one_one_complex ) ).

% complex_cnj_one
thf(fact_10148_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_10149_sums__cnj,axiom,
    ! [F: nat > complex,L: complex] :
      ( ( sums_complex
        @ ^ [X2: nat] : ( cnj @ ( F @ X2 ) )
        @ ( cnj @ L ) )
      = ( sums_complex @ F @ L ) ) ).

% sums_cnj
thf(fact_10150_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_10151_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_10152_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_10153_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_10154_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_10155_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_10156_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_10157_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_10158_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_10159_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_10160_summable__reindex,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
       => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
         => ( summable_real @ ( comp_nat_real_nat @ F @ G ) ) ) ) ) ).

% summable_reindex
thf(fact_10161_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_10162_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_10163_suminf__reindex__mono,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
       => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
         => ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G ) ) @ ( suminf_real @ F ) ) ) ) ) ).

% suminf_reindex_mono
thf(fact_10164_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_10165_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_10166_suminf__reindex,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
       => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
         => ( ! [X3: nat] :
                ( ~ ( member_nat @ X3 @ ( image_nat_nat @ G @ top_top_set_nat ) )
               => ( ( F @ X3 )
                  = zero_zero_real ) )
           => ( ( suminf_real @ ( comp_nat_real_nat @ F @ G ) )
              = ( suminf_real @ F ) ) ) ) ) ) ).

% suminf_reindex
thf(fact_10167_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_10168_complex__div__cnj,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A4: complex,B4: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A4 @ ( cnj @ B4 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% complex_div_cnj
thf(fact_10169_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_10170_Code__Target__Int_Onegative__def,axiom,
    ( code_Target_negative
    = ( comp_int_int_num @ uminus_uminus_int @ numeral_numeral_int ) ) ).

% Code_Target_Int.negative_def
thf(fact_10171_divmod__integer__eq__cases,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L2: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
            @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
              @ ( if_Pro6119634080678213985nteger
                @ ( ( sgn_sgn_Code_integer @ K3 )
                  = ( sgn_sgn_Code_integer @ L2 ) )
                @ ( code_divmod_abs @ K3 @ L2 )
                @ ( produc6916734918728496179nteger
                  @ ^ [R5: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L2 ) @ S4 ) ) )
                  @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ).

% divmod_integer_eq_cases
thf(fact_10172_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] :
      ( ( code_divmod_abs @ zero_z3403309356797280102nteger @ J )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ) ).

% divmod_abs_code(6)
thf(fact_10173_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] :
      ( ( code_divmod_abs @ J @ zero_z3403309356797280102nteger )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ J ) ) ) ).

% divmod_abs_code(5)
thf(fact_10174_divmod__integer__code,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L2: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L2 )
            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L2 )
              @ ( produc6916734918728496179nteger
                @ ^ [R5: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L2 @ S4 ) ) )
                @ ( code_divmod_abs @ K3 @ L2 ) ) )
            @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
              @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
                @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L2 )
                  @ ( produc6916734918728496179nteger
                    @ ^ [R5: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L2 ) @ S4 ) ) )
                    @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ) ).

% divmod_integer_code
thf(fact_10175_bit__cut__integer__code,axiom,
    ( code_bit_cut_integer
    = ( ^ [K3: code_integer] :
          ( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
          @ ( produc9125791028180074456eger_o
            @ ^ [R5: code_integer,S4: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S4 ) ) @ ( S4 = one_one_Code_integer ) )
            @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_cut_integer_code
thf(fact_10176_integer__of__uint32__code,axiom,
    ( integer_of_uint32
    = ( ^ [N4: uint32] : ( if_Code_integer @ ( bit_se5367290876889521763uint32 @ N4 @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( bit_se1080825931792720795nteger @ ( intege5370686899274169573signed @ ( bit_se6294004230839889034uint32 @ N4 @ ( numera9087168376688890119uint32 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( intege5370686899274169573signed @ N4 ) ) ) ) ).

% integer_of_uint32_code
thf(fact_10177_integer__of__uint32__signed__def,axiom,
    ( intege5370686899274169573signed
    = ( ^ [N4: uint32] : ( if_Code_integer @ ( bit_se5367290876889521763uint32 @ N4 @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( undefi3580195557576403463nteger @ integer_of_uint32 @ N4 ) @ ( integer_of_uint32 @ N4 ) ) ) ) ).

% integer_of_uint32_signed_def
thf(fact_10178_bit__cut__integer__def,axiom,
    ( code_bit_cut_integer
    = ( ^ [K3: code_integer] :
          ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K3 ) ) ) ) ).

% bit_cut_integer_def
thf(fact_10179_integer__of__uint32__signed__code,axiom,
    ( intege5370686899274169573signed
    = ( ^ [N4: uint32] : ( if_Code_integer @ ( bit_se5367290876889521763uint32 @ N4 @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( undefi3580195557576403463nteger @ integer_of_uint32 @ N4 ) @ ( code_integer_of_int @ ( semiri7338730514057886004m1_int @ ( rep_uint32 @ N4 ) ) ) ) ) ) ).

% integer_of_uint32_signed_code
thf(fact_10180_uint32__sdiv__def,axiom,
    ( uint32_sdiv
    = ( ^ [X2: uint32,Y2: uint32] : ( if_uint32 @ ( Y2 = zero_zero_uint32 ) @ ( undefi332904144742839227uint32 @ divide_divide_uint32 @ X2 @ zero_zero_uint32 ) @ ( abs_uint32 @ ( signed6753297604338940182l_num1 @ ( rep_uint322 @ X2 ) @ ( rep_uint322 @ Y2 ) ) ) ) ) ) ).

% uint32_sdiv_def
thf(fact_10181_one__uint32__def,axiom,
    ( one_one_uint32
    = ( abs_uint32 @ one_on7727431528512463931l_num1 ) ) ).

% one_uint32_def
thf(fact_10182_zero__uint32__def,axiom,
    ( zero_zero_uint32
    = ( abs_uint32 @ zero_z3563351764282998399l_num1 ) ) ).

% zero_uint32_def
thf(fact_10183_less__uint32_Oabs__eq,axiom,
    ! [Xa: word_N3645301735248828278l_num1,X: word_N3645301735248828278l_num1] :
      ( ( ord_less_uint32 @ ( abs_uint32 @ Xa ) @ ( abs_uint32 @ X ) )
      = ( ord_le750835935415966154l_num1 @ Xa @ X ) ) ).

% less_uint32.abs_eq
thf(fact_10184_Rep__uint32_H__code,axiom,
    ( rep_uint32
    = ( ^ [X2: uint32] : ( bit_bi5746210779246519537l_num1 @ ( bit_se5367290876889521763uint32 @ X2 ) ) ) ) ).

% Rep_uint32'_code
thf(fact_10185_card__UNIV__unit,axiom,
    ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
    = one_one_nat ) ).

% card_UNIV_unit
thf(fact_10186_card__Collect__less__nat,axiom,
    ! [N: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) )
      = N ) ).

% card_Collect_less_nat
thf(fact_10187_card__Collect__le__nat,axiom,
    ! [N: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I4: nat] : ( ord_less_eq_nat @ I4 @ N ) ) )
      = ( suc @ N ) ) ).

% card_Collect_le_nat
thf(fact_10188_card__UNIV__bool,axiom,
    ( ( finite_card_o @ top_top_set_o )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% card_UNIV_bool
thf(fact_10189_card__atLeastAtMost__int,axiom,
    ! [L: int,U2: int] :
      ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L @ U2 ) )
      = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U2 @ L ) @ one_one_int ) ) ) ).

% card_atLeastAtMost_int
thf(fact_10190_card__greaterThanLessThan__int,axiom,
    ! [L: int,U2: int] :
      ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L @ U2 ) )
      = ( nat2 @ ( minus_minus_int @ U2 @ ( plus_plus_int @ L @ one_one_int ) ) ) ) ).

% card_greaterThanLessThan_int
thf(fact_10191_card__atLeastZeroLessThan__int,axiom,
    ! [U2: int] :
      ( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U2 ) )
      = ( nat2 @ U2 ) ) ).

% card_atLeastZeroLessThan_int
thf(fact_10192_card__less,axiom,
    ! [M7: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_10193_card__less__Suc,axiom,
    ! [M7: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K3: nat] :
                  ( ( member_nat @ ( suc @ K3 ) @ M7 )
                  & ( ord_less_nat @ K3 @ I ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_10194_card__less__Suc2,axiom,
    ! [M7: set_nat,I: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ ( suc @ K3 ) @ M7 )
                & ( ord_less_nat @ K3 @ I ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_10195_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N3: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
     => ( ord_less_eq_nat @ ( finite_card_nat @ N3 ) @ N ) ) ).

% subset_eq_atLeast0_lessThan_card
thf(fact_10196_card__sum__le__nat__sum,axiom,
    ! [S: set_nat] :
      ( ord_less_eq_nat
      @ ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S ) ) )
      @ ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ S ) ) ).

% card_sum_le_nat_sum
thf(fact_10197_card__nth__roots,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( finite_card_complex
            @ ( collect_complex
              @ ^ [Z4: complex] :
                  ( ( power_power_complex @ Z4 @ N )
                  = C ) ) )
          = N ) ) ) ).

% card_nth_roots
thf(fact_10198_card__roots__unity__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( finite_card_complex
          @ ( collect_complex
            @ ^ [Z4: complex] :
                ( ( power_power_complex @ Z4 @ N )
                = one_one_complex ) ) )
        = N ) ) ).

% card_roots_unity_eq
thf(fact_10199_card__num0,axiom,
    ( ( finite6454714172617411596l_num0 @ top_to3689904424835650196l_num0 )
    = zero_zero_nat ) ).

% card_num0
thf(fact_10200_card__num1,axiom,
    ( ( finite6454714172617411597l_num1 @ top_to3689904429138878997l_num1 )
    = one_one_nat ) ).

% card_num1
thf(fact_10201_card__nat,axiom,
    ( ( finite_card_nat @ top_top_set_nat )
    = zero_zero_nat ) ).

% card_nat
thf(fact_10202_card__literal,axiom,
    ( ( finite_card_literal @ top_top_set_literal )
    = zero_zero_nat ) ).

% card_literal
thf(fact_10203_UNIV__bool,axiom,
    ( top_top_set_o
    = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).

% UNIV_bool

% Helper facts (38)
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_If_2_1_If_001t__Uint32__Ouint32_T,axiom,
    ! [X: uint32,Y: uint32] :
      ( ( if_uint32 @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Uint32__Ouint32_T,axiom,
    ! [X: uint32,Y: uint32] :
      ( ( if_uint32 @ $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__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__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__Heap____Time____Monad__OHeap_I_Eo_J_T,axiom,
    ! [X: heap_Time_Heap_o,Y: heap_Time_Heap_o] :
      ( ( if_Heap_Time_Heap_o @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Heap____Time____Monad__OHeap_I_Eo_J_T,axiom,
    ! [X: heap_Time_Heap_o,Y: heap_Time_Heap_o] :
      ( ( if_Heap_Time_Heap_o @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_T,axiom,
    ! [X: heap_T8145700208782473153_VEBTi,Y: heap_T8145700208782473153_VEBTi] :
      ( ( if_Hea8453224502484754311_VEBTi @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_T,axiom,
    ! [X: heap_T8145700208782473153_VEBTi,Y: heap_T8145700208782473153_VEBTi] :
      ( ( if_Hea8453224502484754311_VEBTi @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_T,axiom,
    ! [X: produc827990862158126777uint32,Y: produc827990862158126777uint32] :
      ( ( if_Pro1135515155860407935uint32 @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_T,axiom,
    ! [X: produc827990862158126777uint32,Y: produc827990862158126777uint32] :
      ( ( if_Pro1135515155860407935uint32 @ $true @ X @ Y )
      = X ) ).

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 ) ).

thf(help_If_3_1_If_001t__Word__Oword_It__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_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Word__Oword_It__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_J_T,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( if_wor5778924947035936048l_num1 @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Word__Oword_It__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_J_T,axiom,
    ! [X: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( if_wor5778924947035936048l_num1 @ $true @ X @ Y )
      = X ) ).

% Conjectures (2)
thf(conj_0,hypothesis,
    ! [X6: nat] :
      ( ( member_nat @ X6 @ ( set_nat2 @ xs ) )
     => ( ord_less_nat @ X6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ n ) ) ) ).

thf(conj_1,conjecture,
    ( hoare_1429296392585015714_VEBTi @ ( vEBT_Intf_vebt_assn @ n @ s @ t )
    @ ( vEBT_E6105538542217078229_VEBTi
      @ ^ [X2: nat,S4: vEBT_VEBTi] : ( vEBT_vebt_inserti @ S4 @ X2 )
      @ xs
      @ t )
    @ ( vEBT_Intf_vebt_assn @ n @ ( sup_sup_set_nat @ s @ ( set_nat2 @ xs ) ) ) ) ).

%------------------------------------------------------------------------------