%------------------------------------------------------------------------------
% File     : ITP238^3 : TPTP v8.2.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Pred 00333_016090
% 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    : 0069_VEBT_Pred_00333_016090 [Des22]

% Status   : Theorem
% Rating   : 0.90 v8.2.0, 0.85 v8.1.0
% Syntax   : Number of formulae    : 11333 (5578 unt;1084 typ;   0 def)
%            Number of atoms       : 28549 (12091 equ;   0 cnn)
%            Maximal formula atoms :   71 (   2 avg)
%            Number of connectives : 115442 (2700   ~; 518   |;1769   &;99439   @)
%                                         (   0 <=>;11016  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   39 (   6 avg)
%            Number of types       :  110 ( 109 usr)
%            Number of type conns  : 4888 (4888   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  978 ( 975 usr;  64 con; 0-8 aty)
%            Number of variables   : 26530 (2393   ^;23368   !; 769   ?;26530   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-17 22:49:49.367
%------------------------------------------------------------------------------
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thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
    archim7802044766580827645g_real: real > int ).

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

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

thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
    bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
    bNF_re3403563459893282935_int_o: ( int > int > $o ) > ( ( int > $o ) > ( int > $o ) > $o ) > ( int > int > $o ) > ( int > int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re711492959462206631nt_int: ( int > int > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( int > int > int ) > ( int > int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    bNF_re157797125943740599nt_int: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > product_prod_int_int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Rat__Orat_J,type,
    bNF_re3461391660133120880nt_rat: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > rat ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_Eo_001_Eo,type,
    bNF_re5089333283451836215nt_o_o: ( int > int > $o ) > ( $o > $o > $o ) > ( int > $o ) > ( int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    bNF_re4712519889275205905nt_int: ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > ( int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re6250860962936578807nt_int: ( int > int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
    bNF_re2214769303045360666nt_rat: ( int > int > $o ) > ( product_prod_int_int > rat > $o ) > ( int > product_prod_int_int ) > ( int > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
    bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_re1345281282404953727at_nat: ( nat > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_Eo_001_Eo,type,
    bNF_re4705727531993890431at_o_o: ( nat > nat > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    bNF_re6830278522597306478at_int: ( nat > nat > $o ) > ( product_prod_nat_nat > int > $o ) > ( nat > product_prod_nat_nat ) > ( nat > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001_062_It__Num__Onum_Mt__Int__Oint_J_001_062_It__Num__Onum_Mt__Int__Oint_J,type,
    bNF_re8402795839162346335um_int: ( num > num > $o ) > ( ( num > int ) > ( num > int ) > $o ) > ( num > num > int ) > ( num > num > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Int__Oint,type,
    bNF_re1822329894187522285nt_int: ( num > num > $o ) > ( int > int > $o ) > ( num > int ) > ( num > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    bNF_re5228765855967844073nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo_001_Eo,type,
    bNF_re8699439704749558557nt_o_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re7145576690424134365nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
    bNF_re7627151682743391978at_rat: ( product_prod_int_int > rat > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( rat > rat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_Eo_001_Eo,type,
    bNF_re1494630372529172596at_o_o: ( product_prod_int_int > rat > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( rat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
    bNF_re8279943556446156061nt_rat: ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
    bNF_re717283939379294677_int_o: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( int > int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re7408651293131936558nt_int: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( int > int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_Eo_001_Eo,type,
    bNF_re6644619430987730960nt_o_o: ( product_prod_nat_nat > int > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re4555766996558763186at_nat: ( product_prod_nat_nat > int > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( int > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    bNF_re7400052026677387805at_int: ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    bNF_re4202695980764964119_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bNF_re3099431351363272937at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001_Eo,type,
    bNF_re3666534408544137501at_o_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re8246922863344978751at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( product_prod_nat_nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_re2241393799969408733at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ).

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

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

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

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

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

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

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

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

thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
    bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o ).

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

thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
    bit_or_not_num_neg: num > num > num ).

thf(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
    bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
    bit_se6528837805403552850or_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
    bit_se1146084159140164899it_int: int > nat > $o ).

thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
    bit_se1148574629649215175it_nat: nat > nat > $o ).

thf(sy_c_Bit__Operations_Otake__bit__num,type,
    bit_take_bit_num: nat > num > option_num ).

thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oand__num,type,
    bit_un1837492267222099188nd_num: num > num > option_num ).

thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oand__num__rel,type,
    bit_un5425074673868309765um_rel: product_prod_num_num > product_prod_num_num > $o ).

thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oxor__num,type,
    bit_un6178654185764691216or_num: num > num > option_num ).

thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oxor__num__rel,type,
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thf(sy_c_Int_Ointrel,type,
    intrel: product_prod_nat_nat > product_prod_nat_nat > $o ).

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

thf(sy_c_Int_Opcr__int,type,
    pcr_int: product_prod_nat_nat > int > $o ).

thf(sy_c_Int_Opower__int_001t__Real__Oreal,type,
    power_int_real: real > int > real ).

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__Complex__Ocomplex,type,
    ring_17405671764205052669omplex: int > complex ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
    ring_1_of_int_int: int > int ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
    ring_1_of_int_rat: int > rat ).

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

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Extended____Nat__Oenat,type,
    inf_in1870772243966228564d_enat: extended_enat > extended_enat > extended_enat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
    inf_inf_nat: nat > nat > nat ).

thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
    semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

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

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

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

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

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

thf(sy_c_Limits_Oat__infinity_001t__Real__Oreal,type,
    at_infinity_real: filter_real ).

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

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

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

thf(sy_c_List_Ofold_001t__Int__Oint_001t__Int__Oint,type,
    fold_int_int: ( int > int > int ) > list_int > int > int ).

thf(sy_c_List_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
    fold_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).

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

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

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

thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
    nil_int: list_int ).

thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

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

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

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

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

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

thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
    set_list_nat2: list_list_nat > set_list_nat ).

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
    nth_list_nat: list_list_nat > nat > list_nat ).

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

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

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

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_M_Eo_J,type,
    nth_Pr7514405829937366042_int_o: list_P5087981734274514673_int_o > nat > product_prod_int_o ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    nth_Pr4439495888332055232nt_int: list_P5707943133018811711nt_int > nat > product_prod_int_int ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
    nth_Pr8617346907841251940nt_nat: list_P8198026277950538467nt_nat > nat > product_prod_int_nat ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr3474266648193625910T_VEBT: list_P7524865323317820941T_VEBT > nat > produc1531783533982839933T_VEBT ).

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

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

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

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

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

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

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

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

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

thf(sy_c_List_Oproduct_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).

thf(sy_c_List_Oproduct_001t__Int__Oint_001_Eo,type,
    product_int_o: list_int > list_o > list_P5087981734274514673_int_o ).

thf(sy_c_List_Oproduct_001t__Int__Oint_001t__Int__Oint,type,
    product_int_int: list_int > list_int > list_P5707943133018811711nt_int ).

thf(sy_c_List_Oproduct_001t__Int__Oint_001t__Nat__Onat,type,
    product_int_nat: list_int > list_nat > list_P8198026277950538467nt_nat ).

thf(sy_c_List_Oproduct_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    produc662631939642741121T_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT ).

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

thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
    product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Nat__Bijection_Olist__encode,type,
    nat_list_encode: list_nat > nat ).

thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
    nat_list_encode_rel: list_nat > list_nat > $o ).

thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).

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

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

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

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

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

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

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

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

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

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
    neg_nu8804712462038260780nteger: code_integer > code_integer ).

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

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thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
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thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
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thf(sy_c_Num_Onum_OBit0,type,
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thf(sy_c_Num_Onum_OBit1,type,
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thf(sy_c_Num_Onum_OOne,type,
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thf(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
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thf(sy_c_Num_Onum__of__nat,type,
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thf(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
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thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
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thf(sy_c_Num_Opow,type,
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thf(sy_c_Num_Opred__numeral,type,
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thf(sy_c_Num_Osqr,type,
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thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
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thf(sy_c_Option_Ooption_Osize__option_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Rat__Orat,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Orderings_Oord__class_Omin_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Real__Oreal,type,
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thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Real__Oreal,type,
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thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Num__Onum_M_062_It__Num__Onum_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_Eo,type,
    produc6253627499356882019eger_o: ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > produc1908205239877642774nteger > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo,type,
    produc1573362020775583542_int_o: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ) > produc2285326912895808259nt_int > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001_Eo,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    produc8580519160106071146omplex: ( int > int > set_complex ) > product_prod_int_int > set_complex ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J,type,
    produc73460835934605544et_int: ( int > int > set_int ) > product_prod_int_int > set_int ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Nat__Onat_J,type,
    produc4251311855443802252et_nat: ( int > int > set_nat ) > product_prod_int_int > set_nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Real__Oreal_J,type,
    produc6452406959799940328t_real: ( int > int > set_real ) > product_prod_int_int > set_real ).

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

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

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Complex__Ocomplex,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint,type,
    produc6840382203811409530at_int: ( nat > nat > int ) > product_prod_nat_nat > int ).

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

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat,type,
    produc6207742614233964070at_rat: ( nat > nat > rat ) > product_prod_nat_nat > rat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
    produc1703576794950452218t_real: ( nat > nat > real ) > product_prod_nat_nat > real ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
    produc5414030515140494994real_o: ( real > real > $o ) > produc2422161461964618553l_real > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_It__Complex__Ocomplex_J_001t__Set__Oset_It__Complex__Ocomplex_J_001_Eo,type,
    produc3914248068834153634plex_o: ( set_complex > set_complex > $o ) > produc8064648209034914857omplex > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J_001_Eo,type,
    produc4109468873575309990_int_o: ( set_int > set_int > $o ) > produc2115011035271226405et_int > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
    produc6247414631856714078_nat_o: ( set_nat > set_nat > $o ) > produc7819656566062154093et_nat > $o ).

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

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

thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
    product_snd_int_int: product_prod_int_int > int ).

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

thf(sy_c_Rat_OAbs__Rat,type,
    abs_Rat: product_prod_int_int > rat ).

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

thf(sy_c_Rat_OFrct,type,
    frct: product_prod_int_int > rat ).

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

thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
    field_5140801741446780682s_real: set_real ).

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

thf(sy_c_Rat_Onormalize,type,
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thf(sy_c_Rat_Oof__int,type,
    of_int: int > rat ).

thf(sy_c_Rat_Opcr__rat,type,
    pcr_rat: product_prod_int_int > rat > $o ).

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

thf(sy_c_Rat_Oquotient__of,type,
    quotient_of: rat > product_prod_int_int ).

thf(sy_c_Rat_Oratrel,type,
    ratrel: product_prod_int_int > product_prod_int_int > $o ).

thf(sy_c_Real_Ovanishes,type,
    vanishes: ( nat > rat ) > $o ).

thf(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
    real_V2521375963428798218omplex: set_complex ).

thf(sy_c_Real__Vector__Spaces_Obounded__linear_001t__Real__Oreal_001t__Real__Oreal,type,
    real_V5970128139526366754l_real: ( real > real ) > $o ).

thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
    real_V3694042436643373181omplex: complex > complex > real ).

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

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

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

thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
    real_V4546457046886955230omplex: real > complex ).

thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
    real_V2046097035970521341omplex: real > complex > complex ).

thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
    real_V1485227260804924795R_real: real > real > real ).

thf(sy_c_Relation_OField_001t__Nat__Onat,type,
    field_nat: set_Pr1261947904930325089at_nat > set_nat ).

thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Int__Oint,type,
    algebr932160517623751201me_int: int > int > $o ).

thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Nat__Onat,type,
    algebr934650988132801477me_nat: nat > nat > $o ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
    divide6298287555418463151nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
    divide1717551699836669952omplex: complex > complex > complex ).

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

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

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

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

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

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

thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
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thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
    modulo_modulo_int: int > int > int ).

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

thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
    zero_n1201886186963655149omplex: $o > complex ).

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

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

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

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

thf(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
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thf(sy_c_Series_Osuminf_001t__Int__Oint,type,
    suminf_int: ( nat > int ) > int ).

thf(sy_c_Series_Osuminf_001t__Nat__Onat,type,
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thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
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thf(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
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thf(sy_c_Series_Osummable_001t__Int__Oint,type,
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thf(sy_c_Series_Osummable_001t__Nat__Onat,type,
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thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
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thf(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
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    collect_num: ( num > $o ) > set_num ).

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

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    collec3799799289383736868l_real: ( produc2422161461964618553l_real > $o ) > set_Pr6218003697084177305l_real ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_It__Complex__Ocomplex_J_Mt__Set__Oset_It__Complex__Ocomplex_J_J,type,
    collec5108298041176329748omplex: ( produc8064648209034914857omplex > $o ) > set_Pr6308028481084910985omplex ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
    collec957716948307931664et_int: ( produc2115011035271226405et_int > $o ) > set_Pr2522554150109002629et_int ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    collec6662362479098859352et_nat: ( produc7819656566062154093et_nat > $o ) > set_Pr5488025237498180813et_nat ).

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

thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
    collect_real: ( real > $o ) > set_real ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    collect_set_complex: ( set_complex > $o ) > set_set_complex ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
    collect_set_int: ( set_int > $o ) > set_set_int ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
    collect_set_nat: ( set_nat > $o ) > set_set_nat ).

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

thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
    image_int_int: ( int > int ) > set_int > set_int ).

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

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

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

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

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

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

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

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

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

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

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

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__rel_001t__Nat__Onat,type,
    set_fo3699595496184130361el_nat: produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o ).

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

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

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

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

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

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Int__Oint_J,type,
    set_or370866239135849197et_int: set_int > set_int > set_set_int ).

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

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

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

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

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

thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Num__Onum,type,
    set_ord_atMost_num: num > set_num ).

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

thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
    set_ord_atMost_real: real > set_real ).

thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Int__Oint_J,type,
    set_or58775011639299419et_int: set_int > set_set_int ).

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

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

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

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

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

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

thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
    set_ord_lessThan_int: int > set_int ).

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

thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
    set_ord_lessThan_num: num > set_num ).

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

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

thf(sy_c_String_Oascii__of,type,
    ascii_of: char > char ).

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Int__Oint_J,type,
    topolo3100542954746470799et_int: ( nat > set_int ) > $o ).

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

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

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

thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
    topolo896644834953643431omplex: filter6041513312241820739omplex ).

thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
    topolo1511823702728130853y_real: filter2146258269922977983l_real ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Transcendental_Opi,type,
    pi: real ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Nat__Onat,type,
    vEBT_V3895251965096974666el_nat: produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Num__Onum,type,
    vEBT_V452583751252753300el_num: produc1193250871479095198on_num > produc1193250871479095198on_num > $o ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    vEBT_V7235779383477046023at_nat: produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o ).

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

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

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

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

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

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

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

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

thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
    accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).

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

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    accp_P6019419558468335806at_nat: ( produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o ) > produc4471711990508489141at_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_J,type,
    accp_P5496254298877145759on_nat: ( produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o ) > produc8306885398267862888on_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Num__Onum_M_062_It__Num__Onum_Mt__Num__Onum_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J_J,type,
    accp_P7605991808943153877on_num: ( produc1193250871479095198on_num > produc1193250871479095198on_num > $o ) > produc1193250871479095198on_num > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    accp_P3267385326087170368at_nat: ( produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o ) > produc5542196010084753463at_nat > $o ).

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

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

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

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

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

thf(sy_c_Wellfounded_Ofinite__psubset_001t__Complex__Ocomplex,type,
    finite8643634255014194347omplex: set_Pr6308028481084910985omplex ).

thf(sy_c_Wellfounded_Ofinite__psubset_001t__Int__Oint,type,
    finite_psubset_int: set_Pr2522554150109002629et_int ).

thf(sy_c_Wellfounded_Ofinite__psubset_001t__Nat__Onat,type,
    finite_psubset_nat: set_Pr5488025237498180813et_nat ).

thf(sy_c_Wellfounded_Omeasure_001t__Int__Oint,type,
    measure_int: ( int > nat ) > set_Pr958786334691620121nt_int ).

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__List__Olist_I_Eo_J,type,
    member_list_o: list_o > set_list_o > $o ).

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

thf(sy_c_member_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o ).

thf(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > set_nat > $o ).

thf(sy_c_member_001t__Num__Onum,type,
    member_num: num > set_num > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    member3068662437193594005nteger: produc8763457246119570046nteger > set_Pr8056137968301705908nteger > $o ).

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

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

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

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Complex__Ocomplex_J_Mt__Set__Oset_It__Complex__Ocomplex_J_J,type,
    member351165363924911826omplex: produc8064648209034914857omplex > set_Pr6308028481084910985omplex > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
    member2572552093476627150et_int: produc2115011035271226405et_int > set_Pr2522554150109002629et_int > $o ).

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

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

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

thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
    member_set_int: set_int > set_set_int > $o ).

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

thf(sy_v_deg____,type,
    deg: nat ).

thf(sy_v_m____,type,
    m: nat ).

thf(sy_v_ma____,type,
    ma: nat ).

thf(sy_v_mi____,type,
    mi: nat ).

thf(sy_v_na____,type,
    na: nat ).

thf(sy_v_summary____,type,
    summary: vEBT_VEBT ).

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

thf(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

thf(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (10209)
thf(fact_0_False,axiom,
    ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
   != none_nat ) ).

% False
thf(fact_1__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062,axiom,
    ( ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = na ) ).

% \<open>deg div 2 = n\<close>
thf(fact_2_semiring__norm_I85_J,axiom,
    ! [M: num] :
      ( ( bit0 @ M )
     != one ) ).

% semiring_norm(85)
thf(fact_3_semiring__norm_I83_J,axiom,
    ! [N: num] :
      ( one
     != ( bit0 @ N ) ) ).

% semiring_norm(83)
thf(fact_4__092_060open_0622_A_092_060le_062_Adeg_092_060close_062,axiom,
    ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ).

% \<open>2 \<le> deg\<close>
thf(fact_5_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_6_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_7_high__def,axiom,
    ( vEBT_VEBT_high
    = ( ^ [X: nat,N2: nat] : ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% high_def
thf(fact_8_True,axiom,
    ord_less_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ).

% True
thf(fact_9_i1,axiom,
    ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
      = none_nat )
    | ~ ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% i1
thf(fact_10_divide__numeral__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_11_divide__numeral__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_12_divide__numeral__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_13_option_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( some_nat @ X2 )
        = ( some_nat @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_14_option_Oinject,axiom,
    ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( ( some_P7363390416028606310at_nat @ X2 )
        = ( some_P7363390416028606310at_nat @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_15_option_Oinject,axiom,
    ! [X2: num,Y2: num] :
      ( ( ( some_num @ X2 )
        = ( some_num @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_16_verit__eq__simplify_I8_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( ( bit0 @ X2 )
        = ( bit0 @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% verit_eq_simplify(8)
thf(fact_17_semiring__norm_I87_J,axiom,
    ! [M: num,N: num] :
      ( ( ( bit0 @ M )
        = ( bit0 @ N ) )
      = ( M = N ) ) ).

% semiring_norm(87)
thf(fact_18__C4_Ohyps_C_I5_J,axiom,
    m = na ).

% "4.hyps"(5)
thf(fact_19_max__in__set__def,axiom,
    ( vEBT_VEBT_max_in_set
    = ( ^ [Xs: set_nat,X: nat] :
          ( ( member_nat @ X @ Xs )
          & ! [Y: nat] :
              ( ( member_nat @ Y @ Xs )
             => ( ord_less_eq_nat @ Y @ X ) ) ) ) ) ).

% max_in_set_def
thf(fact_20_greater__shift,axiom,
    ( ord_less_nat
    = ( ^ [Y: nat,X: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).

% greater_shift
thf(fact_21_min__in__set__def,axiom,
    ( vEBT_VEBT_min_in_set
    = ( ^ [Xs: set_nat,X: nat] :
          ( ( member_nat @ X @ Xs )
          & ! [Y: nat] :
              ( ( member_nat @ Y @ Xs )
             => ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).

% min_in_set_def
thf(fact_22_power__shift,axiom,
    ! [X3: nat,Y3: nat,Z: nat] :
      ( ( ( power_power_nat @ X3 @ Y3 )
        = Z )
      = ( ( vEBT_VEBT_power @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) )
        = ( some_nat @ Z ) ) ) ).

% power_shift
thf(fact_23_bit__split__inv,axiom,
    ! [X3: nat,D: nat] :
      ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X3 @ D ) @ ( vEBT_VEBT_low @ X3 @ D ) @ D )
      = X3 ) ).

% bit_split_inv
thf(fact_24_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numera6690914467698888265omplex @ M )
        = ( numera6690914467698888265omplex @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_25_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_real @ M )
        = ( numeral_numeral_real @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_26_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_rat @ M )
        = ( numeral_numeral_rat @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_27_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_nat @ M )
        = ( numeral_numeral_nat @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_28_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_int @ M )
        = ( numeral_numeral_int @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_29_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_30_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_31_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_32_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_33_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_34_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_35_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_36_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_37_not__Some__eq,axiom,
    ! [X3: option_nat] :
      ( ( ! [Y: nat] :
            ( X3
           != ( some_nat @ Y ) ) )
      = ( X3 = none_nat ) ) ).

% not_Some_eq
thf(fact_38_not__Some__eq,axiom,
    ! [X3: option4927543243414619207at_nat] :
      ( ( ! [Y: product_prod_nat_nat] :
            ( X3
           != ( some_P7363390416028606310at_nat @ Y ) ) )
      = ( X3 = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
thf(fact_39_not__Some__eq,axiom,
    ! [X3: option_num] :
      ( ( ! [Y: num] :
            ( X3
           != ( some_num @ Y ) ) )
      = ( X3 = none_num ) ) ).

% not_Some_eq
thf(fact_40_not__None__eq,axiom,
    ! [X3: option_nat] :
      ( ( X3 != none_nat )
      = ( ? [Y: nat] :
            ( X3
            = ( some_nat @ Y ) ) ) ) ).

% not_None_eq
thf(fact_41_not__None__eq,axiom,
    ! [X3: option4927543243414619207at_nat] :
      ( ( X3 != none_P5556105721700978146at_nat )
      = ( ? [Y: product_prod_nat_nat] :
            ( X3
            = ( some_P7363390416028606310at_nat @ Y ) ) ) ) ).

% not_None_eq
thf(fact_42_not__None__eq,axiom,
    ! [X3: option_num] :
      ( ( X3 != none_num )
      = ( ? [Y: num] :
            ( X3
            = ( some_num @ Y ) ) ) ) ).

% not_None_eq
thf(fact_43__C4_Ohyps_C_I4_J,axiom,
    ( ( size_s6755466524823107622T_VEBT @ treeList )
    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).

% "4.hyps"(4)
thf(fact_44__092_060open_062length_AtreeList_A_061_A2_A_094_An_092_060close_062,axiom,
    ( ( size_s6755466524823107622T_VEBT @ treeList )
    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ).

% \<open>length treeList = 2 ^ n\<close>
thf(fact_45__092_060open_062high_Ax_An_A_060_A2_A_094_An_A_092_060and_062_Alow_Ax_An_A_060_A2_A_094_An_092_060close_062,axiom,
    ( ( ord_less_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) )
    & ( ord_less_nat @ ( vEBT_VEBT_low @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ) ).

% \<open>high x n < 2 ^ n \<and> low x n < 2 ^ n\<close>
thf(fact_46__C4_Ohyps_C_I10_J,axiom,
    ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).

% "4.hyps"(10)
thf(fact_47_less__shift,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_less @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).

% less_shift
thf(fact_48_lesseq__shift,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).

% lesseq_shift
thf(fact_49_verit__comp__simplify1_I3_J,axiom,
    ! [B: real,A2: real] :
      ( ( ~ ( ord_less_eq_real @ B @ A2 ) )
      = ( ord_less_real @ A2 @ B ) ) ).

% verit_comp_simplify1(3)
thf(fact_50_verit__comp__simplify1_I3_J,axiom,
    ! [B: rat,A2: rat] :
      ( ( ~ ( ord_less_eq_rat @ B @ A2 ) )
      = ( ord_less_rat @ A2 @ B ) ) ).

% verit_comp_simplify1(3)
thf(fact_51_verit__comp__simplify1_I3_J,axiom,
    ! [B: num,A2: num] :
      ( ( ~ ( ord_less_eq_num @ B @ A2 ) )
      = ( ord_less_num @ A2 @ B ) ) ).

% verit_comp_simplify1(3)
thf(fact_52_verit__comp__simplify1_I3_J,axiom,
    ! [B: nat,A2: nat] :
      ( ( ~ ( ord_less_eq_nat @ B @ A2 ) )
      = ( ord_less_nat @ A2 @ B ) ) ).

% verit_comp_simplify1(3)
thf(fact_53_verit__comp__simplify1_I3_J,axiom,
    ! [B: int,A2: int] :
      ( ( ~ ( ord_less_eq_int @ B @ A2 ) )
      = ( ord_less_int @ A2 @ B ) ) ).

% verit_comp_simplify1(3)
thf(fact_54_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_55_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_56_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_57_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_58_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_59_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_60_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_61_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_62_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_63_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_64_verit__la__disequality,axiom,
    ! [A: rat,B2: rat] :
      ( ( A = B2 )
      | ~ ( ord_less_eq_rat @ A @ B2 )
      | ~ ( ord_less_eq_rat @ B2 @ A ) ) ).

% verit_la_disequality
thf(fact_65_verit__la__disequality,axiom,
    ! [A: num,B2: num] :
      ( ( A = B2 )
      | ~ ( ord_less_eq_num @ A @ B2 )
      | ~ ( ord_less_eq_num @ B2 @ A ) ) ).

% verit_la_disequality
thf(fact_66_verit__la__disequality,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = B2 )
      | ~ ( ord_less_eq_nat @ A @ B2 )
      | ~ ( ord_less_eq_nat @ B2 @ A ) ) ).

% verit_la_disequality
thf(fact_67_verit__la__disequality,axiom,
    ! [A: int,B2: int] :
      ( ( A = B2 )
      | ~ ( ord_less_eq_int @ A @ B2 )
      | ~ ( ord_less_eq_int @ B2 @ A ) ) ).

% verit_la_disequality
thf(fact_68_div__le__dividend,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).

% div_le_dividend
thf(fact_69_div__le__mono,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).

% div_le_mono
thf(fact_70_combine__options__cases,axiom,
    ! [X3: option_nat,P: option_nat > option_nat > $o,Y3: option_nat] :
      ( ( ( X3 = none_nat )
       => ( P @ X3 @ Y3 ) )
     => ( ( ( Y3 = none_nat )
         => ( P @ X3 @ Y3 ) )
       => ( ! [A3: nat,B3: nat] :
              ( ( X3
                = ( some_nat @ A3 ) )
             => ( ( Y3
                  = ( some_nat @ B3 ) )
               => ( P @ X3 @ Y3 ) ) )
         => ( P @ X3 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_71_combine__options__cases,axiom,
    ! [X3: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
      ( ( ( X3 = none_nat )
       => ( P @ X3 @ Y3 ) )
     => ( ( ( Y3 = none_P5556105721700978146at_nat )
         => ( P @ X3 @ Y3 ) )
       => ( ! [A3: nat,B3: product_prod_nat_nat] :
              ( ( X3
                = ( some_nat @ A3 ) )
             => ( ( Y3
                  = ( some_P7363390416028606310at_nat @ B3 ) )
               => ( P @ X3 @ Y3 ) ) )
         => ( P @ X3 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_72_combine__options__cases,axiom,
    ! [X3: option_nat,P: option_nat > option_num > $o,Y3: option_num] :
      ( ( ( X3 = none_nat )
       => ( P @ X3 @ Y3 ) )
     => ( ( ( Y3 = none_num )
         => ( P @ X3 @ Y3 ) )
       => ( ! [A3: nat,B3: num] :
              ( ( X3
                = ( some_nat @ A3 ) )
             => ( ( Y3
                  = ( some_num @ B3 ) )
               => ( P @ X3 @ Y3 ) ) )
         => ( P @ X3 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_73_combine__options__cases,axiom,
    ! [X3: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y3: option_nat] :
      ( ( ( X3 = none_P5556105721700978146at_nat )
       => ( P @ X3 @ Y3 ) )
     => ( ( ( Y3 = none_nat )
         => ( P @ X3 @ Y3 ) )
       => ( ! [A3: product_prod_nat_nat,B3: nat] :
              ( ( X3
                = ( some_P7363390416028606310at_nat @ A3 ) )
             => ( ( Y3
                  = ( some_nat @ B3 ) )
               => ( P @ X3 @ Y3 ) ) )
         => ( P @ X3 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_74_combine__options__cases,axiom,
    ! [X3: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
      ( ( ( X3 = none_P5556105721700978146at_nat )
       => ( P @ X3 @ Y3 ) )
     => ( ( ( Y3 = none_P5556105721700978146at_nat )
         => ( P @ X3 @ Y3 ) )
       => ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
              ( ( X3
                = ( some_P7363390416028606310at_nat @ A3 ) )
             => ( ( Y3
                  = ( some_P7363390416028606310at_nat @ B3 ) )
               => ( P @ X3 @ Y3 ) ) )
         => ( P @ X3 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_75_combine__options__cases,axiom,
    ! [X3: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y3: option_num] :
      ( ( ( X3 = none_P5556105721700978146at_nat )
       => ( P @ X3 @ Y3 ) )
     => ( ( ( Y3 = none_num )
         => ( P @ X3 @ Y3 ) )
       => ( ! [A3: product_prod_nat_nat,B3: num] :
              ( ( X3
                = ( some_P7363390416028606310at_nat @ A3 ) )
             => ( ( Y3
                  = ( some_num @ B3 ) )
               => ( P @ X3 @ Y3 ) ) )
         => ( P @ X3 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_76_combine__options__cases,axiom,
    ! [X3: option_num,P: option_num > option_nat > $o,Y3: option_nat] :
      ( ( ( X3 = none_num )
       => ( P @ X3 @ Y3 ) )
     => ( ( ( Y3 = none_nat )
         => ( P @ X3 @ Y3 ) )
       => ( ! [A3: num,B3: nat] :
              ( ( X3
                = ( some_num @ A3 ) )
             => ( ( Y3
                  = ( some_nat @ B3 ) )
               => ( P @ X3 @ Y3 ) ) )
         => ( P @ X3 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_77_combine__options__cases,axiom,
    ! [X3: option_num,P: option_num > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
      ( ( ( X3 = none_num )
       => ( P @ X3 @ Y3 ) )
     => ( ( ( Y3 = none_P5556105721700978146at_nat )
         => ( P @ X3 @ Y3 ) )
       => ( ! [A3: num,B3: product_prod_nat_nat] :
              ( ( X3
                = ( some_num @ A3 ) )
             => ( ( Y3
                  = ( some_P7363390416028606310at_nat @ B3 ) )
               => ( P @ X3 @ Y3 ) ) )
         => ( P @ X3 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_78_combine__options__cases,axiom,
    ! [X3: option_num,P: option_num > option_num > $o,Y3: option_num] :
      ( ( ( X3 = none_num )
       => ( P @ X3 @ Y3 ) )
     => ( ( ( Y3 = none_num )
         => ( P @ X3 @ Y3 ) )
       => ( ! [A3: num,B3: num] :
              ( ( X3
                = ( some_num @ A3 ) )
             => ( ( Y3
                  = ( some_num @ B3 ) )
               => ( P @ X3 @ Y3 ) ) )
         => ( P @ X3 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_79_split__option__all,axiom,
    ( ( ^ [P2: option_nat > $o] :
        ! [X4: option_nat] : ( P2 @ X4 ) )
    = ( ^ [P3: option_nat > $o] :
          ( ( P3 @ none_nat )
          & ! [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).

% split_option_all
thf(fact_80_split__option__all,axiom,
    ( ( ^ [P2: option4927543243414619207at_nat > $o] :
        ! [X4: option4927543243414619207at_nat] : ( P2 @ X4 ) )
    = ( ^ [P3: option4927543243414619207at_nat > $o] :
          ( ( P3 @ none_P5556105721700978146at_nat )
          & ! [X: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).

% split_option_all
thf(fact_81_split__option__all,axiom,
    ( ( ^ [P2: option_num > $o] :
        ! [X4: option_num] : ( P2 @ X4 ) )
    = ( ^ [P3: option_num > $o] :
          ( ( P3 @ none_num )
          & ! [X: num] : ( P3 @ ( some_num @ X ) ) ) ) ) ).

% split_option_all
thf(fact_82_split__option__ex,axiom,
    ( ( ^ [P2: option_nat > $o] :
        ? [X4: option_nat] : ( P2 @ X4 ) )
    = ( ^ [P3: option_nat > $o] :
          ( ( P3 @ none_nat )
          | ? [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_83_split__option__ex,axiom,
    ( ( ^ [P2: option4927543243414619207at_nat > $o] :
        ? [X4: option4927543243414619207at_nat] : ( P2 @ X4 ) )
    = ( ^ [P3: option4927543243414619207at_nat > $o] :
          ( ( P3 @ none_P5556105721700978146at_nat )
          | ? [X: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_84_split__option__ex,axiom,
    ( ( ^ [P2: option_num > $o] :
        ? [X4: option_num] : ( P2 @ X4 ) )
    = ( ^ [P3: option_num > $o] :
          ( ( P3 @ none_num )
          | ? [X: num] : ( P3 @ ( some_num @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_85_option_Oexhaust,axiom,
    ! [Y3: option_nat] :
      ( ( Y3 != none_nat )
     => ~ ! [X22: nat] :
            ( Y3
           != ( some_nat @ X22 ) ) ) ).

% option.exhaust
thf(fact_86_option_Oexhaust,axiom,
    ! [Y3: option4927543243414619207at_nat] :
      ( ( Y3 != none_P5556105721700978146at_nat )
     => ~ ! [X22: product_prod_nat_nat] :
            ( Y3
           != ( some_P7363390416028606310at_nat @ X22 ) ) ) ).

% option.exhaust
thf(fact_87_option_Oexhaust,axiom,
    ! [Y3: option_num] :
      ( ( Y3 != none_num )
     => ~ ! [X22: num] :
            ( Y3
           != ( some_num @ X22 ) ) ) ).

% option.exhaust
thf(fact_88_option_OdiscI,axiom,
    ! [Option: option_nat,X2: nat] :
      ( ( Option
        = ( some_nat @ X2 ) )
     => ( Option != none_nat ) ) ).

% option.discI
thf(fact_89_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X2: product_prod_nat_nat] :
      ( ( Option
        = ( some_P7363390416028606310at_nat @ X2 ) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
thf(fact_90_option_OdiscI,axiom,
    ! [Option: option_num,X2: num] :
      ( ( Option
        = ( some_num @ X2 ) )
     => ( Option != none_num ) ) ).

% option.discI
thf(fact_91_option_Odistinct_I1_J,axiom,
    ! [X2: nat] :
      ( none_nat
     != ( some_nat @ X2 ) ) ).

% option.distinct(1)
thf(fact_92_option_Odistinct_I1_J,axiom,
    ! [X2: product_prod_nat_nat] :
      ( none_P5556105721700978146at_nat
     != ( some_P7363390416028606310at_nat @ X2 ) ) ).

% option.distinct(1)
thf(fact_93_option_Odistinct_I1_J,axiom,
    ! [X2: num] :
      ( none_num
     != ( some_num @ X2 ) ) ).

% option.distinct(1)
thf(fact_94_mem__Collect__eq,axiom,
    ! [A: complex,P: complex > $o] :
      ( ( member_complex @ A @ ( collect_complex @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_95_mem__Collect__eq,axiom,
    ! [A: real,P: real > $o] :
      ( ( member_real @ A @ ( collect_real @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_96_mem__Collect__eq,axiom,
    ! [A: list_nat,P: list_nat > $o] :
      ( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_97_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_98_mem__Collect__eq,axiom,
    ! [A: int,P: int > $o] :
      ( ( member_int @ A @ ( collect_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_99_Collect__mem__eq,axiom,
    ! [A4: set_complex] :
      ( ( collect_complex
        @ ^ [X: complex] : ( member_complex @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_100_Collect__mem__eq,axiom,
    ! [A4: set_real] :
      ( ( collect_real
        @ ^ [X: real] : ( member_real @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_101_Collect__mem__eq,axiom,
    ! [A4: set_list_nat] :
      ( ( collect_list_nat
        @ ^ [X: list_nat] : ( member_list_nat @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_102_Collect__mem__eq,axiom,
    ! [A4: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_103_Collect__mem__eq,axiom,
    ! [A4: set_int] :
      ( ( collect_int
        @ ^ [X: int] : ( member_int @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_104_Collect__cong,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ! [X5: complex] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_complex @ P )
        = ( collect_complex @ Q ) ) ) ).

% Collect_cong
thf(fact_105_Collect__cong,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ! [X5: real] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_real @ P )
        = ( collect_real @ Q ) ) ) ).

% Collect_cong
thf(fact_106_Collect__cong,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ! [X5: list_nat] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_list_nat @ P )
        = ( collect_list_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_107_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X5: nat] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_108_Collect__cong,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X5: int] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_int @ P )
        = ( collect_int @ Q ) ) ) ).

% Collect_cong
thf(fact_109_verit__eq__simplify_I10_J,axiom,
    ! [X2: num] :
      ( one
     != ( bit0 @ X2 ) ) ).

% verit_eq_simplify(10)
thf(fact_110__092_060open_062invar__vebt_A_ItreeList_A_B_Ahigh_Ax_A_Ideg_Adiv_A2_J_J_An_092_060close_062,axiom,
    vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ na ).

% \<open>invar_vebt (treeList ! high x (deg div 2)) n\<close>
thf(fact_111_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( ( vEBT_vebt_mint @ T )
        = none_nat )
     => ( vEBT_VEBT_minNull @ T ) ) ).

% minminNull
thf(fact_112_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ T )
     => ( ( vEBT_vebt_mint @ T )
        = none_nat ) ) ).

% minNullmin
thf(fact_113_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_114_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_115_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_116_less__exp,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% less_exp
thf(fact_117_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
     => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% high_bound_aux
thf(fact_118_power__minus__is__div,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A @ B2 ) )
        = ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).

% power_minus_is_div
thf(fact_119_misiz,axiom,
    ! [T: vEBT_VEBT,N: nat,M: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( some_nat @ M )
          = ( vEBT_vebt_mint @ T ) )
       => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% misiz
thf(fact_120_pow__sum,axiom,
    ! [A: nat,B2: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ).

% pow_sum
thf(fact_121__092_060open_062x_A_092_060le_062_Ama_092_060close_062,axiom,
    ord_less_eq_nat @ xa @ ma ).

% \<open>x \<le> ma\<close>
thf(fact_122__C4_Ohyps_C_I6_J,axiom,
    ( deg
    = ( plus_plus_nat @ na @ m ) ) ).

% "4.hyps"(6)
thf(fact_123__C4_Ohyps_C_I9_J,axiom,
    ord_less_eq_nat @ mi @ ma ).

% "4.hyps"(9)
thf(fact_124__C4_Ohyps_C_I2_J,axiom,
    vEBT_invar_vebt @ summary @ m ).

% "4.hyps"(2)
thf(fact_125_add__numeral__left,axiom,
    ! [V: num,W: num,Z: complex] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_126_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_127_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_128_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_129_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_130_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_131_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_132_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_133_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_134_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_135_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_136_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_137_semiring__norm_I75_J,axiom,
    ! [M: num] :
      ~ ( ord_less_num @ M @ one ) ).

% semiring_norm(75)
thf(fact_138_semiring__norm_I68_J,axiom,
    ! [N: num] : ( ord_less_eq_num @ one @ N ) ).

% semiring_norm(68)
thf(fact_139_semiring__norm_I76_J,axiom,
    ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).

% semiring_norm(76)
thf(fact_140_semiring__norm_I69_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).

% semiring_norm(69)
thf(fact_141_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_142_local_Opower__def,axiom,
    ( vEBT_VEBT_power
    = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).

% local.power_def
thf(fact_143_is__num__normalize_I1_J,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B2 ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_144_is__num__normalize_I1_J,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_145_is__num__normalize_I1_J,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_146_le__num__One__iff,axiom,
    ! [X3: num] :
      ( ( ord_less_eq_num @ X3 @ one )
      = ( X3 = one ) ) ).

% le_num_One_iff
thf(fact_147_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).

% numeral_Bit0
thf(fact_148_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_149_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_150_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_151_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_152_power2__commute,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( power_power_complex @ ( minus_minus_complex @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_complex @ ( minus_minus_complex @ Y3 @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_153_power2__commute,axiom,
    ! [X3: real,Y3: real] :
      ( ( power_power_real @ ( minus_minus_real @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ ( minus_minus_real @ Y3 @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_154_power2__commute,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( power_power_rat @ ( minus_minus_rat @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_rat @ ( minus_minus_rat @ Y3 @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_155_power2__commute,axiom,
    ! [X3: int,Y3: int] :
      ( ( power_power_int @ ( minus_minus_int @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ ( minus_minus_int @ Y3 @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_156_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_157_power__divide,axiom,
    ! [A: complex,B2: complex,N: nat] :
      ( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B2 ) @ N )
      = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B2 @ N ) ) ) ).

% power_divide
thf(fact_158_power__divide,axiom,
    ! [A: real,B2: real,N: nat] :
      ( ( power_power_real @ ( divide_divide_real @ A @ B2 ) @ N )
      = ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ).

% power_divide
thf(fact_159_power__divide,axiom,
    ! [A: rat,B2: rat,N: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ A @ B2 ) @ N )
      = ( divide_divide_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ).

% power_divide
thf(fact_160_mint__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X3 ) )
       => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 ) ) ) ).

% mint_corr
thf(fact_161_mint__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 )
       => ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X3 ) ) ) ) ).

% mint_sound
thf(fact_162_add__shift,axiom,
    ! [X3: nat,Y3: nat,Z: nat] :
      ( ( ( plus_plus_nat @ X3 @ Y3 )
        = Z )
      = ( ( vEBT_VEBT_add @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) )
        = ( some_nat @ Z ) ) ) ).

% add_shift
thf(fact_163_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_164_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_165_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_166_le__add__diff__inverse,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ( plus_plus_real @ B2 @ ( minus_minus_real @ A @ B2 ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_167_le__add__diff__inverse,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_eq_rat @ B2 @ A )
     => ( ( plus_plus_rat @ B2 @ ( minus_minus_rat @ A @ B2 ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_168_le__add__diff__inverse,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A @ B2 ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_169_le__add__diff__inverse,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A @ B2 ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_170_le__add__diff__inverse2,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ B2 )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_171_le__add__diff__inverse2,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_eq_rat @ B2 @ A )
     => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B2 ) @ B2 )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_172_le__add__diff__inverse2,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B2 ) @ B2 )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_173_le__add__diff__inverse2,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ B2 )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_174_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X3: nat,N: nat] :
      ( ( vEBT_vebt_member @ Tree @ X3 )
     => ( ( vEBT_invar_vebt @ Tree @ N )
       => ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% member_bound
thf(fact_175__C4_Ohyps_C_I11_J,axiom,
    ( ( mi != ma )
   => ! [I2: nat] :
        ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
       => ( ( ( ( vEBT_VEBT_high @ ma @ na )
              = I2 )
           => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I2 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
          & ! [X6: nat] :
              ( ( ( ( vEBT_VEBT_high @ X6 @ na )
                  = I2 )
                & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I2 ) @ ( vEBT_VEBT_low @ X6 @ na ) ) )
             => ( ( ord_less_nat @ mi @ X6 )
                & ( ord_less_eq_nat @ X6 @ ma ) ) ) ) ) ) ).

% "4.hyps"(11)
thf(fact_176_div__exp__eq,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).

% div_exp_eq
thf(fact_177_div__exp__eq,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).

% div_exp_eq
thf(fact_178_field__less__half__sum,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ( ord_less_real @ X3 @ ( divide_divide_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_179_field__less__half__sum,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ X3 @ Y3 )
     => ( ord_less_rat @ X3 @ ( divide_divide_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_180_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ T )
     => ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 ) ) ).

% not_min_Null_member
thf(fact_181_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X3: nat] :
      ( ( vEBT_VEBT_minNull @ T )
     => ~ ( vEBT_vebt_member @ T @ X3 ) ) ).

% min_Null_member
thf(fact_182_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X3 )
        = ( vEBT_vebt_member @ T @ X3 ) ) ) ).

% both_member_options_equiv_member
thf(fact_183_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X3 )
       => ( vEBT_vebt_member @ T @ X3 ) ) ) ).

% valid_member_both_member_options
thf(fact_184_add__def,axiom,
    ( vEBT_VEBT_add
    = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).

% add_def
thf(fact_185_mint__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).

% mint_member
thf(fact_186_member__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_vebt_member @ T @ X3 )
        = ( member_nat @ X3 @ ( vEBT_set_vebt @ T ) ) ) ) ).

% member_correct
thf(fact_187_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Mini: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Mini ) )
       => ( ( vEBT_vebt_member @ T @ X3 )
         => ( ord_less_eq_nat @ Mini @ X3 ) ) ) ) ).

% mint_corr_help
thf(fact_188_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_set_vebt @ T )
        = ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_set_vebt'_valid
thf(fact_189_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_190_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_191_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_192_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_193_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_194_semiring__norm_I2_J,axiom,
    ( ( plus_plus_num @ one @ one )
    = ( bit0 @ one ) ) ).

% semiring_norm(2)
thf(fact_195_pred__member,axiom,
    ! [T: vEBT_VEBT,X3: nat,Y3: nat] :
      ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 @ Y3 )
      = ( ( vEBT_vebt_member @ T @ Y3 )
        & ( ord_less_nat @ Y3 @ X3 )
        & ! [Z2: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z2 )
              & ( ord_less_nat @ Z2 @ X3 ) )
           => ( ord_less_eq_nat @ Z2 @ Y3 ) ) ) ) ).

% pred_member
thf(fact_196__C4_Ohyps_C_I7_J,axiom,
    ! [I2: nat] :
      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
     => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I2 ) @ X7 ) )
        = ( vEBT_V8194947554948674370ptions @ summary @ I2 ) ) ) ).

% "4.hyps"(7)
thf(fact_197_add__One__commute,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ N )
      = ( plus_plus_num @ N @ one ) ) ).

% add_One_commute
thf(fact_198_linorder__neqE__linordered__idom,axiom,
    ! [X3: real,Y3: real] :
      ( ( X3 != Y3 )
     => ( ~ ( ord_less_real @ X3 @ Y3 )
       => ( ord_less_real @ Y3 @ X3 ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_199_linorder__neqE__linordered__idom,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( X3 != Y3 )
     => ( ~ ( ord_less_rat @ X3 @ Y3 )
       => ( ord_less_rat @ Y3 @ X3 ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_200_linorder__neqE__linordered__idom,axiom,
    ! [X3: int,Y3: int] :
      ( ( X3 != Y3 )
     => ( ~ ( ord_less_int @ X3 @ Y3 )
       => ( ord_less_int @ Y3 @ X3 ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_201_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_202_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_203_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( M != N ) ) ).

% less_not_refl2
thf(fact_204_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_205_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_206_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N3 )
             => ( P @ M2 ) )
         => ( P @ N3 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_207_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ~ ( P @ N3 )
         => ? [M2: nat] :
              ( ( ord_less_nat @ M2 @ N3 )
              & ~ ( P @ M2 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_208_linorder__neqE__nat,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( X3 != Y3 )
     => ( ~ ( ord_less_nat @ X3 @ Y3 )
       => ( ord_less_nat @ Y3 @ X3 ) ) ) ).

% linorder_neqE_nat
thf(fact_209_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_210_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_211_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_212_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_213_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_214_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 ) )
       => ? [X5: nat] :
            ( ( P @ X5 )
            & ! [Y5: nat] :
                ( ( P @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_215_size__neq__size__imp__neq,axiom,
    ! [X3: list_VEBT_VEBT,Y3: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ X3 )
       != ( size_s6755466524823107622T_VEBT @ Y3 ) )
     => ( X3 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_216_size__neq__size__imp__neq,axiom,
    ! [X3: num,Y3: num] :
      ( ( ( size_size_num @ X3 )
       != ( size_size_num @ Y3 ) )
     => ( X3 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_217_size__neq__size__imp__neq,axiom,
    ! [X3: list_o,Y3: list_o] :
      ( ( ( size_size_list_o @ X3 )
       != ( size_size_list_o @ Y3 ) )
     => ( X3 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_218_size__neq__size__imp__neq,axiom,
    ! [X3: list_nat,Y3: list_nat] :
      ( ( ( size_size_list_nat @ X3 )
       != ( size_size_list_nat @ Y3 ) )
     => ( X3 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_219_size__neq__size__imp__neq,axiom,
    ! [X3: vEBT_VEBT,Y3: vEBT_VEBT] :
      ( ( ( size_size_VEBT_VEBT @ X3 )
       != ( size_size_VEBT_VEBT @ Y3 ) )
     => ( X3 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_220_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_221_add__diff__add,axiom,
    ! [A: complex,C: complex,B2: complex,D: complex] :
      ( ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B2 @ D ) )
      = ( plus_plus_complex @ ( minus_minus_complex @ A @ B2 ) @ ( minus_minus_complex @ C @ D ) ) ) ).

% add_diff_add
thf(fact_222_add__diff__add,axiom,
    ! [A: real,C: real,B2: real,D: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) )
      = ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ ( minus_minus_real @ C @ D ) ) ) ).

% add_diff_add
thf(fact_223_add__diff__add,axiom,
    ! [A: rat,C: rat,B2: rat,D: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) )
      = ( plus_plus_rat @ ( minus_minus_rat @ A @ B2 ) @ ( minus_minus_rat @ C @ D ) ) ) ).

% add_diff_add
thf(fact_224_add__diff__add,axiom,
    ! [A: int,C: int,B2: int,D: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) )
      = ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ ( minus_minus_int @ C @ D ) ) ) ).

% add_diff_add
thf(fact_225_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M3 @ N2 )
          & ( M3 != N2 ) ) ) ) ).

% nat_less_le
thf(fact_226_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_227_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N2: nat] :
          ( ( ord_less_nat @ M3 @ N2 )
          | ( M3 = N2 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_228_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_229_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_230_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_231_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_232_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_233_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_234_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_235_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_236_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_237_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_238_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_239_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_240_le__add1,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).

% le_add1
thf(fact_241_le__add2,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).

% le_add2
thf(fact_242_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_243_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_244_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N3: nat] :
          ( L
          = ( plus_plus_nat @ K @ N3 ) ) ) ).

% le_Suc_ex
thf(fact_245_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_246_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_247_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_248_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_249_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N2: nat] :
        ? [K2: nat] :
          ( N2
          = ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).

% nat_le_iff_add
thf(fact_250_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_251_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_252_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_253_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_254_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_255_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_256_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_257_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B2 ) )
          = ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_258_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_259_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_260_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_261_diff__add__inverse,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
      = M ) ).

% diff_add_inverse
thf(fact_262_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
      = M ) ).

% diff_add_inverse2
thf(fact_263_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_264_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_265_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_266_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_267_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_268_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_269_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_270_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_271_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B2: real] :
      ( ~ ( ord_less_real @ A @ B2 )
     => ( ( plus_plus_real @ B2 @ ( minus_minus_real @ A @ B2 ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_272_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B2: rat] :
      ( ~ ( ord_less_rat @ A @ B2 )
     => ( ( plus_plus_rat @ B2 @ ( minus_minus_rat @ A @ B2 ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_273_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B2: nat] :
      ( ~ ( ord_less_nat @ A @ B2 )
     => ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A @ B2 ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_274_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B2: int] :
      ( ~ ( ord_less_int @ A @ B2 )
     => ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A @ B2 ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_275_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_nat @ M4 @ N3 )
         => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_276_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_277_diff__less__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).

% diff_less_mono
thf(fact_278_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_279_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_280_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_281_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_282_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_283_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_284_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_285_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_286_field__sum__of__halves,axiom,
    ! [X3: real] :
      ( ( plus_plus_real @ ( divide_divide_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = X3 ) ).

% field_sum_of_halves
thf(fact_287_field__sum__of__halves,axiom,
    ! [X3: rat] :
      ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X3 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
      = X3 ) ).

% field_sum_of_halves
thf(fact_288_in__children__def,axiom,
    ( vEBT_V5917875025757280293ildren
    = ( ^ [N2: nat,TreeList: list_VEBT_VEBT,X: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ N2 ) ) @ ( vEBT_VEBT_low @ X @ N2 ) ) ) ) ).

% in_children_def
thf(fact_289_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 )
       => ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X3 ) ) ) ) ).

% maxt_sound
thf(fact_290_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X3 ) )
       => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 ) ) ) ).

% maxt_corr
thf(fact_291_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat,Y3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
         => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X3 ) @ Y3 )
           => ( ( vEBT_vebt_member @ T @ Y3 )
              | ( X3 = Y3 ) ) ) ) ) ) ).

% post_member_pre_member
thf(fact_292_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat,Y3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
         => ( ( vEBT_V8194947554948674370ptions @ T @ X3 )
           => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y3 ) @ X3 ) ) ) ) ) ).

% valid_insert_both_member_options_pres
thf(fact_293_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X3 ) @ X3 ) ) ) ).

% valid_insert_both_member_options_add
thf(fact_294_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( ( vEBT_vebt_member @ T @ X3 )
         => ( ord_less_eq_nat @ X3 @ Maxi ) ) ) ) ).

% maxt_corr_help
thf(fact_295_maxt__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).

% maxt_member
thf(fact_296_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
     => ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ X3 ) ) ) ) ).

% both_member_options_ding
thf(fact_297_maxbmo,axiom,
    ! [T: vEBT_VEBT,X3: nat] :
      ( ( ( vEBT_vebt_maxt @ T )
        = ( some_nat @ X3 ) )
     => ( vEBT_V8194947554948674370ptions @ T @ X3 ) ) ).

% maxbmo
thf(fact_298_bit__concat__def,axiom,
    ( vEBT_VEBT_bit_concat
    = ( ^ [H: nat,L2: nat,D2: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D2 ) ) @ L2 ) ) ) ).

% bit_concat_def
thf(fact_299_low__inv,axiom,
    ! [X3: nat,N: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X3 ) @ N )
        = X3 ) ) ).

% low_inv
thf(fact_300_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
     => ( Deg = N ) ) ).

% deg_deg_n
thf(fact_301_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_302_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_303_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_304_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_305_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_306_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
      = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_307_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_308_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_309_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_310_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_311_high__inv,axiom,
    ! [X3: nat,N: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X3 ) @ N )
        = Y3 ) ) ).

% high_inv
thf(fact_312_distrib__right__numeral,axiom,
    ! [A: complex,B2: complex,V: num] :
      ( ( times_times_complex @ ( plus_plus_complex @ A @ B2 ) @ ( numera6690914467698888265omplex @ V ) )
      = ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B2 @ ( numera6690914467698888265omplex @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_313_distrib__right__numeral,axiom,
    ! [A: real,B2: real,V: num] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ ( numeral_numeral_real @ V ) )
      = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B2 @ ( numeral_numeral_real @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_314_distrib__right__numeral,axiom,
    ! [A: rat,B2: rat,V: num] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ ( numeral_numeral_rat @ V ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B2 @ ( numeral_numeral_rat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_315_distrib__right__numeral,axiom,
    ! [A: nat,B2: nat,V: num] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ ( numeral_numeral_nat @ V ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B2 @ ( numeral_numeral_nat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_316_distrib__right__numeral,axiom,
    ! [A: int,B2: int,V: num] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ ( numeral_numeral_int @ V ) )
      = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B2 @ ( numeral_numeral_int @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_317_distrib__left__numeral,axiom,
    ! [V: num,B2: complex,C: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B2 @ C ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B2 ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_318_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_319_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_320_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_321_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_322_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: complex,C: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( minus_minus_complex @ B2 @ C ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B2 ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_323_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_324_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_325_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_326_left__diff__distrib__numeral,axiom,
    ! [A: complex,B2: complex,V: num] :
      ( ( times_times_complex @ ( minus_minus_complex @ A @ B2 ) @ ( numera6690914467698888265omplex @ V ) )
      = ( minus_minus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B2 @ ( numera6690914467698888265omplex @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_327_left__diff__distrib__numeral,axiom,
    ! [A: real,B2: real,V: num] :
      ( ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ ( numeral_numeral_real @ V ) )
      = ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B2 @ ( numeral_numeral_real @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_328_left__diff__distrib__numeral,axiom,
    ! [A: rat,B2: rat,V: num] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ ( numeral_numeral_rat @ V ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B2 @ ( numeral_numeral_rat @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_329_left__diff__distrib__numeral,axiom,
    ! [A: int,B2: int,V: num] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ ( numeral_numeral_int @ V ) )
      = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B2 @ ( numeral_numeral_int @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_330_divide__le__eq__numeral1_I1_J,axiom,
    ! [B2: real,W: num,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) @ A )
      = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_331_divide__le__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W: num,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) @ A )
      = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_332_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B2: real,W: num] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) )
      = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B2 ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_333_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B2: rat,W: num] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) )
      = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B2 ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_334_divide__less__eq__numeral1_I1_J,axiom,
    ! [B2: real,W: num,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) @ A )
      = ( ord_less_real @ B2 @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_335_divide__less__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W: num,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) @ A )
      = ( ord_less_rat @ B2 @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_336_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B2: real,W: num] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) )
      = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B2 ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_337_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B2: rat,W: num] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) )
      = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B2 ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_338_power__add__numeral2,axiom,
    ! [A: complex,M: num,N: num,B2: complex] :
      ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
      = ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_339_power__add__numeral2,axiom,
    ! [A: real,M: num,N: num,B2: real] :
      ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
      = ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_340_power__add__numeral2,axiom,
    ! [A: rat,M: num,N: num,B2: rat] :
      ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
      = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_341_power__add__numeral2,axiom,
    ! [A: nat,M: num,N: num,B2: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
      = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_342_power__add__numeral2,axiom,
    ! [A: int,M: num,N: num,B2: int] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
      = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_343_power__add__numeral,axiom,
    ! [A: complex,M: num,N: num] :
      ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_344_power__add__numeral,axiom,
    ! [A: real,M: num,N: num] :
      ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_345_power__add__numeral,axiom,
    ! [A: rat,M: num,N: num] :
      ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_346_power__add__numeral,axiom,
    ! [A: nat,M: num,N: num] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_347_power__add__numeral,axiom,
    ! [A: int,M: num,N: num] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_348__C4_Ohyps_C_I8_J,axiom,
    ( ( mi = ma )
   => ! [X6: vEBT_VEBT] :
        ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ treeList ) )
       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_12 ) ) ) ).

% "4.hyps"(8)
thf(fact_349_combine__common__factor,axiom,
    ! [A: real,E: real,B2: real,C: real] :
      ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_350_combine__common__factor,axiom,
    ! [A: rat,E: rat,B2: rat,C: rat] :
      ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_351_combine__common__factor,axiom,
    ! [A: nat,E: nat,B2: nat,C: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B2 @ E ) @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_352_combine__common__factor,axiom,
    ! [A: int,E: int,B2: int,C: int] :
      ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_353_distrib__right,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).

% distrib_right
thf(fact_354_distrib__right,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).

% distrib_right
thf(fact_355_distrib__right,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).

% distrib_right
thf(fact_356_distrib__right,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).

% distrib_right
thf(fact_357_distrib__left,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B2 @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).

% distrib_left
thf(fact_358_distrib__left,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ).

% distrib_left
thf(fact_359_distrib__left,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) ) ) ).

% distrib_left
thf(fact_360_distrib__left,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B2 @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).

% distrib_left
thf(fact_361_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_362_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_363_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_364_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_365_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B2 @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_366_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_367_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B2 @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_368_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_369_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_370_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_371_right__diff__distrib_H,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( times_times_complex @ A @ ( minus_minus_complex @ B2 @ C ) )
      = ( minus_minus_complex @ ( times_times_complex @ A @ B2 ) @ ( times_times_complex @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_372_right__diff__distrib_H,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ B2 @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_373_right__diff__distrib_H,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ A @ ( minus_minus_rat @ B2 @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_374_right__diff__distrib_H,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ A @ ( minus_minus_nat @ B2 @ C ) )
      = ( minus_minus_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_375_right__diff__distrib_H,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_376_left__diff__distrib_H,axiom,
    ! [B2: complex,C: complex,A: complex] :
      ( ( times_times_complex @ ( minus_minus_complex @ B2 @ C ) @ A )
      = ( minus_minus_complex @ ( times_times_complex @ B2 @ A ) @ ( times_times_complex @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_377_left__diff__distrib_H,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( times_times_real @ ( minus_minus_real @ B2 @ C ) @ A )
      = ( minus_minus_real @ ( times_times_real @ B2 @ A ) @ ( times_times_real @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_378_left__diff__distrib_H,axiom,
    ! [B2: rat,C: rat,A: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ B2 @ C ) @ A )
      = ( minus_minus_rat @ ( times_times_rat @ B2 @ A ) @ ( times_times_rat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_379_left__diff__distrib_H,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ B2 @ C ) @ A )
      = ( minus_minus_nat @ ( times_times_nat @ B2 @ A ) @ ( times_times_nat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_380_left__diff__distrib_H,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( times_times_int @ ( minus_minus_int @ B2 @ C ) @ A )
      = ( minus_minus_int @ ( times_times_int @ B2 @ A ) @ ( times_times_int @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_381_right__diff__distrib,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( times_times_complex @ A @ ( minus_minus_complex @ B2 @ C ) )
      = ( minus_minus_complex @ ( times_times_complex @ A @ B2 ) @ ( times_times_complex @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_382_right__diff__distrib,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ B2 @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_383_right__diff__distrib,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ A @ ( minus_minus_rat @ B2 @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_384_right__diff__distrib,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_385_left__diff__distrib,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( times_times_complex @ ( minus_minus_complex @ A @ B2 ) @ C )
      = ( minus_minus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B2 @ C ) ) ) ).

% left_diff_distrib
thf(fact_386_left__diff__distrib,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ C )
      = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).

% left_diff_distrib
thf(fact_387_left__diff__distrib,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
      = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).

% left_diff_distrib
thf(fact_388_left__diff__distrib,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ C )
      = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).

% left_diff_distrib
thf(fact_389_power__commuting__commutes,axiom,
    ! [X3: complex,Y3: complex,N: nat] :
      ( ( ( times_times_complex @ X3 @ Y3 )
        = ( times_times_complex @ Y3 @ X3 ) )
     => ( ( times_times_complex @ ( power_power_complex @ X3 @ N ) @ Y3 )
        = ( times_times_complex @ Y3 @ ( power_power_complex @ X3 @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_390_power__commuting__commutes,axiom,
    ! [X3: real,Y3: real,N: nat] :
      ( ( ( times_times_real @ X3 @ Y3 )
        = ( times_times_real @ Y3 @ X3 ) )
     => ( ( times_times_real @ ( power_power_real @ X3 @ N ) @ Y3 )
        = ( times_times_real @ Y3 @ ( power_power_real @ X3 @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_391_power__commuting__commutes,axiom,
    ! [X3: rat,Y3: rat,N: nat] :
      ( ( ( times_times_rat @ X3 @ Y3 )
        = ( times_times_rat @ Y3 @ X3 ) )
     => ( ( times_times_rat @ ( power_power_rat @ X3 @ N ) @ Y3 )
        = ( times_times_rat @ Y3 @ ( power_power_rat @ X3 @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_392_power__commuting__commutes,axiom,
    ! [X3: nat,Y3: nat,N: nat] :
      ( ( ( times_times_nat @ X3 @ Y3 )
        = ( times_times_nat @ Y3 @ X3 ) )
     => ( ( times_times_nat @ ( power_power_nat @ X3 @ N ) @ Y3 )
        = ( times_times_nat @ Y3 @ ( power_power_nat @ X3 @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_393_power__commuting__commutes,axiom,
    ! [X3: int,Y3: int,N: nat] :
      ( ( ( times_times_int @ X3 @ Y3 )
        = ( times_times_int @ Y3 @ X3 ) )
     => ( ( times_times_int @ ( power_power_int @ X3 @ N ) @ Y3 )
        = ( times_times_int @ Y3 @ ( power_power_int @ X3 @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_394_power__mult__distrib,axiom,
    ! [A: complex,B2: complex,N: nat] :
      ( ( power_power_complex @ ( times_times_complex @ A @ B2 ) @ N )
      = ( times_times_complex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B2 @ N ) ) ) ).

% power_mult_distrib
thf(fact_395_power__mult__distrib,axiom,
    ! [A: real,B2: real,N: nat] :
      ( ( power_power_real @ ( times_times_real @ A @ B2 ) @ N )
      = ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ).

% power_mult_distrib
thf(fact_396_power__mult__distrib,axiom,
    ! [A: rat,B2: rat,N: nat] :
      ( ( power_power_rat @ ( times_times_rat @ A @ B2 ) @ N )
      = ( times_times_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ).

% power_mult_distrib
thf(fact_397_power__mult__distrib,axiom,
    ! [A: nat,B2: nat,N: nat] :
      ( ( power_power_nat @ ( times_times_nat @ A @ B2 ) @ N )
      = ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ).

% power_mult_distrib
thf(fact_398_power__mult__distrib,axiom,
    ! [A: int,B2: int,N: nat] :
      ( ( power_power_int @ ( times_times_int @ A @ B2 ) @ N )
      = ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ).

% power_mult_distrib
thf(fact_399_power__commutes,axiom,
    ! [A: complex,N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ A @ N ) @ A )
      = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).

% power_commutes
thf(fact_400_power__commutes,axiom,
    ! [A: real,N: nat] :
      ( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
      = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).

% power_commutes
thf(fact_401_power__commutes,axiom,
    ! [A: rat,N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ A @ N ) @ A )
      = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).

% power_commutes
thf(fact_402_power__commutes,axiom,
    ! [A: nat,N: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).

% power_commutes
thf(fact_403_power__commutes,axiom,
    ! [A: int,N: nat] :
      ( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
      = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).

% power_commutes
thf(fact_404_power__mult,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
      = ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).

% power_mult
thf(fact_405_power__mult,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
      = ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).

% power_mult
thf(fact_406_power__mult,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
      = ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).

% power_mult
thf(fact_407_power__mult,axiom,
    ! [A: complex,M: nat,N: nat] :
      ( ( power_power_complex @ A @ ( times_times_nat @ M @ N ) )
      = ( power_power_complex @ ( power_power_complex @ A @ M ) @ N ) ) ).

% power_mult
thf(fact_408_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_409_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_410_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_411_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_412_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_413_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).

% left_add_mult_distrib
thf(fact_414_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_415_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_416_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_417_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_418_div__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q2 ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) ).

% div_mult2_eq
thf(fact_419_mult__numeral__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_420_mult__numeral__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_421_mult__numeral__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_422_mult__numeral__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_423_mult__numeral__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_424_mult__numeral__1__right,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_425_mult__numeral__1__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_426_mult__numeral__1__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_427_mult__numeral__1__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_428_mult__numeral__1__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_429_mult__diff__mult,axiom,
    ! [X3: complex,Y3: complex,A: complex,B2: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X3 @ Y3 ) @ ( times_times_complex @ A @ B2 ) )
      = ( plus_plus_complex @ ( times_times_complex @ X3 @ ( minus_minus_complex @ Y3 @ B2 ) ) @ ( times_times_complex @ ( minus_minus_complex @ X3 @ A ) @ B2 ) ) ) ).

% mult_diff_mult
thf(fact_430_mult__diff__mult,axiom,
    ! [X3: real,Y3: real,A: real,B2: real] :
      ( ( minus_minus_real @ ( times_times_real @ X3 @ Y3 ) @ ( times_times_real @ A @ B2 ) )
      = ( plus_plus_real @ ( times_times_real @ X3 @ ( minus_minus_real @ Y3 @ B2 ) ) @ ( times_times_real @ ( minus_minus_real @ X3 @ A ) @ B2 ) ) ) ).

% mult_diff_mult
thf(fact_431_mult__diff__mult,axiom,
    ! [X3: rat,Y3: rat,A: rat,B2: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X3 @ Y3 ) @ ( times_times_rat @ A @ B2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ X3 @ ( minus_minus_rat @ Y3 @ B2 ) ) @ ( times_times_rat @ ( minus_minus_rat @ X3 @ A ) @ B2 ) ) ) ).

% mult_diff_mult
thf(fact_432_mult__diff__mult,axiom,
    ! [X3: int,Y3: int,A: int,B2: int] :
      ( ( minus_minus_int @ ( times_times_int @ X3 @ Y3 ) @ ( times_times_int @ A @ B2 ) )
      = ( plus_plus_int @ ( times_times_int @ X3 @ ( minus_minus_int @ Y3 @ B2 ) ) @ ( times_times_int @ ( minus_minus_int @ X3 @ A ) @ B2 ) ) ) ).

% mult_diff_mult
thf(fact_433_square__diff__square__factored,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X3 @ X3 ) @ ( times_times_complex @ Y3 @ Y3 ) )
      = ( times_times_complex @ ( plus_plus_complex @ X3 @ Y3 ) @ ( minus_minus_complex @ X3 @ Y3 ) ) ) ).

% square_diff_square_factored
thf(fact_434_square__diff__square__factored,axiom,
    ! [X3: real,Y3: real] :
      ( ( minus_minus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) )
      = ( times_times_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( minus_minus_real @ X3 @ Y3 ) ) ) ).

% square_diff_square_factored
thf(fact_435_square__diff__square__factored,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) )
      = ( times_times_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( minus_minus_rat @ X3 @ Y3 ) ) ) ).

% square_diff_square_factored
thf(fact_436_square__diff__square__factored,axiom,
    ! [X3: int,Y3: int] :
      ( ( minus_minus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) )
      = ( times_times_int @ ( plus_plus_int @ X3 @ Y3 ) @ ( minus_minus_int @ X3 @ Y3 ) ) ) ).

% square_diff_square_factored
thf(fact_437_eq__add__iff2,axiom,
    ! [A: complex,E: complex,C: complex,B2: complex,D: complex] :
      ( ( ( plus_plus_complex @ ( times_times_complex @ A @ E ) @ C )
        = ( plus_plus_complex @ ( times_times_complex @ B2 @ E ) @ D ) )
      = ( C
        = ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B2 @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_438_eq__add__iff2,axiom,
    ! [A: real,E: real,C: real,B2: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
      = ( C
        = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_439_eq__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
      = ( C
        = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_440_eq__add__iff2,axiom,
    ! [A: int,E: int,C: int,B2: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
      = ( C
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_441_eq__add__iff1,axiom,
    ! [A: complex,E: complex,C: complex,B2: complex,D: complex] :
      ( ( ( plus_plus_complex @ ( times_times_complex @ A @ E ) @ C )
        = ( plus_plus_complex @ ( times_times_complex @ B2 @ E ) @ D ) )
      = ( ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A @ B2 ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_442_eq__add__iff1,axiom,
    ! [A: real,E: real,C: real,B2: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
      = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_443_eq__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_444_eq__add__iff1,axiom,
    ! [A: int,E: int,C: int,B2: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_445_power__add,axiom,
    ! [A: complex,M: nat,N: nat] :
      ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).

% power_add
thf(fact_446_power__add,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).

% power_add
thf(fact_447_power__add,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ).

% power_add
thf(fact_448_power__add,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).

% power_add
thf(fact_449_power__add,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).

% power_add
thf(fact_450_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_451_times__div__less__eq__dividend,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).

% times_div_less_eq_dividend
thf(fact_452_div__times__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).

% div_times_less_eq_dividend
thf(fact_453_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E: real,C: real,B2: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ E ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_454_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ E ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_455_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E: int,C: int,B2: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ E ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_456_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E: real,C: real,B2: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
      = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ E ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_457_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
      = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A ) @ E ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_458_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E: int,C: int,B2: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
      = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A ) @ E ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_459_less__add__iff1,axiom,
    ! [A: real,E: real,C: real,B2: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
      = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ E ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_460_less__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
      = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ E ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_461_less__add__iff1,axiom,
    ! [A: int,E: int,C: int,B2: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ E ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_462_less__add__iff2,axiom,
    ! [A: real,E: real,C: real,B2: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
      = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_463_less__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
      = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_464_less__add__iff2,axiom,
    ! [A: int,E: int,C: int,B2: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
      = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_465_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
          = N ) ) ) ).

% nat_eq_add_iff1
thf(fact_466_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( M
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_467_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_le_add_iff1
thf(fact_468_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_le_add_iff2
thf(fact_469_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_diff_add_eq1
thf(fact_470_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_471_mult__2,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_complex @ Z @ Z ) ) ).

% mult_2
thf(fact_472_mult__2,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_real @ Z @ Z ) ) ).

% mult_2
thf(fact_473_mult__2,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2
thf(fact_474_mult__2,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2
thf(fact_475_mult__2,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2
thf(fact_476_mult__2__right,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ Z @ Z ) ) ).

% mult_2_right
thf(fact_477_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_478_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_479_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_480_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_481_left__add__twice,axiom,
    ! [A: complex,B2: complex] :
      ( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B2 ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).

% left_add_twice
thf(fact_482_left__add__twice,axiom,
    ! [A: real,B2: real] :
      ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B2 ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).

% left_add_twice
thf(fact_483_left__add__twice,axiom,
    ! [A: rat,B2: rat] :
      ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).

% left_add_twice
thf(fact_484_left__add__twice,axiom,
    ! [A: nat,B2: nat] :
      ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).

% left_add_twice
thf(fact_485_left__add__twice,axiom,
    ! [A: int,B2: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B2 ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).

% left_add_twice
thf(fact_486_power4__eq__xxxx,axiom,
    ! [X3: complex] :
      ( ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X3 @ X3 ) @ X3 ) @ X3 ) ) ).

% power4_eq_xxxx
thf(fact_487_power4__eq__xxxx,axiom,
    ! [X3: real] :
      ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_real @ ( times_times_real @ ( times_times_real @ X3 @ X3 ) @ X3 ) @ X3 ) ) ).

% power4_eq_xxxx
thf(fact_488_power4__eq__xxxx,axiom,
    ! [X3: rat] :
      ( ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X3 @ X3 ) @ X3 ) @ X3 ) ) ).

% power4_eq_xxxx
thf(fact_489_power4__eq__xxxx,axiom,
    ! [X3: nat] :
      ( ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X3 @ X3 ) @ X3 ) @ X3 ) ) ).

% power4_eq_xxxx
thf(fact_490_power4__eq__xxxx,axiom,
    ! [X3: int] :
      ( ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_int @ ( times_times_int @ ( times_times_int @ X3 @ X3 ) @ X3 ) @ X3 ) ) ).

% power4_eq_xxxx
thf(fact_491_power2__eq__square,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_complex @ A @ A ) ) ).

% power2_eq_square
thf(fact_492_power2__eq__square,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_real @ A @ A ) ) ).

% power2_eq_square
thf(fact_493_power2__eq__square,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_rat @ A @ A ) ) ).

% power2_eq_square
thf(fact_494_power2__eq__square,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_nat @ A @ A ) ) ).

% power2_eq_square
thf(fact_495_power2__eq__square,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_int @ A @ A ) ) ).

% power2_eq_square
thf(fact_496_power__even__eq,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_497_power__even__eq,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_498_power__even__eq,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_499_power__even__eq,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_500_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_less_add_iff2
thf(fact_501_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_less_add_iff1
thf(fact_502_is__pred__in__set__def,axiom,
    ( vEBT_is_pred_in_set
    = ( ^ [Xs: set_nat,X: nat,Y: nat] :
          ( ( member_nat @ Y @ Xs )
          & ( ord_less_nat @ Y @ X )
          & ! [Z2: nat] :
              ( ( member_nat @ Z2 @ Xs )
             => ( ( ord_less_nat @ Z2 @ X )
               => ( ord_less_eq_nat @ Z2 @ Y ) ) ) ) ) ) ).

% is_pred_in_set_def
thf(fact_503_power2__sum,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( power_power_complex @ ( plus_plus_complex @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).

% power2_sum
thf(fact_504_power2__sum,axiom,
    ! [X3: real,Y3: real] :
      ( ( power_power_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).

% power2_sum
thf(fact_505_power2__sum,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( power_power_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).

% power2_sum
thf(fact_506_power2__sum,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).

% power2_sum
thf(fact_507_power2__sum,axiom,
    ! [X3: int,Y3: int] :
      ( ( power_power_int @ ( plus_plus_int @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).

% power2_sum
thf(fact_508_power2__diff,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( power_power_complex @ ( minus_minus_complex @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).

% power2_diff
thf(fact_509_power2__diff,axiom,
    ! [X3: real,Y3: real] :
      ( ( power_power_real @ ( minus_minus_real @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).

% power2_diff
thf(fact_510_power2__diff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( power_power_rat @ ( minus_minus_rat @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).

% power2_diff
thf(fact_511_power2__diff,axiom,
    ! [X3: int,Y3: int] :
      ( ( power_power_int @ ( minus_minus_int @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).

% power2_diff
thf(fact_512_mul__def,axiom,
    ( vEBT_VEBT_mul
    = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).

% mul_def
thf(fact_513_mul__shift,axiom,
    ! [X3: nat,Y3: nat,Z: nat] :
      ( ( ( times_times_nat @ X3 @ Y3 )
        = Z )
      = ( ( vEBT_VEBT_mul @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) )
        = ( some_nat @ Z ) ) ) ).

% mul_shift
thf(fact_514_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
        & ( ( X3 = Mi )
          | ( X3 = Ma )
          | ( ( ord_less_nat @ X3 @ Ma )
            & ( ord_less_nat @ Mi @ X3 )
            & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% member_inv
thf(fact_515_thisvalid,axiom,
    vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ deg ).

% thisvalid
thf(fact_516_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
     => ( ( Mi != Ma )
       => ( ( ord_less_nat @ Mi @ Ma )
          & ? [M4: nat] :
              ( ( ( some_nat @ M4 )
                = ( vEBT_vebt_mint @ Summary ) )
              & ( ord_less_nat @ M4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% mintlistlength
thf(fact_517_sum__squares__bound,axiom,
    ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_518_sum__squares__bound,axiom,
    ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_519__C4_Ohyps_C_I3_J,axiom,
    ! [X3: nat,Px: nat] :
      ( ( ( vEBT_vebt_pred @ summary @ X3 )
        = ( some_nat @ Px ) )
      = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ summary ) @ X3 @ Px ) ) ).

% "4.hyps"(3)
thf(fact_520_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
     => ( ( ord_less_eq_nat @ Mi @ Ma )
        & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).

% mi_ma_2_deg
thf(fact_521_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_522_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_523_add__diff__cancel__right_H,axiom,
    ! [A: complex,B2: complex] :
      ( ( minus_minus_complex @ ( plus_plus_complex @ A @ B2 ) @ B2 )
      = A ) ).

% add_diff_cancel_right'
thf(fact_524_add__diff__cancel__right_H,axiom,
    ! [A: real,B2: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
      = A ) ).

% add_diff_cancel_right'
thf(fact_525_add__diff__cancel__right_H,axiom,
    ! [A: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ B2 )
      = A ) ).

% add_diff_cancel_right'
thf(fact_526_add__diff__cancel__right_H,axiom,
    ! [A: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
      = A ) ).

% add_diff_cancel_right'
thf(fact_527_add__diff__cancel__right_H,axiom,
    ! [A: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
      = A ) ).

% add_diff_cancel_right'
thf(fact_528_inthall,axiom,
    ! [Xs2: list_real,P: real > $o,N: nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ ( set_real2 @ Xs2 ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
       => ( P @ ( nth_real @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_529_inthall,axiom,
    ! [Xs2: list_complex,P: complex > $o,N: nat] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ ( set_complex2 @ Xs2 ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
       => ( P @ ( nth_complex @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_530_inthall,axiom,
    ! [Xs2: list_int,P: int > $o,N: nat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ ( set_int2 @ Xs2 ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
       => ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_531_inthall,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_532_inthall,axiom,
    ! [Xs2: list_o,P: $o > $o,N: nat] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ ( set_o2 @ Xs2 ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
       => ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_533_inthall,axiom,
    ! [Xs2: list_nat,P: nat > $o,N: nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
       => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_534_add__left__cancel,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ( plus_plus_real @ A @ B2 )
        = ( plus_plus_real @ A @ C ) )
      = ( B2 = C ) ) ).

% add_left_cancel
thf(fact_535_add__left__cancel,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B2 )
        = ( plus_plus_rat @ A @ C ) )
      = ( B2 = C ) ) ).

% add_left_cancel
thf(fact_536_add__left__cancel,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B2 )
        = ( plus_plus_nat @ A @ C ) )
      = ( B2 = C ) ) ).

% add_left_cancel
thf(fact_537_add__left__cancel,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ( plus_plus_int @ A @ B2 )
        = ( plus_plus_int @ A @ C ) )
      = ( B2 = C ) ) ).

% add_left_cancel
thf(fact_538_add__right__cancel,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B2 @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B2 = C ) ) ).

% add_right_cancel
thf(fact_539_add__right__cancel,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B2 @ A )
        = ( plus_plus_rat @ C @ A ) )
      = ( B2 = C ) ) ).

% add_right_cancel
thf(fact_540_add__right__cancel,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B2 @ A )
        = ( plus_plus_nat @ C @ A ) )
      = ( B2 = C ) ) ).

% add_right_cancel
thf(fact_541_add__right__cancel,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B2 @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B2 = C ) ) ).

% add_right_cancel
thf(fact_542__C4_Ohyps_C_I1_J,axiom,
    ! [X6: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ treeList ) )
     => ( ( vEBT_invar_vebt @ X6 @ na )
        & ! [Xa: nat,Xb: nat] :
            ( ( ( vEBT_vebt_pred @ X6 @ Xa )
              = ( some_nat @ Xb ) )
            = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ X6 ) @ Xa @ Xb ) ) ) ) ).

% "4.hyps"(1)
thf(fact_543_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
     => ( ( Mi = Ma )
       => ( ! [X6: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
             => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_12 ) )
          & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).

% mi_eq_ma_no_ch
thf(fact_544_add__le__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
      = ( ord_less_eq_real @ A @ B2 ) ) ).

% add_le_cancel_left
thf(fact_545_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
      = ( ord_less_eq_rat @ A @ B2 ) ) ).

% add_le_cancel_left
thf(fact_546_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
      = ( ord_less_eq_nat @ A @ B2 ) ) ).

% add_le_cancel_left
thf(fact_547_add__le__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
      = ( ord_less_eq_int @ A @ B2 ) ) ).

% add_le_cancel_left
thf(fact_548_add__le__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
      = ( ord_less_eq_real @ A @ B2 ) ) ).

% add_le_cancel_right
thf(fact_549_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
      = ( ord_less_eq_rat @ A @ B2 ) ) ).

% add_le_cancel_right
thf(fact_550_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
      = ( ord_less_eq_nat @ A @ B2 ) ) ).

% add_le_cancel_right
thf(fact_551_add__le__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
      = ( ord_less_eq_int @ A @ B2 ) ) ).

% add_le_cancel_right
thf(fact_552_add__less__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
      = ( ord_less_real @ A @ B2 ) ) ).

% add_less_cancel_left
thf(fact_553_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
      = ( ord_less_rat @ A @ B2 ) ) ).

% add_less_cancel_left
thf(fact_554_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
      = ( ord_less_nat @ A @ B2 ) ) ).

% add_less_cancel_left
thf(fact_555_add__less__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
      = ( ord_less_int @ A @ B2 ) ) ).

% add_less_cancel_left
thf(fact_556_add__less__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
      = ( ord_less_real @ A @ B2 ) ) ).

% add_less_cancel_right
thf(fact_557_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
      = ( ord_less_rat @ A @ B2 ) ) ).

% add_less_cancel_right
thf(fact_558_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
      = ( ord_less_nat @ A @ B2 ) ) ).

% add_less_cancel_right
thf(fact_559_add__less__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
      = ( ord_less_int @ A @ B2 ) ) ).

% add_less_cancel_right
thf(fact_560_add__diff__cancel,axiom,
    ! [A: complex,B2: complex] :
      ( ( minus_minus_complex @ ( plus_plus_complex @ A @ B2 ) @ B2 )
      = A ) ).

% add_diff_cancel
thf(fact_561_add__diff__cancel,axiom,
    ! [A: real,B2: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
      = A ) ).

% add_diff_cancel
thf(fact_562_add__diff__cancel,axiom,
    ! [A: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ B2 )
      = A ) ).

% add_diff_cancel
thf(fact_563_add__diff__cancel,axiom,
    ! [A: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
      = A ) ).

% add_diff_cancel
thf(fact_564_diff__add__cancel,axiom,
    ! [A: complex,B2: complex] :
      ( ( plus_plus_complex @ ( minus_minus_complex @ A @ B2 ) @ B2 )
      = A ) ).

% diff_add_cancel
thf(fact_565_diff__add__cancel,axiom,
    ! [A: real,B2: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ B2 )
      = A ) ).

% diff_add_cancel
thf(fact_566_diff__add__cancel,axiom,
    ! [A: rat,B2: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B2 ) @ B2 )
      = A ) ).

% diff_add_cancel
thf(fact_567_diff__add__cancel,axiom,
    ! [A: int,B2: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ B2 )
      = A ) ).

% diff_add_cancel
thf(fact_568_add__diff__cancel__left,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( minus_minus_complex @ ( plus_plus_complex @ C @ A ) @ ( plus_plus_complex @ C @ B2 ) )
      = ( minus_minus_complex @ A @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_569_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
      = ( minus_minus_real @ A @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_570_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
      = ( minus_minus_rat @ A @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_571_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
      = ( minus_minus_nat @ A @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_572_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
      = ( minus_minus_int @ A @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_573_add__diff__cancel__left_H,axiom,
    ! [A: complex,B2: complex] :
      ( ( minus_minus_complex @ ( plus_plus_complex @ A @ B2 ) @ A )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_574_add__diff__cancel__left_H,axiom,
    ! [A: real,B2: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ A )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_575_add__diff__cancel__left_H,axiom,
    ! [A: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ A )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_576_add__diff__cancel__left_H,axiom,
    ! [A: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ A )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_577_add__diff__cancel__left_H,axiom,
    ! [A: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ A )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_578_add__diff__cancel__right,axiom,
    ! [A: complex,C: complex,B2: complex] :
      ( ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B2 @ C ) )
      = ( minus_minus_complex @ A @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_579_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
      = ( minus_minus_real @ A @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_580_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
      = ( minus_minus_rat @ A @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_581_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
      = ( minus_minus_nat @ A @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_582_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
      = ( minus_minus_int @ A @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_583_insert__simp__mima,axiom,
    ! [X3: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X3 = Mi )
        | ( X3 = Ma ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).

% insert_simp_mima
thf(fact_584_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_585_semiring__norm_I11_J,axiom,
    ! [M: num] :
      ( ( times_times_num @ M @ one )
      = M ) ).

% semiring_norm(11)
thf(fact_586_semiring__norm_I12_J,axiom,
    ! [N: num] :
      ( ( times_times_num @ one @ N )
      = N ) ).

% semiring_norm(12)
thf(fact_587_pred__max,axiom,
    ! [Deg: nat,Ma: nat,X3: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_nat @ Ma @ X3 )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
          = ( some_nat @ Ma ) ) ) ) ).

% pred_max
thf(fact_588_num__double,axiom,
    ! [N: num] :
      ( ( times_times_num @ ( bit0 @ one ) @ N )
      = ( bit0 @ N ) ) ).

% num_double
thf(fact_589_power__mult__numeral,axiom,
    ! [A: nat,M: num,N: num] :
      ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_590_power__mult__numeral,axiom,
    ! [A: real,M: num,N: num] :
      ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_591_power__mult__numeral,axiom,
    ! [A: int,M: num,N: num] :
      ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_592_power__mult__numeral,axiom,
    ! [A: complex,M: num,N: num] :
      ( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_593_pred__list__to__short,axiom,
    ! [Deg: nat,X3: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X3 @ Ma )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
            = none_nat ) ) ) ) ).

% pred_list_to_short
thf(fact_594_enat__less__induct,axiom,
    ! [P: extended_enat > $o,N: extended_enat] :
      ( ! [N3: extended_enat] :
          ( ! [M2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ M2 @ N3 )
             => ( P @ M2 ) )
         => ( P @ N3 ) )
     => ( P @ N ) ) ).

% enat_less_induct
thf(fact_595_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y3: extended_enat,X3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Z @ Y3 )
     => ( ( plus_p3455044024723400733d_enat @ X3 @ ( minus_3235023915231533773d_enat @ Y3 @ Z ) )
        = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X3 @ Y3 ) @ Z ) ) ) ).

% add_diff_assoc_enat
thf(fact_596_L2__set__mult__ineq__lemma,axiom,
    ! [A: 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 @ A @ C ) ) @ ( times_times_real @ B2 @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% L2_set_mult_ineq_lemma
thf(fact_597_four__x__squared,axiom,
    ! [X3: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% four_x_squared
thf(fact_598_div__mult2__numeral__eq,axiom,
    ! [A: nat,K: num,L: num] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
      = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_599_div__mult2__numeral__eq,axiom,
    ! [A: int,K: num,L: num] :
      ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L ) )
      = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_600_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va ) @ Vb )
      = none_nat ) ).

% vebt_pred.simps(4)
thf(fact_601_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B2 ) @ C )
      = ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_602_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B2 ) @ C )
      = ( times_times_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_603_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B2 ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_604_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B2 ) @ C )
      = ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_605_mult_Oassoc,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B2 ) @ C )
      = ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).

% mult.assoc
thf(fact_606_mult_Oassoc,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B2 ) @ C )
      = ( times_times_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).

% mult.assoc
thf(fact_607_mult_Oassoc,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B2 ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).

% mult.assoc
thf(fact_608_mult_Oassoc,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B2 ) @ C )
      = ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).

% mult.assoc
thf(fact_609_mult_Ocommute,axiom,
    ( times_times_real
    = ( ^ [A5: real,B4: real] : ( times_times_real @ B4 @ A5 ) ) ) ).

% mult.commute
thf(fact_610_mult_Ocommute,axiom,
    ( times_times_rat
    = ( ^ [A5: rat,B4: rat] : ( times_times_rat @ B4 @ A5 ) ) ) ).

% mult.commute
thf(fact_611_mult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A5: nat,B4: nat] : ( times_times_nat @ B4 @ A5 ) ) ) ).

% mult.commute
thf(fact_612_mult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A5: int,B4: int] : ( times_times_int @ B4 @ A5 ) ) ) ).

% mult.commute
thf(fact_613_mult_Oleft__commute,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( times_times_real @ B2 @ ( times_times_real @ A @ C ) )
      = ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).

% mult.left_commute
thf(fact_614_mult_Oleft__commute,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( times_times_rat @ B2 @ ( times_times_rat @ A @ C ) )
      = ( times_times_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).

% mult.left_commute
thf(fact_615_mult_Oleft__commute,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( times_times_nat @ B2 @ ( times_times_nat @ A @ C ) )
      = ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).

% mult.left_commute
thf(fact_616_mult_Oleft__commute,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( times_times_int @ B2 @ ( times_times_int @ A @ C ) )
      = ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).

% mult.left_commute
thf(fact_617_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B2 ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_618_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_619_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_620_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_621_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_622_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_623_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_624_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_625_group__cancel_Oadd1,axiom,
    ! [A4: real,K: real,A: real,B2: real] :
      ( ( A4
        = ( plus_plus_real @ K @ A ) )
     => ( ( plus_plus_real @ A4 @ B2 )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_626_group__cancel_Oadd1,axiom,
    ! [A4: rat,K: rat,A: rat,B2: rat] :
      ( ( A4
        = ( plus_plus_rat @ K @ A ) )
     => ( ( plus_plus_rat @ A4 @ B2 )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_627_group__cancel_Oadd1,axiom,
    ! [A4: nat,K: nat,A: nat,B2: nat] :
      ( ( A4
        = ( plus_plus_nat @ K @ A ) )
     => ( ( plus_plus_nat @ A4 @ B2 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_628_group__cancel_Oadd1,axiom,
    ! [A4: int,K: int,A: int,B2: int] :
      ( ( A4
        = ( plus_plus_int @ K @ A ) )
     => ( ( plus_plus_int @ A4 @ B2 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_629_group__cancel_Oadd2,axiom,
    ! [B5: real,K: real,B2: real,A: real] :
      ( ( B5
        = ( plus_plus_real @ K @ B2 ) )
     => ( ( plus_plus_real @ A @ B5 )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_630_group__cancel_Oadd2,axiom,
    ! [B5: rat,K: rat,B2: rat,A: rat] :
      ( ( B5
        = ( plus_plus_rat @ K @ B2 ) )
     => ( ( plus_plus_rat @ A @ B5 )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_631_group__cancel_Oadd2,axiom,
    ! [B5: nat,K: nat,B2: nat,A: nat] :
      ( ( B5
        = ( plus_plus_nat @ K @ B2 ) )
     => ( ( plus_plus_nat @ A @ B5 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_632_group__cancel_Oadd2,axiom,
    ! [B5: int,K: int,B2: int,A: int] :
      ( ( B5
        = ( plus_plus_int @ K @ B2 ) )
     => ( ( plus_plus_int @ A @ B5 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_633_add_Oassoc,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B2 ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).

% add.assoc
thf(fact_634_add_Oassoc,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).

% add.assoc
thf(fact_635_add_Oassoc,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add.assoc
thf(fact_636_add_Oassoc,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).

% add.assoc
thf(fact_637_add_Oleft__cancel,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ( plus_plus_real @ A @ B2 )
        = ( plus_plus_real @ A @ C ) )
      = ( B2 = C ) ) ).

% add.left_cancel
thf(fact_638_add_Oleft__cancel,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B2 )
        = ( plus_plus_rat @ A @ C ) )
      = ( B2 = C ) ) ).

% add.left_cancel
thf(fact_639_add_Oleft__cancel,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ( plus_plus_int @ A @ B2 )
        = ( plus_plus_int @ A @ C ) )
      = ( B2 = C ) ) ).

% add.left_cancel
thf(fact_640_add_Oright__cancel,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B2 @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B2 = C ) ) ).

% add.right_cancel
thf(fact_641_add_Oright__cancel,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B2 @ A )
        = ( plus_plus_rat @ C @ A ) )
      = ( B2 = C ) ) ).

% add.right_cancel
thf(fact_642_add_Oright__cancel,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B2 @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B2 = C ) ) ).

% add.right_cancel
thf(fact_643_add_Ocommute,axiom,
    ( plus_plus_real
    = ( ^ [A5: real,B4: real] : ( plus_plus_real @ B4 @ A5 ) ) ) ).

% add.commute
thf(fact_644_add_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ B4 @ A5 ) ) ) ).

% add.commute
thf(fact_645_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).

% add.commute
thf(fact_646_add_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A5: int,B4: int] : ( plus_plus_int @ B4 @ A5 ) ) ) ).

% add.commute
thf(fact_647_add_Oleft__commute,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( plus_plus_real @ B2 @ ( plus_plus_real @ A @ C ) )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).

% add.left_commute
thf(fact_648_add_Oleft__commute,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( plus_plus_rat @ B2 @ ( plus_plus_rat @ A @ C ) )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).

% add.left_commute
thf(fact_649_add_Oleft__commute,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A @ C ) )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add.left_commute
thf(fact_650_add_Oleft__commute,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( plus_plus_int @ B2 @ ( plus_plus_int @ A @ C ) )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).

% add.left_commute
thf(fact_651_add__left__imp__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ( plus_plus_real @ A @ B2 )
        = ( plus_plus_real @ A @ C ) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
thf(fact_652_add__left__imp__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B2 )
        = ( plus_plus_rat @ A @ C ) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
thf(fact_653_add__left__imp__eq,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B2 )
        = ( plus_plus_nat @ A @ C ) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
thf(fact_654_add__left__imp__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ( plus_plus_int @ A @ B2 )
        = ( plus_plus_int @ A @ C ) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
thf(fact_655_add__right__imp__eq,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B2 @ A )
        = ( plus_plus_real @ C @ A ) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
thf(fact_656_add__right__imp__eq,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B2 @ A )
        = ( plus_plus_rat @ C @ A ) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
thf(fact_657_add__right__imp__eq,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B2 @ A )
        = ( plus_plus_nat @ C @ A ) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
thf(fact_658_add__right__imp__eq,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B2 @ A )
        = ( plus_plus_int @ C @ A ) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
thf(fact_659_diff__eq__diff__eq,axiom,
    ! [A: complex,B2: complex,C: complex,D: complex] :
      ( ( ( minus_minus_complex @ A @ B2 )
        = ( minus_minus_complex @ C @ D ) )
     => ( ( A = B2 )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_660_diff__eq__diff__eq,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B2 )
        = ( minus_minus_real @ C @ D ) )
     => ( ( A = B2 )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_661_diff__eq__diff__eq,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B2 )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( A = B2 )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_662_diff__eq__diff__eq,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B2 )
        = ( minus_minus_int @ C @ D ) )
     => ( ( A = B2 )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_663_diff__right__commute,axiom,
    ! [A: complex,C: complex,B2: complex] :
      ( ( minus_minus_complex @ ( minus_minus_complex @ A @ C ) @ B2 )
      = ( minus_minus_complex @ ( minus_minus_complex @ A @ B2 ) @ C ) ) ).

% diff_right_commute
thf(fact_664_diff__right__commute,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B2 )
      = ( minus_minus_real @ ( minus_minus_real @ A @ B2 ) @ C ) ) ).

% diff_right_commute
thf(fact_665_diff__right__commute,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B2 )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ B2 ) @ C ) ) ).

% diff_right_commute
thf(fact_666_diff__right__commute,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B2 )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C ) ) ).

% diff_right_commute
thf(fact_667_diff__right__commute,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B2 )
      = ( minus_minus_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).

% diff_right_commute
thf(fact_668_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_669_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_670_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_671_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_672_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_673_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_674_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_675_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_676_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_677_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_678_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_679_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_680_add__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).

% add_mono
thf(fact_681_add__mono,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).

% add_mono
thf(fact_682_add__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).

% add_mono
thf(fact_683_add__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).

% add_mono
thf(fact_684_add__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) ) ) ).

% add_left_mono
thf(fact_685_add__left__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) ) ) ).

% add_left_mono
thf(fact_686_add__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).

% add_left_mono
thf(fact_687_add__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) ) ) ).

% add_left_mono
thf(fact_688_less__eqE,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ~ ! [C2: nat] :
            ( B2
           != ( plus_plus_nat @ A @ C2 ) ) ) ).

% less_eqE
thf(fact_689_add__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) ) ) ).

% add_right_mono
thf(fact_690_add__right__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) ) ) ).

% add_right_mono
thf(fact_691_add__right__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add_right_mono
thf(fact_692_add__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).

% add_right_mono
thf(fact_693_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] :
        ? [C3: nat] :
          ( B4
          = ( plus_plus_nat @ A5 @ C3 ) ) ) ) ).

% le_iff_add
thf(fact_694_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
     => ( ord_less_eq_real @ A @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_695_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
     => ( ord_less_eq_rat @ A @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_696_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
     => ( ord_less_eq_nat @ A @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_697_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
     => ( ord_less_eq_int @ A @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_698_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
     => ( ord_less_eq_real @ A @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_699_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
     => ( ord_less_eq_rat @ A @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_700_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
     => ( ord_less_eq_nat @ A @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_701_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
     => ( ord_less_eq_int @ A @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_702_diff__mono,axiom,
    ! [A: real,B2: real,D: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ D @ C )
       => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ D ) ) ) ) ).

% diff_mono
thf(fact_703_diff__mono,axiom,
    ! [A: rat,B2: rat,D: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ D @ C )
       => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B2 @ D ) ) ) ) ).

% diff_mono
thf(fact_704_diff__mono,axiom,
    ! [A: int,B2: int,D: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ D @ C )
       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).

% diff_mono
thf(fact_705_diff__left__mono,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B2 ) ) ) ).

% diff_left_mono
thf(fact_706_diff__left__mono,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B2 @ A )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B2 ) ) ) ).

% diff_left_mono
thf(fact_707_diff__left__mono,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B2 ) ) ) ).

% diff_left_mono
thf(fact_708_diff__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ C ) ) ) ).

% diff_right_mono
thf(fact_709_diff__right__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B2 @ C ) ) ) ).

% diff_right_mono
thf(fact_710_diff__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).

% diff_right_mono
thf(fact_711_diff__eq__diff__less__eq,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B2 )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_eq_real @ A @ B2 )
        = ( ord_less_eq_real @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_712_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B2 )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_eq_rat @ A @ B2 )
        = ( ord_less_eq_rat @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_713_diff__eq__diff__less__eq,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B2 )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_eq_int @ A @ B2 )
        = ( ord_less_eq_int @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_714_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_715_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_716_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_717_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_718_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_719_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_720_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_721_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_722_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_723_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_724_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_725_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_726_add__strict__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).

% add_strict_mono
thf(fact_727_add__strict__mono,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).

% add_strict_mono
thf(fact_728_add__strict__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).

% add_strict_mono
thf(fact_729_add__strict__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).

% add_strict_mono
thf(fact_730_add__strict__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_731_add__strict__left__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_732_add__strict__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_733_add__strict__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_734_add__strict__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_735_add__strict__right__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_736_add__strict__right__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_737_add__strict__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_738_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
     => ( ord_less_real @ A @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_739_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
     => ( ord_less_rat @ A @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_740_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
     => ( ord_less_nat @ A @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_741_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
     => ( ord_less_int @ A @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_742_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
     => ( ord_less_real @ A @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_743_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
     => ( ord_less_rat @ A @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_744_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
     => ( ord_less_nat @ A @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_745_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
     => ( ord_less_int @ A @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_746_diff__strict__mono,axiom,
    ! [A: real,B2: real,D: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ D @ C )
       => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_747_diff__strict__mono,axiom,
    ! [A: rat,B2: rat,D: rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_rat @ D @ C )
       => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B2 @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_748_diff__strict__mono,axiom,
    ! [A: int,B2: int,D: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_int @ D @ C )
       => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_749_diff__eq__diff__less,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B2 )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_real @ A @ B2 )
        = ( ord_less_real @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_750_diff__eq__diff__less,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B2 )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_rat @ A @ B2 )
        = ( ord_less_rat @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_751_diff__eq__diff__less,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B2 )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_int @ A @ B2 )
        = ( ord_less_int @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_752_diff__strict__left__mono,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B2 ) ) ) ).

% diff_strict_left_mono
thf(fact_753_diff__strict__left__mono,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B2 @ A )
     => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B2 ) ) ) ).

% diff_strict_left_mono
thf(fact_754_diff__strict__left__mono,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_int @ B2 @ A )
     => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B2 ) ) ) ).

% diff_strict_left_mono
thf(fact_755_diff__strict__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_756_diff__strict__right__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B2 @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_757_diff__strict__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_758_group__cancel_Osub1,axiom,
    ! [A4: complex,K: complex,A: complex,B2: complex] :
      ( ( A4
        = ( plus_plus_complex @ K @ A ) )
     => ( ( minus_minus_complex @ A4 @ B2 )
        = ( plus_plus_complex @ K @ ( minus_minus_complex @ A @ B2 ) ) ) ) ).

% group_cancel.sub1
thf(fact_759_group__cancel_Osub1,axiom,
    ! [A4: real,K: real,A: real,B2: real] :
      ( ( A4
        = ( plus_plus_real @ K @ A ) )
     => ( ( minus_minus_real @ A4 @ B2 )
        = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B2 ) ) ) ) ).

% group_cancel.sub1
thf(fact_760_group__cancel_Osub1,axiom,
    ! [A4: rat,K: rat,A: rat,B2: rat] :
      ( ( A4
        = ( plus_plus_rat @ K @ A ) )
     => ( ( minus_minus_rat @ A4 @ B2 )
        = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B2 ) ) ) ) ).

% group_cancel.sub1
thf(fact_761_group__cancel_Osub1,axiom,
    ! [A4: int,K: int,A: int,B2: int] :
      ( ( A4
        = ( plus_plus_int @ K @ A ) )
     => ( ( minus_minus_int @ A4 @ B2 )
        = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B2 ) ) ) ) ).

% group_cancel.sub1
thf(fact_762_diff__eq__eq,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( ( minus_minus_complex @ A @ B2 )
        = C )
      = ( A
        = ( plus_plus_complex @ C @ B2 ) ) ) ).

% diff_eq_eq
thf(fact_763_diff__eq__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ( minus_minus_real @ A @ B2 )
        = C )
      = ( A
        = ( plus_plus_real @ C @ B2 ) ) ) ).

% diff_eq_eq
thf(fact_764_diff__eq__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ( minus_minus_rat @ A @ B2 )
        = C )
      = ( A
        = ( plus_plus_rat @ C @ B2 ) ) ) ).

% diff_eq_eq
thf(fact_765_diff__eq__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ( minus_minus_int @ A @ B2 )
        = C )
      = ( A
        = ( plus_plus_int @ C @ B2 ) ) ) ).

% diff_eq_eq
thf(fact_766_eq__diff__eq,axiom,
    ! [A: complex,C: complex,B2: complex] :
      ( ( A
        = ( minus_minus_complex @ C @ B2 ) )
      = ( ( plus_plus_complex @ A @ B2 )
        = C ) ) ).

% eq_diff_eq
thf(fact_767_eq__diff__eq,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( A
        = ( minus_minus_real @ C @ B2 ) )
      = ( ( plus_plus_real @ A @ B2 )
        = C ) ) ).

% eq_diff_eq
thf(fact_768_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( A
        = ( minus_minus_rat @ C @ B2 ) )
      = ( ( plus_plus_rat @ A @ B2 )
        = C ) ) ).

% eq_diff_eq
thf(fact_769_eq__diff__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( A
        = ( minus_minus_int @ C @ B2 ) )
      = ( ( plus_plus_int @ A @ B2 )
        = C ) ) ).

% eq_diff_eq
thf(fact_770_add__diff__eq,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( plus_plus_complex @ A @ ( minus_minus_complex @ B2 @ C ) )
      = ( minus_minus_complex @ ( plus_plus_complex @ A @ B2 ) @ C ) ) ).

% add_diff_eq
thf(fact_771_add__diff__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( plus_plus_real @ A @ ( minus_minus_real @ B2 @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ C ) ) ).

% add_diff_eq
thf(fact_772_add__diff__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B2 @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ C ) ) ).

% add_diff_eq
thf(fact_773_add__diff__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( plus_plus_int @ A @ ( minus_minus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).

% add_diff_eq
thf(fact_774_diff__diff__eq2,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( minus_minus_complex @ A @ ( minus_minus_complex @ B2 @ C ) )
      = ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ B2 ) ) ).

% diff_diff_eq2
thf(fact_775_diff__diff__eq2,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( minus_minus_real @ A @ ( minus_minus_real @ B2 @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ).

% diff_diff_eq2
thf(fact_776_diff__diff__eq2,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B2 @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B2 ) ) ).

% diff_diff_eq2
thf(fact_777_diff__diff__eq2,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( minus_minus_int @ A @ ( minus_minus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ).

% diff_diff_eq2
thf(fact_778_diff__add__eq,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( plus_plus_complex @ ( minus_minus_complex @ A @ B2 ) @ C )
      = ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ B2 ) ) ).

% diff_add_eq
thf(fact_779_diff__add__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ C )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ).

% diff_add_eq
thf(fact_780_diff__add__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B2 ) ) ).

% diff_add_eq
thf(fact_781_diff__add__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ C )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ).

% diff_add_eq
thf(fact_782_diff__add__eq__diff__diff__swap,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( minus_minus_complex @ A @ ( plus_plus_complex @ B2 @ C ) )
      = ( minus_minus_complex @ ( minus_minus_complex @ A @ C ) @ B2 ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_783_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( minus_minus_real @ A @ ( plus_plus_real @ B2 @ C ) )
      = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B2 ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_784_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B2 ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_785_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( minus_minus_int @ A @ ( plus_plus_int @ B2 @ C ) )
      = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B2 ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_786_add__implies__diff,axiom,
    ! [C: complex,B2: complex,A: complex] :
      ( ( ( plus_plus_complex @ C @ B2 )
        = A )
     => ( C
        = ( minus_minus_complex @ A @ B2 ) ) ) ).

% add_implies_diff
thf(fact_787_add__implies__diff,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ( plus_plus_real @ C @ B2 )
        = A )
     => ( C
        = ( minus_minus_real @ A @ B2 ) ) ) ).

% add_implies_diff
thf(fact_788_add__implies__diff,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( ( plus_plus_rat @ C @ B2 )
        = A )
     => ( C
        = ( minus_minus_rat @ A @ B2 ) ) ) ).

% add_implies_diff
thf(fact_789_add__implies__diff,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( ( plus_plus_nat @ C @ B2 )
        = A )
     => ( C
        = ( minus_minus_nat @ A @ B2 ) ) ) ).

% add_implies_diff
thf(fact_790_add__implies__diff,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( ( plus_plus_int @ C @ B2 )
        = A )
     => ( C
        = ( minus_minus_int @ A @ B2 ) ) ) ).

% add_implies_diff
thf(fact_791_diff__diff__eq,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( minus_minus_complex @ ( minus_minus_complex @ A @ B2 ) @ C )
      = ( minus_minus_complex @ A @ ( plus_plus_complex @ B2 @ C ) ) ) ).

% diff_diff_eq
thf(fact_792_diff__diff__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ B2 ) @ C )
      = ( minus_minus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).

% diff_diff_eq
thf(fact_793_diff__diff__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
      = ( minus_minus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).

% diff_diff_eq
thf(fact_794_diff__diff__eq,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% diff_diff_eq
thf(fact_795_diff__diff__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ B2 ) @ C )
      = ( minus_minus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).

% diff_diff_eq
thf(fact_796_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_797_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_798_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_799_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_800_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_801_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_802_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_803_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_804_add__le__less__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_805_add__le__less__mono,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_806_add__le__less__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_807_add__le__less__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_808_add__less__le__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_809_add__less__le__mono,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_810_add__less__le__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_811_add__less__le__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_812_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ A @ B2 )
       => ( ( ( minus_minus_nat @ B2 @ A )
            = C )
          = ( B2
            = ( plus_plus_nat @ C @ A ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_813_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B2 @ A ) )
        = B2 ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_814_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_815_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A )
        = ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ C ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_816_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ C )
        = ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_817_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A )
        = ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_818_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_819_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_820_le__add__diff,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A ) ) ) ).

% le_add_diff
thf(fact_821_diff__add,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ A )
        = B2 ) ) ).

% diff_add
thf(fact_822_le__diff__eq,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B2 ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ C ) ) ).

% le_diff_eq
thf(fact_823_le__diff__eq,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B2 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B2 ) @ C ) ) ).

% le_diff_eq
thf(fact_824_le__diff__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B2 ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).

% le_diff_eq
thf(fact_825_diff__le__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B2 ) @ C )
      = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B2 ) ) ) ).

% diff_le_eq
thf(fact_826_diff__le__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
      = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B2 ) ) ) ).

% diff_le_eq
thf(fact_827_diff__le__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B2 ) @ C )
      = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B2 ) ) ) ).

% diff_le_eq
thf(fact_828_less__diff__eq,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B2 ) )
      = ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ C ) ) ).

% less_diff_eq
thf(fact_829_less__diff__eq,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B2 ) )
      = ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ C ) ) ).

% less_diff_eq
thf(fact_830_less__diff__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B2 ) )
      = ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).

% less_diff_eq
thf(fact_831_diff__less__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ ( minus_minus_real @ A @ B2 ) @ C )
      = ( ord_less_real @ A @ ( plus_plus_real @ C @ B2 ) ) ) ).

% diff_less_eq
thf(fact_832_diff__less__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
      = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B2 ) ) ) ).

% diff_less_eq
thf(fact_833_diff__less__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ ( minus_minus_int @ A @ B2 ) @ C )
      = ( ord_less_int @ A @ ( plus_plus_int @ C @ B2 ) ) ) ).

% diff_less_eq
thf(fact_834__C2_C,axiom,
    ( ( ( ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = none_nat )
     => ( ( ( ord_less_nat @ mi @ xa )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
            = ( some_nat @ mi ) ) )
        & ( ~ ( ord_less_nat @ mi @ xa )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
            = none_nat ) ) ) )
    & ( ( ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
       != none_nat )
     => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
        = ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% "2"
thf(fact_835_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X5 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X7 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
                                    = I3 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
                                & ! [X5: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X5 @ N )
                                        = I3 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
                                   => ( ( ord_less_nat @ Mi @ X5 )
                                      & ( ord_less_eq_nat @ X5 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
thf(fact_836_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X3: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 ) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
thf(fact_837_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ Deg )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          | ( X3 = Mi )
          | ( X3 = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
thf(fact_838_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X5 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
               => ( ! [X5: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
thf(fact_839_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
     => ( ( Mi != Ma )
       => ( ( the_nat @ ( vEBT_vebt_maxt @ Summary ) )
          = ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% summaxma
thf(fact_840_set__n__deg__not__0,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,M: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X5 @ N ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
       => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).

% set_n_deg_not_0
thf(fact_841_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( some_nat @ Ma ) ) ).

% vebt_maxt.simps(3)
thf(fact_842_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( some_nat @ Mi ) ) ).

% vebt_mint.simps(3)
thf(fact_843_times__divide__eq__right,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B2 @ C ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B2 ) @ C ) ) ).

% times_divide_eq_right
thf(fact_844_times__divide__eq__right,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( times_times_real @ A @ ( divide_divide_real @ B2 @ C ) )
      = ( divide_divide_real @ ( times_times_real @ A @ B2 ) @ C ) ) ).

% times_divide_eq_right
thf(fact_845_times__divide__eq__right,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( times_times_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
      = ( divide_divide_rat @ ( times_times_rat @ A @ B2 ) @ C ) ) ).

% times_divide_eq_right
thf(fact_846_divide__divide__eq__right,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B2 @ C ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B2 ) ) ).

% divide_divide_eq_right
thf(fact_847_divide__divide__eq__right,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( divide_divide_real @ A @ ( divide_divide_real @ B2 @ C ) )
      = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B2 ) ) ).

% divide_divide_eq_right
thf(fact_848_divide__divide__eq__right,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
      = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ).

% divide_divide_eq_right
thf(fact_849__092_060open_0621_A_092_060le_062_An_092_060close_062,axiom,
    ord_less_eq_nat @ one_one_nat @ na ).

% \<open>1 \<le> n\<close>
thf(fact_850_real__divide__square__eq,axiom,
    ! [R: real,A: real] :
      ( ( divide_divide_real @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ R ) )
      = ( divide_divide_real @ A @ R ) ) ).

% real_divide_square_eq
thf(fact_851_VEBT_Oinject_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
      ( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
        = ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
      = ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
thf(fact_852_mult__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ one_one_complex @ A )
      = A ) ).

% mult_1
thf(fact_853_mult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% mult_1
thf(fact_854_mult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% mult_1
thf(fact_855_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_856_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_857_mult_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.right_neutral
thf(fact_858_mult_Oright__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.right_neutral
thf(fact_859_mult_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.right_neutral
thf(fact_860_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_861_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_862_times__divide__eq__left,axiom,
    ! [B2: complex,C: complex,A: complex] :
      ( ( times_times_complex @ ( divide1717551699836669952omplex @ B2 @ C ) @ A )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ B2 @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_863_times__divide__eq__left,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( times_times_real @ ( divide_divide_real @ B2 @ C ) @ A )
      = ( divide_divide_real @ ( times_times_real @ B2 @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_864_times__divide__eq__left,axiom,
    ! [B2: rat,C: rat,A: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
      = ( divide_divide_rat @ ( times_times_rat @ B2 @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_865_divide__divide__eq__left,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B2 ) @ C )
      = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B2 @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_866_divide__divide__eq__left,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ A @ B2 ) @ C )
      = ( divide_divide_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_867_divide__divide__eq__left,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B2 ) @ C )
      = ( divide_divide_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_868_div__by__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ one_one_complex )
      = A ) ).

% div_by_1
thf(fact_869_div__by__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ one_one_real )
      = A ) ).

% div_by_1
thf(fact_870_div__by__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ one_one_rat )
      = A ) ).

% div_by_1
thf(fact_871_div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% div_by_1
thf(fact_872_div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% div_by_1
thf(fact_873_bits__div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% bits_div_by_1
thf(fact_874_bits__div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% bits_div_by_1
thf(fact_875_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ one_one_rat @ N )
      = one_one_rat ) ).

% power_one
thf(fact_876_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ one_one_nat @ N )
      = one_one_nat ) ).

% power_one
thf(fact_877_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_real @ one_one_real @ N )
      = one_one_real ) ).

% power_one
thf(fact_878_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_int @ one_one_int @ N )
      = one_one_int ) ).

% power_one
thf(fact_879_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ one_one_complex @ N )
      = one_one_complex ) ).

% power_one
thf(fact_880_power__one__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_881_power__one__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_882_power__one__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_883_power__one__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_884_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_885_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_886_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numera6690914467698888265omplex @ N )
        = one_one_complex )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_887_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_real @ N )
        = one_one_real )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_888_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_rat @ N )
        = one_one_rat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_889_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_nat @ N )
        = one_one_nat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_890_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_int @ N )
        = one_one_int )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_891_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_complex
        = ( numera6690914467698888265omplex @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_892_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_real
        = ( numeral_numeral_real @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_893_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_rat
        = ( numeral_numeral_rat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_894_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_nat
        = ( numeral_numeral_nat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_895_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_int
        = ( numeral_numeral_int @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_896_power__inject__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( power_power_real @ A @ M )
          = ( power_power_real @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_897_power__inject__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ( power_power_rat @ A @ M )
          = ( power_power_rat @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_898_power__inject__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ( power_power_nat @ A @ M )
          = ( power_power_nat @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_899_power__inject__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ( power_power_int @ A @ M )
          = ( power_power_int @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_900_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( ( some_nat @ ( the_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_901_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_902_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( ( some_num @ ( the_num @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_903_power__strict__increasing__iff,axiom,
    ! [B2: real,X3: nat,Y3: nat] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ ( power_power_real @ B2 @ X3 ) @ ( power_power_real @ B2 @ Y3 ) )
        = ( ord_less_nat @ X3 @ Y3 ) ) ) ).

% power_strict_increasing_iff
thf(fact_904_power__strict__increasing__iff,axiom,
    ! [B2: rat,X3: nat,Y3: nat] :
      ( ( ord_less_rat @ one_one_rat @ B2 )
     => ( ( ord_less_rat @ ( power_power_rat @ B2 @ X3 ) @ ( power_power_rat @ B2 @ Y3 ) )
        = ( ord_less_nat @ X3 @ Y3 ) ) ) ).

% power_strict_increasing_iff
thf(fact_905_power__strict__increasing__iff,axiom,
    ! [B2: nat,X3: nat,Y3: nat] :
      ( ( ord_less_nat @ one_one_nat @ B2 )
     => ( ( ord_less_nat @ ( power_power_nat @ B2 @ X3 ) @ ( power_power_nat @ B2 @ Y3 ) )
        = ( ord_less_nat @ X3 @ Y3 ) ) ) ).

% power_strict_increasing_iff
thf(fact_906_power__strict__increasing__iff,axiom,
    ! [B2: int,X3: nat,Y3: nat] :
      ( ( ord_less_int @ one_one_int @ B2 )
     => ( ( ord_less_int @ ( power_power_int @ B2 @ X3 ) @ ( power_power_int @ B2 @ Y3 ) )
        = ( ord_less_nat @ X3 @ Y3 ) ) ) ).

% power_strict_increasing_iff
thf(fact_907_one__add__one,axiom,
    ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_908_one__add__one,axiom,
    ( ( plus_plus_real @ one_one_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_909_one__add__one,axiom,
    ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_910_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_911_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_912_power__increasing__iff,axiom,
    ! [B2: real,X3: nat,Y3: nat] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_eq_real @ ( power_power_real @ B2 @ X3 ) @ ( power_power_real @ B2 @ Y3 ) )
        = ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).

% power_increasing_iff
thf(fact_913_power__increasing__iff,axiom,
    ! [B2: rat,X3: nat,Y3: nat] :
      ( ( ord_less_rat @ one_one_rat @ B2 )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ B2 @ X3 ) @ ( power_power_rat @ B2 @ Y3 ) )
        = ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).

% power_increasing_iff
thf(fact_914_power__increasing__iff,axiom,
    ! [B2: nat,X3: nat,Y3: nat] :
      ( ( ord_less_nat @ one_one_nat @ B2 )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ X3 ) @ ( power_power_nat @ B2 @ Y3 ) )
        = ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).

% power_increasing_iff
thf(fact_915_power__increasing__iff,axiom,
    ! [B2: int,X3: nat,Y3: nat] :
      ( ( ord_less_int @ one_one_int @ B2 )
     => ( ( ord_less_eq_int @ ( power_power_int @ B2 @ X3 ) @ ( power_power_int @ B2 @ Y3 ) )
        = ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).

% power_increasing_iff
thf(fact_916_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ one_one_complex )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_917_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_918_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_919_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_920_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_921_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_922_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_923_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_924_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_925_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_926_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_927_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_928_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_929_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_930_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_931_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_932_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_933_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_934_one__reorient,axiom,
    ! [X3: complex] :
      ( ( one_one_complex = X3 )
      = ( X3 = one_one_complex ) ) ).

% one_reorient
thf(fact_935_one__reorient,axiom,
    ! [X3: real] :
      ( ( one_one_real = X3 )
      = ( X3 = one_one_real ) ) ).

% one_reorient
thf(fact_936_one__reorient,axiom,
    ! [X3: rat] :
      ( ( one_one_rat = X3 )
      = ( X3 = one_one_rat ) ) ).

% one_reorient
thf(fact_937_one__reorient,axiom,
    ! [X3: nat] :
      ( ( one_one_nat = X3 )
      = ( X3 = one_one_nat ) ) ).

% one_reorient
thf(fact_938_one__reorient,axiom,
    ! [X3: int] :
      ( ( one_one_int = X3 )
      = ( X3 = one_one_int ) ) ).

% one_reorient
thf(fact_939_less__eq__real__def,axiom,
    ( ord_less_eq_real
    = ( ^ [X: real,Y: real] :
          ( ( ord_less_real @ X @ Y )
          | ( X = Y ) ) ) ) ).

% less_eq_real_def
thf(fact_940_complete__real,axiom,
    ! [S2: set_real] :
      ( ? [X6: real] : ( member_real @ X6 @ S2 )
     => ( ? [Z3: real] :
          ! [X5: real] :
            ( ( member_real @ X5 @ S2 )
           => ( ord_less_eq_real @ X5 @ Z3 ) )
       => ? [Y4: real] :
            ( ! [X6: real] :
                ( ( member_real @ X6 @ S2 )
               => ( ord_less_eq_real @ X6 @ Y4 ) )
            & ! [Z3: real] :
                ( ! [X5: real] :
                    ( ( member_real @ X5 @ S2 )
                   => ( ord_less_eq_real @ X5 @ Z3 ) )
               => ( ord_less_eq_real @ Y4 @ Z3 ) ) ) ) ) ).

% complete_real
thf(fact_941_le__numeral__extra_I4_J,axiom,
    ord_less_eq_real @ one_one_real @ one_one_real ).

% le_numeral_extra(4)
thf(fact_942_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_943_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_944_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_945_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ one_one_real ) ).

% less_numeral_extra(4)
thf(fact_946_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_947_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_948_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_949_mult_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.comm_neutral
thf(fact_950_mult_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.comm_neutral
thf(fact_951_mult_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.comm_neutral
thf(fact_952_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_953_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_954_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ one_one_complex @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_955_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_956_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_957_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_958_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_959_nat__mult__1,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ one_one_nat @ N )
      = N ) ).

% nat_mult_1
thf(fact_960_nat__mult__1__right,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ N @ one_one_nat )
      = N ) ).

% nat_mult_1_right
thf(fact_961_option_Osel,axiom,
    ! [X2: nat] :
      ( ( the_nat @ ( some_nat @ X2 ) )
      = X2 ) ).

% option.sel
thf(fact_962_option_Osel,axiom,
    ! [X2: product_prod_nat_nat] :
      ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X2 ) )
      = X2 ) ).

% option.sel
thf(fact_963_option_Osel,axiom,
    ! [X2: num] :
      ( ( the_num @ ( some_num @ X2 ) )
      = X2 ) ).

% option.sel
thf(fact_964_option_Oexpand,axiom,
    ! [Option: option_nat,Option2: option_nat] :
      ( ( ( Option = none_nat )
        = ( Option2 = none_nat ) )
     => ( ( ( Option != none_nat )
         => ( ( Option2 != none_nat )
           => ( ( the_nat @ Option )
              = ( the_nat @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_965_option_Oexpand,axiom,
    ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
      ( ( ( Option = none_P5556105721700978146at_nat )
        = ( Option2 = none_P5556105721700978146at_nat ) )
     => ( ( ( Option != none_P5556105721700978146at_nat )
         => ( ( Option2 != none_P5556105721700978146at_nat )
           => ( ( the_Pr8591224930841456533at_nat @ Option )
              = ( the_Pr8591224930841456533at_nat @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_966_option_Oexpand,axiom,
    ! [Option: option_num,Option2: option_num] :
      ( ( ( Option = none_num )
        = ( Option2 = none_num ) )
     => ( ( ( Option != none_num )
         => ( ( Option2 != none_num )
           => ( ( the_num @ Option )
              = ( the_num @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_967_less__half__sum,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).

% less_half_sum
thf(fact_968_less__half__sum,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B2 ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).

% less_half_sum
thf(fact_969_gt__half__sum,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B2 ) ) ).

% gt_half_sum
thf(fact_970_gt__half__sum,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B2 ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B2 ) ) ).

% gt_half_sum
thf(fact_971_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).

% one_le_numeral
thf(fact_972_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).

% one_le_numeral
thf(fact_973_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).

% one_le_numeral
thf(fact_974_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).

% one_le_numeral
thf(fact_975_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).

% not_numeral_less_one
thf(fact_976_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).

% not_numeral_less_one
thf(fact_977_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_978_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_979_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_980_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_981_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_982_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_983_less__add__one,axiom,
    ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).

% less_add_one
thf(fact_984_less__add__one,axiom,
    ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).

% less_add_one
thf(fact_985_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_986_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_987_add__mono1,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B2 @ one_one_real ) ) ) ).

% add_mono1
thf(fact_988_add__mono1,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B2 @ one_one_rat ) ) ) ).

% add_mono1
thf(fact_989_add__mono1,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B2 @ one_one_nat ) ) ) ).

% add_mono1
thf(fact_990_add__mono1,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B2 @ one_one_int ) ) ) ).

% add_mono1
thf(fact_991_one__plus__numeral__commute,axiom,
    ! [X3: num] :
      ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X3 ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X3 ) @ one_one_complex ) ) ).

% one_plus_numeral_commute
thf(fact_992_one__plus__numeral__commute,axiom,
    ! [X3: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X3 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ X3 ) @ one_one_real ) ) ).

% one_plus_numeral_commute
thf(fact_993_one__plus__numeral__commute,axiom,
    ! [X3: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X3 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X3 ) @ one_one_rat ) ) ).

% one_plus_numeral_commute
thf(fact_994_one__plus__numeral__commute,axiom,
    ! [X3: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X3 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X3 ) @ one_one_nat ) ) ).

% one_plus_numeral_commute
thf(fact_995_one__plus__numeral__commute,axiom,
    ! [X3: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X3 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X3 ) @ one_one_int ) ) ).

% one_plus_numeral_commute
thf(fact_996_numeral__One,axiom,
    ( ( numera6690914467698888265omplex @ one )
    = one_one_complex ) ).

% numeral_One
thf(fact_997_numeral__One,axiom,
    ( ( numeral_numeral_real @ one )
    = one_one_real ) ).

% numeral_One
thf(fact_998_numeral__One,axiom,
    ( ( numeral_numeral_rat @ one )
    = one_one_rat ) ).

% numeral_One
thf(fact_999_numeral__One,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numeral_One
thf(fact_1000_numeral__One,axiom,
    ( ( numeral_numeral_int @ one )
    = one_one_int ) ).

% numeral_One
thf(fact_1001_one__le__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A )
     => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).

% one_le_power
thf(fact_1002_one__le__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).

% one_le_power
thf(fact_1003_one__le__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).

% one_le_power
thf(fact_1004_one__le__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).

% one_le_power
thf(fact_1005_left__right__inverse__power,axiom,
    ! [X3: complex,Y3: complex,N: nat] :
      ( ( ( times_times_complex @ X3 @ Y3 )
        = one_one_complex )
     => ( ( times_times_complex @ ( power_power_complex @ X3 @ N ) @ ( power_power_complex @ Y3 @ N ) )
        = one_one_complex ) ) ).

% left_right_inverse_power
thf(fact_1006_left__right__inverse__power,axiom,
    ! [X3: real,Y3: real,N: nat] :
      ( ( ( times_times_real @ X3 @ Y3 )
        = one_one_real )
     => ( ( times_times_real @ ( power_power_real @ X3 @ N ) @ ( power_power_real @ Y3 @ N ) )
        = one_one_real ) ) ).

% left_right_inverse_power
thf(fact_1007_left__right__inverse__power,axiom,
    ! [X3: rat,Y3: rat,N: nat] :
      ( ( ( times_times_rat @ X3 @ Y3 )
        = one_one_rat )
     => ( ( times_times_rat @ ( power_power_rat @ X3 @ N ) @ ( power_power_rat @ Y3 @ N ) )
        = one_one_rat ) ) ).

% left_right_inverse_power
thf(fact_1008_left__right__inverse__power,axiom,
    ! [X3: nat,Y3: nat,N: nat] :
      ( ( ( times_times_nat @ X3 @ Y3 )
        = one_one_nat )
     => ( ( times_times_nat @ ( power_power_nat @ X3 @ N ) @ ( power_power_nat @ Y3 @ N ) )
        = one_one_nat ) ) ).

% left_right_inverse_power
thf(fact_1009_left__right__inverse__power,axiom,
    ! [X3: int,Y3: int,N: nat] :
      ( ( ( times_times_int @ X3 @ Y3 )
        = one_one_int )
     => ( ( times_times_int @ ( power_power_int @ X3 @ N ) @ ( power_power_int @ Y3 @ N ) )
        = one_one_int ) ) ).

% left_right_inverse_power
thf(fact_1010_power__one__over,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N )
      = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N ) ) ) ).

% power_one_over
thf(fact_1011_power__one__over,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
      = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).

% power_one_over
thf(fact_1012_power__one__over,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N )
      = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).

% power_one_over
thf(fact_1013_numerals_I1_J,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numerals(1)
thf(fact_1014_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option
        = ( some_nat @ ( the_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_1015_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option
        = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_1016_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option
        = ( some_num @ ( the_num @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_1017_linordered__field__no__ub,axiom,
    ! [X6: real] :
    ? [X_1: real] : ( ord_less_real @ X6 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_1018_linordered__field__no__ub,axiom,
    ! [X6: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X6 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_1019_linordered__field__no__lb,axiom,
    ! [X6: real] :
    ? [Y4: real] : ( ord_less_real @ Y4 @ X6 ) ).

% linordered_field_no_lb
thf(fact_1020_linordered__field__no__lb,axiom,
    ! [X6: rat] :
    ? [Y4: rat] : ( ord_less_rat @ Y4 @ X6 ) ).

% linordered_field_no_lb
thf(fact_1021_power__less__power__Suc,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_1022_power__less__power__Suc,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_1023_power__less__power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_1024_power__less__power__Suc,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_1025_power__gt1__lemma,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_1026_power__gt1__lemma,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_1027_power__gt1__lemma,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_1028_power__gt1__lemma,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_1029_square__diff__one__factored,axiom,
    ! [X3: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X3 @ X3 ) @ one_one_complex )
      = ( times_times_complex @ ( plus_plus_complex @ X3 @ one_one_complex ) @ ( minus_minus_complex @ X3 @ one_one_complex ) ) ) ).

% square_diff_one_factored
thf(fact_1030_square__diff__one__factored,axiom,
    ! [X3: real] :
      ( ( minus_minus_real @ ( times_times_real @ X3 @ X3 ) @ one_one_real )
      = ( times_times_real @ ( plus_plus_real @ X3 @ one_one_real ) @ ( minus_minus_real @ X3 @ one_one_real ) ) ) ).

% square_diff_one_factored
thf(fact_1031_square__diff__one__factored,axiom,
    ! [X3: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X3 @ X3 ) @ one_one_rat )
      = ( times_times_rat @ ( plus_plus_rat @ X3 @ one_one_rat ) @ ( minus_minus_rat @ X3 @ one_one_rat ) ) ) ).

% square_diff_one_factored
thf(fact_1032_square__diff__one__factored,axiom,
    ! [X3: int] :
      ( ( minus_minus_int @ ( times_times_int @ X3 @ X3 ) @ one_one_int )
      = ( times_times_int @ ( plus_plus_int @ X3 @ one_one_int ) @ ( minus_minus_int @ X3 @ one_one_int ) ) ) ).

% square_diff_one_factored
thf(fact_1033_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: real] :
      ( ( ord_less_nat @ N @ N4 )
     => ( ( ord_less_real @ one_one_real @ A )
       => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N4 ) ) ) ) ).

% power_strict_increasing
thf(fact_1034_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: rat] :
      ( ( ord_less_nat @ N @ N4 )
     => ( ( ord_less_rat @ one_one_rat @ A )
       => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N4 ) ) ) ) ).

% power_strict_increasing
thf(fact_1035_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( ( ord_less_nat @ N @ N4 )
     => ( ( ord_less_nat @ one_one_nat @ A )
       => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).

% power_strict_increasing
thf(fact_1036_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( ( ord_less_nat @ N @ N4 )
     => ( ( ord_less_int @ one_one_int @ A )
       => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).

% power_strict_increasing
thf(fact_1037_power__less__imp__less__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_1038_power__less__imp__less__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_1039_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_1040_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_1041_power__increasing,axiom,
    ! [N: nat,N4: nat,A: real] :
      ( ( ord_less_eq_nat @ N @ N4 )
     => ( ( ord_less_eq_real @ one_one_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N4 ) ) ) ) ).

% power_increasing
thf(fact_1042_power__increasing,axiom,
    ! [N: nat,N4: nat,A: rat] :
      ( ( ord_less_eq_nat @ N @ N4 )
     => ( ( ord_less_eq_rat @ one_one_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N4 ) ) ) ) ).

% power_increasing
thf(fact_1043_power__increasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ N4 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).

% power_increasing
thf(fact_1044_power__increasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( ( ord_less_eq_nat @ N @ N4 )
     => ( ( ord_less_eq_int @ one_one_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).

% power_increasing
thf(fact_1045_power__le__imp__le__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_1046_power__le__imp__le__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_1047_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_1048_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_1049_one__power2,axiom,
    ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_rat ) ).

% one_power2
thf(fact_1050_one__power2,axiom,
    ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_power2
thf(fact_1051_one__power2,axiom,
    ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_real ) ).

% one_power2
thf(fact_1052_one__power2,axiom,
    ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_power2
thf(fact_1053_one__power2,axiom,
    ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_complex ) ).

% one_power2
thf(fact_1054_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_1055_times__divide__times__eq,axiom,
    ! [X3: complex,Y3: complex,Z: complex,W: complex] :
      ( ( times_times_complex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X3 @ Z ) @ ( times_times_complex @ Y3 @ W ) ) ) ).

% times_divide_times_eq
thf(fact_1056_times__divide__times__eq,axiom,
    ! [X3: real,Y3: real,Z: real,W: real] :
      ( ( times_times_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ Z @ W ) )
      = ( divide_divide_real @ ( times_times_real @ X3 @ Z ) @ ( times_times_real @ Y3 @ W ) ) ) ).

% times_divide_times_eq
thf(fact_1057_times__divide__times__eq,axiom,
    ! [X3: rat,Y3: rat,Z: rat,W: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ Z @ W ) )
      = ( divide_divide_rat @ ( times_times_rat @ X3 @ Z ) @ ( times_times_rat @ Y3 @ W ) ) ) ).

% times_divide_times_eq
thf(fact_1058_divide__divide__times__eq,axiom,
    ! [X3: complex,Y3: complex,Z: complex,W: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X3 @ W ) @ ( times_times_complex @ Y3 @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_1059_divide__divide__times__eq,axiom,
    ! [X3: real,Y3: real,Z: real,W: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ Z @ W ) )
      = ( divide_divide_real @ ( times_times_real @ X3 @ W ) @ ( times_times_real @ Y3 @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_1060_divide__divide__times__eq,axiom,
    ! [X3: rat,Y3: rat,Z: rat,W: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ Z @ W ) )
      = ( divide_divide_rat @ ( times_times_rat @ X3 @ W ) @ ( times_times_rat @ Y3 @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_1061_divide__divide__eq__left_H,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B2 ) @ C )
      = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B2 ) ) ) ).

% divide_divide_eq_left'
thf(fact_1062_divide__divide__eq__left_H,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ A @ B2 ) @ C )
      = ( divide_divide_real @ A @ ( times_times_real @ C @ B2 ) ) ) ).

% divide_divide_eq_left'
thf(fact_1063_divide__divide__eq__left_H,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B2 ) @ C )
      = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B2 ) ) ) ).

% divide_divide_eq_left'
thf(fact_1064_add__divide__distrib,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B2 ) @ C )
      = ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B2 @ C ) ) ) ).

% add_divide_distrib
thf(fact_1065_add__divide__distrib,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ C )
      = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ).

% add_divide_distrib
thf(fact_1066_add__divide__distrib,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
      = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ).

% add_divide_distrib
thf(fact_1067_diff__divide__distrib,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B2 ) @ C )
      = ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B2 @ C ) ) ) ).

% diff_divide_distrib
thf(fact_1068_diff__divide__distrib,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( divide_divide_real @ ( minus_minus_real @ A @ B2 ) @ C )
      = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ).

% diff_divide_distrib
thf(fact_1069_diff__divide__distrib,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
      = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ).

% diff_divide_distrib
thf(fact_1070_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 )
       => ? [N3: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ N3 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_1071_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 )
       => ? [N3: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B2 @ N3 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_1072_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B2 ) )
      = ( some_P7363390416028606310at_nat @ ( F @ A @ B2 ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_1073_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: num > num > num,A: num,B2: num] :
      ( ( vEBT_V819420779217536731ft_num @ F @ ( some_num @ A ) @ ( some_num @ B2 ) )
      = ( some_num @ ( F @ A @ B2 ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_1074_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: nat > nat > nat,A: nat,B2: nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A ) @ ( some_nat @ B2 ) )
      = ( some_nat @ ( F @ A @ B2 ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_1075_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ Uu @ none_P5556105721700978146at_nat @ Uv )
      = none_P5556105721700978146at_nat ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_1076_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: num > num > num,Uv: option_num] :
      ( ( vEBT_V819420779217536731ft_num @ Uu @ none_num @ Uv )
      = none_num ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_1077_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: nat > nat > nat,Uv: option_nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ Uu @ none_nat @ Uv )
      = none_nat ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_1078_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
      = none_nat ) ).

% vebt_mint.simps(2)
thf(fact_1079_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
      = none_nat ) ).

% vebt_maxt.simps(2)
thf(fact_1080_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ Uw @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat )
      = none_P5556105721700978146at_nat ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_1081_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: num > num > num,V: num] :
      ( ( vEBT_V819420779217536731ft_num @ Uw @ ( some_num @ V ) @ none_num )
      = none_num ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_1082_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: nat > nat > nat,V: nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ Uw @ ( some_nat @ V ) @ none_nat )
      = none_nat ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_1083_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb2: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
      ( ( ( vEBT_V1502963449132264192at_nat @ X3 @ Xa2 @ Xb2 )
        = Y3 )
     => ( ( ( Xa2 = none_P5556105721700978146at_nat )
         => ( Y3 != none_P5556105721700978146at_nat ) )
       => ( ( ? [V2: product_prod_nat_nat] :
                ( Xa2
                = ( some_P7363390416028606310at_nat @ V2 ) )
           => ( ( Xb2 = none_P5556105721700978146at_nat )
             => ( Y3 != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A3: product_prod_nat_nat] :
                ( ( Xa2
                  = ( some_P7363390416028606310at_nat @ A3 ) )
               => ! [B3: product_prod_nat_nat] :
                    ( ( Xb2
                      = ( some_P7363390416028606310at_nat @ B3 ) )
                   => ( Y3
                     != ( some_P7363390416028606310at_nat @ ( X3 @ A3 @ B3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_1084_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X3: num > num > num,Xa2: option_num,Xb2: option_num,Y3: option_num] :
      ( ( ( vEBT_V819420779217536731ft_num @ X3 @ Xa2 @ Xb2 )
        = Y3 )
     => ( ( ( Xa2 = none_num )
         => ( Y3 != none_num ) )
       => ( ( ? [V2: num] :
                ( Xa2
                = ( some_num @ V2 ) )
           => ( ( Xb2 = none_num )
             => ( Y3 != none_num ) ) )
         => ~ ! [A3: num] :
                ( ( Xa2
                  = ( some_num @ A3 ) )
               => ! [B3: num] :
                    ( ( Xb2
                      = ( some_num @ B3 ) )
                   => ( Y3
                     != ( some_num @ ( X3 @ A3 @ B3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_1085_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X3: nat > nat > nat,Xa2: option_nat,Xb2: option_nat,Y3: option_nat] :
      ( ( ( vEBT_V4262088993061758097ft_nat @ X3 @ Xa2 @ Xb2 )
        = Y3 )
     => ( ( ( Xa2 = none_nat )
         => ( Y3 != none_nat ) )
       => ( ( ? [V2: nat] :
                ( Xa2
                = ( some_nat @ V2 ) )
           => ( ( Xb2 = none_nat )
             => ( Y3 != none_nat ) ) )
         => ~ ! [A3: nat] :
                ( ( Xa2
                  = ( some_nat @ A3 ) )
               => ! [B3: nat] :
                    ( ( Xb2
                      = ( some_nat @ B3 ) )
                   => ( Y3
                     != ( some_nat @ ( X3 @ A3 @ B3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_1086_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
     => ( ( N
          = ( suc @ ( suc @ Va ) ) )
       => ( ~ ( ord_less_nat @ Ma @ Mi )
         => ( ( Ma != Mi )
           => ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).

% nested_mint
thf(fact_1087_real__average__minus__second,axiom,
    ! [B2: real,A: real] :
      ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B2 @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
      = ( divide_divide_real @ ( minus_minus_real @ B2 @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% real_average_minus_second
thf(fact_1088_real__average__minus__first,axiom,
    ! [A: real,B2: real] :
      ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
      = ( divide_divide_real @ ( minus_minus_real @ B2 @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% real_average_minus_first
thf(fact_1089__C1_C,axiom,
    ( ( ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
         != none_nat )
        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
     => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
        = ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
    & ( ~ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
           != none_nat )
          & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
     => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
        = ( if_option_nat
          @ ( ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
            = none_nat )
          @ ( if_option_nat @ ( ord_less_nat @ mi @ xa ) @ ( some_nat @ mi ) @ none_nat )
          @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% "1"
thf(fact_1090_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X5 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N ) )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X7 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
                                    = I3 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
                                & ! [X5: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X5 @ N )
                                        = I3 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
                                   => ( ( ord_less_nat @ Mi @ X5 )
                                      & ( ord_less_eq_nat @ X5 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
thf(fact_1091_pred__less__length__list,axiom,
    ! [Deg: nat,X3: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X3 @ Ma )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X3 ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
thf(fact_1092_pred__lesseq__max,axiom,
    ! [Deg: nat,X3: nat,Ma: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X3 @ Ma )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X3 ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% pred_lesseq_max
thf(fact_1093_divmod__step__eq,axiom,
    ! [L: num,R: nat,Q2: nat] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q2 @ R ) )
          = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ one_one_nat ) @ ( minus_minus_nat @ R @ ( numeral_numeral_nat @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q2 @ R ) )
          = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ R ) ) ) ) ).

% divmod_step_eq
thf(fact_1094_divmod__step__eq,axiom,
    ! [L: num,R: int,Q2: int] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q2 @ R ) )
          = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ one_one_int ) @ ( minus_minus_int @ R @ ( numeral_numeral_int @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q2 @ R ) )
          = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ R ) ) ) ) ).

% divmod_step_eq
thf(fact_1095_divmod__step__eq,axiom,
    ! [L: num,R: code_integer,Q2: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q2 @ R ) )
          = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R @ ( numera6620942414471956472nteger @ L ) ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q2 @ R ) )
          = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ R ) ) ) ) ).

% divmod_step_eq
thf(fact_1096_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
thf(fact_1097_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
thf(fact_1098_all__set__conv__all__nth,axiom,
    ! [Xs2: list_int,P: int > $o] :
      ( ( ! [X: int] :
            ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
           => ( P @ ( nth_int @ Xs2 @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_1099_all__set__conv__all__nth,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_1100_all__set__conv__all__nth,axiom,
    ! [Xs2: list_o,P: $o > $o] :
      ( ( ! [X: $o] :
            ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
           => ( P @ ( nth_o @ Xs2 @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_1101_all__set__conv__all__nth,axiom,
    ! [Xs2: list_nat,P: nat > $o] :
      ( ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
           => ( P @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_1102_even__odd__cases,axiom,
    ! [X3: nat] :
      ( ! [N3: nat] :
          ( X3
         != ( plus_plus_nat @ N3 @ N3 ) )
     => ~ ! [N3: nat] :
            ( X3
           != ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) ) ) ).

% even_odd_cases
thf(fact_1103_set__vebt_H__def,axiom,
    ( vEBT_VEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).

% set_vebt'_def
thf(fact_1104_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
     => ? [Info2: option4927543243414619207at_nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
          ( Tree
          = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList3 @ S3 ) ) ) ).

% deg_SUcn_Node
thf(fact_1105_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_1106_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_1107_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( suc @ X2 )
        = ( suc @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% nat.inject
thf(fact_1108_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_1109_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_1110_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_1111_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_1112_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_1113_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_1114_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_1115_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_1116_diff__Suc__1,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
      = N ) ).

% diff_Suc_1
thf(fact_1117_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_1118_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_1119_Suc__numeral,axiom,
    ! [N: num] :
      ( ( suc @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).

% Suc_numeral
thf(fact_1120_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_1121_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_1122_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_1123_Suc__1,axiom,
    ( ( suc @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% Suc_1
thf(fact_1124_mult__commute__abs,axiom,
    ! [C: real] :
      ( ( ^ [X: real] : ( times_times_real @ X @ C ) )
      = ( times_times_real @ C ) ) ).

% mult_commute_abs
thf(fact_1125_mult__commute__abs,axiom,
    ! [C: rat] :
      ( ( ^ [X: rat] : ( times_times_rat @ X @ C ) )
      = ( times_times_rat @ C ) ) ).

% mult_commute_abs
thf(fact_1126_mult__commute__abs,axiom,
    ! [C: nat] :
      ( ( ^ [X: nat] : ( times_times_nat @ X @ C ) )
      = ( times_times_nat @ C ) ) ).

% mult_commute_abs
thf(fact_1127_mult__commute__abs,axiom,
    ! [C: int] :
      ( ( ^ [X: int] : ( times_times_int @ X @ C ) )
      = ( times_times_int @ C ) ) ).

% mult_commute_abs
thf(fact_1128_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_1129_Suc__inject,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ( suc @ X3 )
        = ( suc @ Y3 ) )
     => ( X3 = Y3 ) ) ).

% Suc_inject
thf(fact_1130_lambda__one,axiom,
    ( ( ^ [X: complex] : X )
    = ( times_times_complex @ one_one_complex ) ) ).

% lambda_one
thf(fact_1131_lambda__one,axiom,
    ( ( ^ [X: real] : X )
    = ( times_times_real @ one_one_real ) ) ).

% lambda_one
thf(fact_1132_lambda__one,axiom,
    ( ( ^ [X: rat] : X )
    = ( times_times_rat @ one_one_rat ) ) ).

% lambda_one
thf(fact_1133_lambda__one,axiom,
    ( ( ^ [X: nat] : X )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_1134_lambda__one,axiom,
    ( ( ^ [X: int] : X )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_1135_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X3: produc8306885398267862888on_nat] :
      ( ! [Uu2: nat > nat > nat,Uv2: option_nat] :
          ( X3
         != ( produc8929957630744042906on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
     => ( ! [Uw2: nat > nat > nat,V2: nat] :
            ( X3
           != ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
       => ~ ! [F2: nat > nat > nat,A3: nat,B3: nat] :
              ( X3
             != ( produc8929957630744042906on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A3 ) @ ( some_nat @ B3 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_1136_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X3: produc5542196010084753463at_nat] :
      ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
          ( X3
         != ( produc2899441246263362727at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
     => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
            ( X3
           != ( produc2899441246263362727at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
       => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
              ( X3
             != ( produc2899441246263362727at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A3 ) @ ( some_P7363390416028606310at_nat @ B3 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_1137_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X3: produc1193250871479095198on_num] :
      ( ! [Uu2: num > num > num,Uv2: option_num] :
          ( X3
         != ( produc5778274026573060048on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
     => ( ! [Uw2: num > num > num,V2: num] :
            ( X3
           != ( produc5778274026573060048on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
       => ~ ! [F2: num > num > num,A3: num,B3: num] :
              ( X3
             != ( produc5778274026573060048on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ A3 ) @ ( some_num @ B3 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_1138_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X3: produc2233624965454879586on_nat] :
      ( ! [Uu2: nat > nat > $o,Uv2: option_nat] :
          ( X3
         != ( produc4035269172776083154on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
     => ( ! [Uw2: nat > nat > $o,V2: nat] :
            ( X3
           != ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
       => ~ ! [F2: nat > nat > $o,X5: nat,Y4: nat] :
              ( X3
             != ( produc4035269172776083154on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ X5 ) @ ( some_nat @ Y4 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_1139_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X3: produc5491161045314408544at_nat] :
      ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
          ( X3
         != ( produc3994169339658061776at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
     => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
            ( X3
           != ( produc3994169339658061776at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
       => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,X5: product_prod_nat_nat,Y4: product_prod_nat_nat] :
              ( X3
             != ( produc3994169339658061776at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X5 ) @ ( some_P7363390416028606310at_nat @ Y4 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_1140_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X3: produc7036089656553540234on_num] :
      ( ! [Uu2: num > num > $o,Uv2: option_num] :
          ( X3
         != ( produc3576312749637752826on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
     => ( ! [Uw2: num > num > $o,V2: num] :
            ( X3
           != ( produc3576312749637752826on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
       => ~ ! [F2: num > num > $o,X5: num,Y4: num] :
              ( X3
             != ( produc3576312749637752826on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ X5 ) @ ( some_num @ Y4 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_1141_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_1142_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_lessD
thf(fact_1143_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_1144_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_1145_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( M = N ) ) ) ).

% less_SucE
thf(fact_1146_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_1147_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_1148_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_1149_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_1150_All__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 ) ) ) ) ).

% All_less_Suc
thf(fact_1151_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_1152_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
       => ( M = N ) ) ) ).

% less_antisym
thf(fact_1153_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_1154_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_1155_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,K3: nat] :
              ( ( ord_less_nat @ I3 @ J2 )
             => ( ( ord_less_nat @ J2 @ K3 )
               => ( ( P @ I3 @ J2 )
                 => ( ( P @ J2 @ K3 )
                   => ( P @ I3 @ K3 ) ) ) ) )
         => ( P @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_1156_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_1157_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_1158_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_1159_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_1160_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_1161_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_1162_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_1163_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_1164_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_1165_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
             => ( P @ M2 ) )
         => ( P @ N3 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_1166_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( P @ M )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ M @ N3 )
             => ( ( P @ N3 )
               => ( P @ ( suc @ N3 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_1167_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R2: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X5: nat] : ( R2 @ X5 @ X5 )
       => ( ! [X5: nat,Y4: nat,Z4: nat] :
              ( ( R2 @ X5 @ Y4 )
             => ( ( R2 @ Y4 @ Z4 )
               => ( R2 @ X5 @ Z4 ) ) )
         => ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
           => ( R2 @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_1168_nat__arith_Osuc1,axiom,
    ! [A4: nat,K: nat,A: nat] :
      ( ( A4
        = ( plus_plus_nat @ K @ A ) )
     => ( ( suc @ A4 )
        = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_1169_add__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc
thf(fact_1170_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_1171_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I: nat] :
      ( ( P @ K )
     => ( ! [N3: nat] :
            ( ( P @ ( suc @ N3 ) )
           => ( P @ N3 ) )
       => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).

% zero_induct_lemma
thf(fact_1172_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_1173_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).

% numeral_code(2)
thf(fact_1174_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_1175_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_1176_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_1177_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_1178_set__vebt__def,axiom,
    ( vEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).

% set_vebt_def
thf(fact_1179_real__arch__pow,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ X3 )
     => ? [N3: nat] : ( ord_less_real @ Y3 @ ( power_power_real @ X3 @ N3 ) ) ) ).

% real_arch_pow
thf(fact_1180_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_1181_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_1182_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_1183_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_1184_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_1185_set__conv__nth,axiom,
    ( set_complex2
    = ( ^ [Xs: list_complex] :
          ( collect_complex
          @ ^ [Uu3: complex] :
            ? [I4: nat] :
              ( ( Uu3
                = ( nth_complex @ Xs @ I4 ) )
              & ( ord_less_nat @ I4 @ ( size_s3451745648224563538omplex @ Xs ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_1186_set__conv__nth,axiom,
    ( set_real2
    = ( ^ [Xs: list_real] :
          ( collect_real
          @ ^ [Uu3: real] :
            ? [I4: nat] :
              ( ( Uu3
                = ( nth_real @ Xs @ I4 ) )
              & ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_1187_set__conv__nth,axiom,
    ( set_list_nat2
    = ( ^ [Xs: list_list_nat] :
          ( collect_list_nat
          @ ^ [Uu3: list_nat] :
            ? [I4: nat] :
              ( ( Uu3
                = ( nth_list_nat @ Xs @ I4 ) )
              & ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_1188_set__conv__nth,axiom,
    ( set_int2
    = ( ^ [Xs: list_int] :
          ( collect_int
          @ ^ [Uu3: int] :
            ? [I4: nat] :
              ( ( Uu3
                = ( nth_int @ Xs @ I4 ) )
              & ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_1189_set__conv__nth,axiom,
    ( set_VEBT_VEBT2
    = ( ^ [Xs: list_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ^ [Uu3: vEBT_VEBT] :
            ? [I4: nat] :
              ( ( Uu3
                = ( nth_VEBT_VEBT @ Xs @ I4 ) )
              & ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_1190_set__conv__nth,axiom,
    ( set_o2
    = ( ^ [Xs: list_o] :
          ( collect_o
          @ ^ [Uu3: $o] :
            ? [I4: nat] :
              ( ( Uu3
                = ( nth_o @ Xs @ I4 ) )
              & ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_1191_set__conv__nth,axiom,
    ( set_nat2
    = ( ^ [Xs: list_nat] :
          ( collect_nat
          @ ^ [Uu3: nat] :
            ? [I4: nat] :
              ( ( Uu3
                = ( nth_nat @ Xs @ I4 ) )
              & ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_1192_lift__Suc__mono__less,axiom,
    ! [F: nat > real,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N @ N5 )
       => ( ord_less_real @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1193_lift__Suc__mono__less,axiom,
    ! [F: nat > rat,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N @ N5 )
       => ( ord_less_rat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1194_lift__Suc__mono__less,axiom,
    ! [F: nat > num,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N @ N5 )
       => ( ord_less_num @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1195_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N @ N5 )
       => ( ord_less_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1196_lift__Suc__mono__less,axiom,
    ! [F: nat > int,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N @ N5 )
       => ( ord_less_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1197_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > real,N: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1198_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > rat,N: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1199_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > num,N: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1200_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1201_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > int,N: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1202_power__Suc2,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ A @ ( suc @ N ) )
      = ( times_times_complex @ ( power_power_complex @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_1203_power__Suc2,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ A @ ( suc @ N ) )
      = ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_1204_power__Suc2,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ A @ ( suc @ N ) )
      = ( times_times_rat @ ( power_power_rat @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_1205_power__Suc2,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( suc @ N ) )
      = ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_1206_power__Suc2,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( suc @ N ) )
      = ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_1207_power__Suc,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ A @ ( suc @ N ) )
      = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).

% power_Suc
thf(fact_1208_power__Suc,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ A @ ( suc @ N ) )
      = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).

% power_Suc
thf(fact_1209_power__Suc,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ A @ ( suc @ N ) )
      = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).

% power_Suc
thf(fact_1210_power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( suc @ N ) )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).

% power_Suc
thf(fact_1211_power__Suc,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( suc @ N ) )
      = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).

% power_Suc
thf(fact_1212_lift__Suc__mono__le,axiom,
    ! [F: nat > set_int,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1213_lift__Suc__mono__le,axiom,
    ! [F: nat > rat,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1214_lift__Suc__mono__le,axiom,
    ! [F: nat > num,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1215_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1216_lift__Suc__mono__le,axiom,
    ! [F: nat > int,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1217_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_int,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_set_int @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1218_lift__Suc__antimono__le,axiom,
    ! [F: nat > rat,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_rat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1219_lift__Suc__antimono__le,axiom,
    ! [F: nat > num,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_num @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1220_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1221_lift__Suc__antimono__le,axiom,
    ! [F: nat > int,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_int @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1222_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_1223_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_1224_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I @ N3 )
             => ( ( ord_less_nat @ N3 @ J )
               => ( ( P @ N3 )
                 => ( P @ ( suc @ N3 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_1225_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I @ N3 )
             => ( ( ord_less_nat @ N3 @ J )
               => ( ( P @ ( suc @ N3 ) )
                 => ( P @ N3 ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_1226_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_1227_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_1228_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_1229_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_1230_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_1231_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ? [K3: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_1232_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N2: nat] :
        ? [K2: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M3 @ K2 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_1233_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).

% less_add_Suc2
thf(fact_1234_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).

% less_add_Suc1
thf(fact_1235_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ~ ! [Q3: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).

% less_natE
thf(fact_1236_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_1237_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_1238_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_1239_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_1240_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_1241_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_1242_Suc__div__le__mono,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).

% Suc_div_le_mono
thf(fact_1243_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_1244_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_1245_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_1246_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_1247_power__gt1,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_1248_power__gt1,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_1249_power__gt1,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_1250_power__gt1,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_1251_subset__code_I1_J,axiom,
    ! [Xs2: list_real,B5: set_real] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ B5 )
      = ( ! [X: real] :
            ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
           => ( member_real @ X @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_1252_subset__code_I1_J,axiom,
    ! [Xs2: list_complex,B5: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ B5 )
      = ( ! [X: complex] :
            ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
           => ( member_complex @ X @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_1253_subset__code_I1_J,axiom,
    ! [Xs2: list_VEBT_VEBT,B5: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ B5 )
      = ( ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( member_VEBT_VEBT @ X @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_1254_subset__code_I1_J,axiom,
    ! [Xs2: list_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B5 )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
           => ( member_nat @ X @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_1255_subset__code_I1_J,axiom,
    ! [Xs2: list_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ B5 )
      = ( ! [X: int] :
            ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
           => ( member_int @ X @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_1256_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_1257_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_o] :
      ( ( size_size_list_o @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_1258_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_nat] :
      ( ( size_size_list_nat @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_1259_neq__if__length__neq,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
       != ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_1260_neq__if__length__neq,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
       != ( size_size_list_o @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_1261_neq__if__length__neq,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
       != ( size_size_list_nat @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_1262_div__nat__eqI,axiom,
    ! [N: nat,Q2: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q2 ) @ M )
     => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q2 ) ) )
       => ( ( divide_divide_nat @ M @ N )
          = Q2 ) ) ) ).

% div_nat_eqI
thf(fact_1263_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_1264_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_1265_power__odd__eq,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_1266_power__odd__eq,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_1267_power__odd__eq,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_1268_power__odd__eq,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_1269_power__odd__eq,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_1270_length__induct,axiom,
    ! [P: list_VEBT_VEBT > $o,Xs2: list_VEBT_VEBT] :
      ( ! [Xs3: list_VEBT_VEBT] :
          ( ! [Ys2: list_VEBT_VEBT] :
              ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys2 ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_1271_length__induct,axiom,
    ! [P: list_o > $o,Xs2: list_o] :
      ( ! [Xs3: list_o] :
          ( ! [Ys2: list_o] :
              ( ( ord_less_nat @ ( size_size_list_o @ Ys2 ) @ ( size_size_list_o @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_1272_length__induct,axiom,
    ! [P: list_nat > $o,Xs2: list_nat] :
      ( ! [Xs3: list_nat] :
          ( ! [Ys2: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_1273_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X3: nat,Mi: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Ma @ X3 )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X3 )
          = ( some_nat @ Ma ) ) )
      & ( ~ ( ord_less_nat @ Ma @ X3 )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X3 )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X3 ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_pred.simps(7)
thf(fact_1274_nth__equalityI,axiom,
    ! [Xs2: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_size_list_int @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
           => ( ( nth_int @ Xs2 @ I3 )
              = ( nth_int @ Ys @ I3 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_1275_nth__equalityI,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( ( nth_VEBT_VEBT @ Xs2 @ I3 )
              = ( nth_VEBT_VEBT @ Ys @ I3 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_1276_nth__equalityI,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
           => ( ( nth_o @ Xs2 @ I3 )
              = ( nth_o @ Ys @ I3 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_1277_nth__equalityI,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
           => ( ( nth_nat @ Xs2 @ I3 )
              = ( nth_nat @ Ys @ I3 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_1278_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > int > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X7: int] : ( P @ I4 @ X7 ) ) )
      = ( ? [Xs: list_int] :
            ( ( ( size_size_list_int @ Xs )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_int @ Xs @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1279_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > vEBT_VEBT > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X7: vEBT_VEBT] : ( P @ I4 @ X7 ) ) )
      = ( ? [Xs: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ Xs )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_VEBT_VEBT @ Xs @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1280_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > $o > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X7: $o] : ( P @ I4 @ X7 ) ) )
      = ( ? [Xs: list_o] :
            ( ( ( size_size_list_o @ Xs )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_o @ Xs @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1281_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X7: nat] : ( P @ I4 @ X7 ) ) )
      = ( ? [Xs: list_nat] :
            ( ( ( size_size_list_nat @ Xs )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1282_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_int,Z5: list_int] : Y6 = Z5 )
    = ( ^ [Xs: list_int,Ys3: list_int] :
          ( ( ( size_size_list_int @ Xs )
            = ( size_size_list_int @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
             => ( ( nth_int @ Xs @ I4 )
                = ( nth_int @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1283_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_VEBT_VEBT,Z5: list_VEBT_VEBT] : Y6 = Z5 )
    = ( ^ [Xs: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
          ( ( ( size_s6755466524823107622T_VEBT @ Xs )
            = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
             => ( ( nth_VEBT_VEBT @ Xs @ I4 )
                = ( nth_VEBT_VEBT @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1284_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_o,Z5: list_o] : Y6 = Z5 )
    = ( ^ [Xs: list_o,Ys3: list_o] :
          ( ( ( size_size_list_o @ Xs )
            = ( size_size_list_o @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
             => ( ( nth_o @ Xs @ I4 )
                = ( nth_o @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1285_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_nat,Z5: list_nat] : Y6 = Z5 )
    = ( ^ [Xs: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
             => ( ( nth_nat @ Xs @ I4 )
                = ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1286_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X5 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N ) )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
               => ( ! [X5: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
thf(fact_1287_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A5 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_1288_discrete,axiom,
    ( ord_less_int
    = ( ^ [A5: int] : ( ord_less_eq_int @ ( plus_plus_int @ A5 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_1289_nth__mem,axiom,
    ! [N: nat,Xs2: list_real] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
     => ( member_real @ ( nth_real @ Xs2 @ N ) @ ( set_real2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_1290_nth__mem,axiom,
    ! [N: nat,Xs2: list_complex] :
      ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
     => ( member_complex @ ( nth_complex @ Xs2 @ N ) @ ( set_complex2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_1291_nth__mem,axiom,
    ! [N: nat,Xs2: list_int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
     => ( member_int @ ( nth_int @ Xs2 @ N ) @ ( set_int2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_1292_nth__mem,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ N ) @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_1293_nth__mem,axiom,
    ! [N: nat,Xs2: list_o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
     => ( member_o @ ( nth_o @ Xs2 @ N ) @ ( set_o2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_1294_nth__mem,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( member_nat @ ( nth_nat @ Xs2 @ N ) @ ( set_nat2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_1295_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_int,P: int > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ ( set_int2 @ Xs2 ) )
           => ( P @ X5 ) )
       => ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_1296_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ! [X5: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( P @ X5 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_1297_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_o,P: $o > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
     => ( ! [X5: $o] :
            ( ( member_o @ X5 @ ( set_o2 @ Xs2 ) )
           => ( P @ X5 ) )
       => ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_1298_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_nat,P: nat > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
           => ( P @ X5 ) )
       => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_1299_in__set__conv__nth,axiom,
    ! [X3: real,Xs2: list_real] :
      ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs2 ) )
            & ( ( nth_real @ Xs2 @ I4 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_1300_in__set__conv__nth,axiom,
    ! [X3: complex,Xs2: list_complex] :
      ( ( member_complex @ X3 @ ( set_complex2 @ Xs2 ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s3451745648224563538omplex @ Xs2 ) )
            & ( ( nth_complex @ Xs2 @ I4 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_1301_in__set__conv__nth,axiom,
    ! [X3: int,Xs2: list_int] :
      ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
            & ( ( nth_int @ Xs2 @ I4 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_1302_in__set__conv__nth,axiom,
    ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
            & ( ( nth_VEBT_VEBT @ Xs2 @ I4 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_1303_in__set__conv__nth,axiom,
    ! [X3: $o,Xs2: list_o] :
      ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
            & ( ( nth_o @ Xs2 @ I4 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_1304_in__set__conv__nth,axiom,
    ! [X3: nat,Xs2: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
            & ( ( nth_nat @ Xs2 @ I4 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_1305_all__nth__imp__all__set,axiom,
    ! [Xs2: list_real,P: real > $o,X3: real] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs2 ) )
         => ( P @ ( nth_real @ Xs2 @ I3 ) ) )
     => ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_1306_all__nth__imp__all__set,axiom,
    ! [Xs2: list_complex,P: complex > $o,X3: complex] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs2 ) )
         => ( P @ ( nth_complex @ Xs2 @ I3 ) ) )
     => ( ( member_complex @ X3 @ ( set_complex2 @ Xs2 ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_1307_all__nth__imp__all__set,axiom,
    ! [Xs2: list_int,P: int > $o,X3: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
         => ( P @ ( nth_int @ Xs2 @ I3 ) ) )
     => ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_1308_all__nth__imp__all__set,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,X3: vEBT_VEBT] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
         => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I3 ) ) )
     => ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_1309_all__nth__imp__all__set,axiom,
    ! [Xs2: list_o,P: $o > $o,X3: $o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
         => ( P @ ( nth_o @ Xs2 @ I3 ) ) )
     => ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_1310_all__nth__imp__all__set,axiom,
    ! [Xs2: list_nat,P: nat > $o,X3: nat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
         => ( P @ ( nth_nat @ Xs2 @ I3 ) ) )
     => ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_1311_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X3 )
      = ( ( X3 != Mi )
       => ( ( X3 != Ma )
         => ( ~ ( ord_less_nat @ X3 @ Mi )
            & ( ~ ( ord_less_nat @ X3 @ Mi )
             => ( ~ ( ord_less_nat @ Ma @ X3 )
                & ( ~ ( ord_less_nat @ Ma @ X3 )
                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
thf(fact_1312_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) @ X3 )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) ) ).

% vebt_insert.simps(4)
thf(fact_1313_buildup__gives__empty,axiom,
    ! [N: nat] :
      ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
      = bot_bot_set_nat ) ).

% buildup_gives_empty
thf(fact_1314_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_1315_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_1316_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ Vc ) @ X3 )
      = ( ( X3 = Mi )
        | ( X3 = Ma )
        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
          & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).

% VEBT_internal.membermima.simps(4)
thf(fact_1317_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S ) @ X3 )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).

% VEBT_internal.naive_member.simps(3)
thf(fact_1318_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X3: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) @ X3 )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).

% VEBT_internal.membermima.simps(5)
thf(fact_1319_vebt__pred_Oelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y3: option_nat] :
      ( ( ( vEBT_vebt_pred @ X3 @ Xa2 )
        = Y3 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X3
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( ( Xa2 = zero_zero_nat )
           => ( Y3 != none_nat ) ) )
       => ( ! [A3: $o] :
              ( ? [Uw2: $o] :
                  ( X3
                  = ( vEBT_Leaf @ A3 @ Uw2 ) )
             => ( ( Xa2
                  = ( suc @ zero_zero_nat ) )
               => ~ ( ( A3
                     => ( Y3
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A3
                     => ( Y3 = none_nat ) ) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( X3
                  = ( vEBT_Leaf @ A3 @ B3 ) )
               => ( ? [Va2: nat] :
                      ( Xa2
                      = ( suc @ ( suc @ Va2 ) ) )
                 => ~ ( ( B3
                       => ( Y3
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B3
                       => ( ( A3
                           => ( Y3
                              = ( some_nat @ zero_zero_nat ) ) )
                          & ( ~ A3
                           => ( Y3 = none_nat ) ) ) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( X3
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
               => ( Y3 != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( X3
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) )
                 => ( Y3 != none_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
                        ( X3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) )
                   => ( Y3 != none_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X3
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                       => ~ ( ( ( ord_less_nat @ Ma2 @ Xa2 )
                             => ( Y3
                                = ( some_nat @ Ma2 ) ) )
                            & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                             => ( Y3
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                        = none_nat )
                                      @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
thf(fact_1320_low__def,axiom,
    ( vEBT_VEBT_low
    = ( ^ [X: nat,N2: nat] : ( modulo_modulo_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% low_def
thf(fact_1321_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ).

% VEBT_internal.minNull.simps(4)
thf(fact_1322_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_tree_deg_neq_0
thf(fact_1323_valid__0__not,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_0_not
thf(fact_1324_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X3: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X3 ) ).

% buildup_nothing_in_min_max
thf(fact_1325_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X3: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X3 ) ).

% buildup_nothing_in_leaf
thf(fact_1326_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% deg_not_0
thf(fact_1327_Leaf__0__not,axiom,
    ! [A: $o,B2: $o] :
      ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B2 ) @ zero_zero_nat ) ).

% Leaf_0_not
thf(fact_1328_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( N = one_one_nat )
       => ? [A3: $o,B3: $o] :
            ( T
            = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ).

% deg_1_Leafy
thf(fact_1329_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
     => ? [A3: $o,B3: $o] :
          ( T
          = ( vEBT_Leaf @ A3 @ B3 ) ) ) ).

% deg_1_Leaf
thf(fact_1330_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
      = ( ? [A5: $o,B4: $o] :
            ( T
            = ( vEBT_Leaf @ A5 @ B4 ) ) ) ) ).

% deg1Leaf
thf(fact_1331_both__member__options__def,axiom,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [T2: vEBT_VEBT,X: nat] :
          ( ( vEBT_V5719532721284313246member @ T2 @ X )
          | ( vEBT_VEBT_membermima @ T2 @ X ) ) ) ) ).

% both_member_options_def
thf(fact_1332_buildup__gives__valid,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).

% buildup_gives_valid
thf(fact_1333_mod__mod__trivial,axiom,
    ! [A: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
      = ( modulo_modulo_nat @ A @ B2 ) ) ).

% mod_mod_trivial
thf(fact_1334_mod__mod__trivial,axiom,
    ! [A: int,B2: int] :
      ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% mod_mod_trivial
thf(fact_1335_mod__mod__trivial,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ B2 )
      = ( modulo364778990260209775nteger @ A @ B2 ) ) ).

% mod_mod_trivial
thf(fact_1336_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ Tree @ N )
     => ( ( vEBT_vebt_member @ Tree @ X3 )
       => ( ( vEBT_V5719532721284313246member @ Tree @ X3 )
          | ( vEBT_VEBT_membermima @ Tree @ X3 ) ) ) ) ).

% member_valid_both_member_options
thf(fact_1337_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_1338_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_1339_mult__zero__left,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% mult_zero_left
thf(fact_1340_mult__zero__left,axiom,
    ! [A: real] :
      ( ( times_times_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% mult_zero_left
thf(fact_1341_mult__zero__left,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% mult_zero_left
thf(fact_1342_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_1343_mult__zero__left,axiom,
    ! [A: int] :
      ( ( times_times_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_1344_mult__zero__right,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% mult_zero_right
thf(fact_1345_mult__zero__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% mult_zero_right
thf(fact_1346_mult__zero__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% mult_zero_right
thf(fact_1347_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_1348_mult__zero__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_1349_mult__eq__0__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( times_times_complex @ A @ B2 )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        | ( B2 = zero_zero_complex ) ) ) ).

% mult_eq_0_iff
thf(fact_1350_mult__eq__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ( times_times_real @ A @ B2 )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% mult_eq_0_iff
thf(fact_1351_mult__eq__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( times_times_rat @ A @ B2 )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
thf(fact_1352_mult__eq__0__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( times_times_nat @ A @ B2 )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        | ( B2 = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
thf(fact_1353_mult__eq__0__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ( times_times_int @ A @ B2 )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        | ( B2 = zero_zero_int ) ) ) ).

% mult_eq_0_iff
thf(fact_1354_mult__cancel__left,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( ( times_times_complex @ C @ A )
        = ( times_times_complex @ C @ B2 ) )
      = ( ( C = zero_zero_complex )
        | ( A = B2 ) ) ) ).

% mult_cancel_left
thf(fact_1355_mult__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ( times_times_real @ C @ A )
        = ( times_times_real @ C @ B2 ) )
      = ( ( C = zero_zero_real )
        | ( A = B2 ) ) ) ).

% mult_cancel_left
thf(fact_1356_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ( times_times_rat @ C @ A )
        = ( times_times_rat @ C @ B2 ) )
      = ( ( C = zero_zero_rat )
        | ( A = B2 ) ) ) ).

% mult_cancel_left
thf(fact_1357_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ( times_times_nat @ C @ A )
        = ( times_times_nat @ C @ B2 ) )
      = ( ( C = zero_zero_nat )
        | ( A = B2 ) ) ) ).

% mult_cancel_left
thf(fact_1358_mult__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ( times_times_int @ C @ A )
        = ( times_times_int @ C @ B2 ) )
      = ( ( C = zero_zero_int )
        | ( A = B2 ) ) ) ).

% mult_cancel_left
thf(fact_1359_mult__cancel__right,axiom,
    ! [A: complex,C: complex,B2: complex] :
      ( ( ( times_times_complex @ A @ C )
        = ( times_times_complex @ B2 @ C ) )
      = ( ( C = zero_zero_complex )
        | ( A = B2 ) ) ) ).

% mult_cancel_right
thf(fact_1360_mult__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ( times_times_real @ A @ C )
        = ( times_times_real @ B2 @ C ) )
      = ( ( C = zero_zero_real )
        | ( A = B2 ) ) ) ).

% mult_cancel_right
thf(fact_1361_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ( times_times_rat @ A @ C )
        = ( times_times_rat @ B2 @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A = B2 ) ) ) ).

% mult_cancel_right
thf(fact_1362_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ( times_times_nat @ A @ C )
        = ( times_times_nat @ B2 @ C ) )
      = ( ( C = zero_zero_nat )
        | ( A = B2 ) ) ) ).

% mult_cancel_right
thf(fact_1363_mult__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ( times_times_int @ A @ C )
        = ( times_times_int @ B2 @ C ) )
      = ( ( C = zero_zero_int )
        | ( A = B2 ) ) ) ).

% mult_cancel_right
thf(fact_1364_add__0,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% add_0
thf(fact_1365_add__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add_0
thf(fact_1366_add__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add_0
thf(fact_1367_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_1368_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_1369_zero__eq__add__iff__both__eq__0,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X3 @ Y3 ) )
      = ( ( X3 = zero_zero_nat )
        & ( Y3 = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_1370_add__eq__0__iff__both__eq__0,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ( plus_plus_nat @ X3 @ Y3 )
        = zero_zero_nat )
      = ( ( X3 = zero_zero_nat )
        & ( Y3 = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_1371_add__cancel__right__right,axiom,
    ! [A: complex,B2: complex] :
      ( ( A
        = ( plus_plus_complex @ A @ B2 ) )
      = ( B2 = zero_zero_complex ) ) ).

% add_cancel_right_right
thf(fact_1372_add__cancel__right__right,axiom,
    ! [A: real,B2: real] :
      ( ( A
        = ( plus_plus_real @ A @ B2 ) )
      = ( B2 = zero_zero_real ) ) ).

% add_cancel_right_right
thf(fact_1373_add__cancel__right__right,axiom,
    ! [A: rat,B2: rat] :
      ( ( A
        = ( plus_plus_rat @ A @ B2 ) )
      = ( B2 = zero_zero_rat ) ) ).

% add_cancel_right_right
thf(fact_1374_add__cancel__right__right,axiom,
    ! [A: nat,B2: nat] :
      ( ( A
        = ( plus_plus_nat @ A @ B2 ) )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_1375_add__cancel__right__right,axiom,
    ! [A: int,B2: int] :
      ( ( A
        = ( plus_plus_int @ A @ B2 ) )
      = ( B2 = zero_zero_int ) ) ).

% add_cancel_right_right
thf(fact_1376_add__cancel__right__left,axiom,
    ! [A: complex,B2: complex] :
      ( ( A
        = ( plus_plus_complex @ B2 @ A ) )
      = ( B2 = zero_zero_complex ) ) ).

% add_cancel_right_left
thf(fact_1377_add__cancel__right__left,axiom,
    ! [A: real,B2: real] :
      ( ( A
        = ( plus_plus_real @ B2 @ A ) )
      = ( B2 = zero_zero_real ) ) ).

% add_cancel_right_left
thf(fact_1378_add__cancel__right__left,axiom,
    ! [A: rat,B2: rat] :
      ( ( A
        = ( plus_plus_rat @ B2 @ A ) )
      = ( B2 = zero_zero_rat ) ) ).

% add_cancel_right_left
thf(fact_1379_add__cancel__right__left,axiom,
    ! [A: nat,B2: nat] :
      ( ( A
        = ( plus_plus_nat @ B2 @ A ) )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_1380_add__cancel__right__left,axiom,
    ! [A: int,B2: int] :
      ( ( A
        = ( plus_plus_int @ B2 @ A ) )
      = ( B2 = zero_zero_int ) ) ).

% add_cancel_right_left
thf(fact_1381_add__cancel__left__right,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( plus_plus_complex @ A @ B2 )
        = A )
      = ( B2 = zero_zero_complex ) ) ).

% add_cancel_left_right
thf(fact_1382_add__cancel__left__right,axiom,
    ! [A: real,B2: real] :
      ( ( ( plus_plus_real @ A @ B2 )
        = A )
      = ( B2 = zero_zero_real ) ) ).

% add_cancel_left_right
thf(fact_1383_add__cancel__left__right,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( plus_plus_rat @ A @ B2 )
        = A )
      = ( B2 = zero_zero_rat ) ) ).

% add_cancel_left_right
thf(fact_1384_add__cancel__left__right,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( plus_plus_nat @ A @ B2 )
        = A )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_1385_add__cancel__left__right,axiom,
    ! [A: int,B2: int] :
      ( ( ( plus_plus_int @ A @ B2 )
        = A )
      = ( B2 = zero_zero_int ) ) ).

% add_cancel_left_right
thf(fact_1386_add__cancel__left__left,axiom,
    ! [B2: complex,A: complex] :
      ( ( ( plus_plus_complex @ B2 @ A )
        = A )
      = ( B2 = zero_zero_complex ) ) ).

% add_cancel_left_left
thf(fact_1387_add__cancel__left__left,axiom,
    ! [B2: real,A: real] :
      ( ( ( plus_plus_real @ B2 @ A )
        = A )
      = ( B2 = zero_zero_real ) ) ).

% add_cancel_left_left
thf(fact_1388_add__cancel__left__left,axiom,
    ! [B2: rat,A: rat] :
      ( ( ( plus_plus_rat @ B2 @ A )
        = A )
      = ( B2 = zero_zero_rat ) ) ).

% add_cancel_left_left
thf(fact_1389_add__cancel__left__left,axiom,
    ! [B2: nat,A: nat] :
      ( ( ( plus_plus_nat @ B2 @ A )
        = A )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_1390_add__cancel__left__left,axiom,
    ! [B2: int,A: int] :
      ( ( ( plus_plus_int @ B2 @ A )
        = A )
      = ( B2 = zero_zero_int ) ) ).

% add_cancel_left_left
thf(fact_1391_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A @ A ) )
      = ( A = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_1392_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A @ A ) )
      = ( A = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_1393_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_1394_add_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% add.right_neutral
thf(fact_1395_add_Oright__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.right_neutral
thf(fact_1396_add_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.right_neutral
thf(fact_1397_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_1398_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_1399_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ A )
      = zero_zero_complex ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1400_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1401_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1402_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ A )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1403_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1404_diff__zero,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ zero_zero_complex )
      = A ) ).

% diff_zero
thf(fact_1405_diff__zero,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_zero
thf(fact_1406_diff__zero,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_zero
thf(fact_1407_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_1408_diff__zero,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_zero
thf(fact_1409_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_1410_diff__0__right,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ zero_zero_complex )
      = A ) ).

% diff_0_right
thf(fact_1411_diff__0__right,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_0_right
thf(fact_1412_diff__0__right,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_0_right
thf(fact_1413_diff__0__right,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_0_right
thf(fact_1414_diff__self,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ A )
      = zero_zero_complex ) ).

% diff_self
thf(fact_1415_diff__self,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% diff_self
thf(fact_1416_diff__self,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% diff_self
thf(fact_1417_diff__self,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% diff_self
thf(fact_1418_divide__eq__0__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ B2 )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        | ( B2 = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
thf(fact_1419_divide__eq__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ( divide_divide_real @ A @ B2 )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% divide_eq_0_iff
thf(fact_1420_divide__eq__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( divide_divide_rat @ A @ B2 )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
thf(fact_1421_divide__cancel__left,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( ( divide1717551699836669952omplex @ C @ A )
        = ( divide1717551699836669952omplex @ C @ B2 ) )
      = ( ( C = zero_zero_complex )
        | ( A = B2 ) ) ) ).

% divide_cancel_left
thf(fact_1422_divide__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ( divide_divide_real @ C @ A )
        = ( divide_divide_real @ C @ B2 ) )
      = ( ( C = zero_zero_real )
        | ( A = B2 ) ) ) ).

% divide_cancel_left
thf(fact_1423_divide__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ( divide_divide_rat @ C @ A )
        = ( divide_divide_rat @ C @ B2 ) )
      = ( ( C = zero_zero_rat )
        | ( A = B2 ) ) ) ).

% divide_cancel_left
thf(fact_1424_divide__cancel__right,axiom,
    ! [A: complex,C: complex,B2: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ C )
        = ( divide1717551699836669952omplex @ B2 @ C ) )
      = ( ( C = zero_zero_complex )
        | ( A = B2 ) ) ) ).

% divide_cancel_right
thf(fact_1425_divide__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ( divide_divide_real @ A @ C )
        = ( divide_divide_real @ B2 @ C ) )
      = ( ( C = zero_zero_real )
        | ( A = B2 ) ) ) ).

% divide_cancel_right
thf(fact_1426_divide__cancel__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ( divide_divide_rat @ A @ C )
        = ( divide_divide_rat @ B2 @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A = B2 ) ) ) ).

% divide_cancel_right
thf(fact_1427_division__ring__divide__zero,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% division_ring_divide_zero
thf(fact_1428_division__ring__divide__zero,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% division_ring_divide_zero
thf(fact_1429_division__ring__divide__zero,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% division_ring_divide_zero
thf(fact_1430_div__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% div_0
thf(fact_1431_div__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% div_0
thf(fact_1432_div__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% div_0
thf(fact_1433_div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% div_0
thf(fact_1434_div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% div_0
thf(fact_1435_div__by__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% div_by_0
thf(fact_1436_div__by__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% div_by_0
thf(fact_1437_div__by__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% div_by_0
thf(fact_1438_div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% div_by_0
thf(fact_1439_div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% div_by_0
thf(fact_1440_bits__div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_div_0
thf(fact_1441_bits__div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_div_0
thf(fact_1442_bits__div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% bits_div_by_0
thf(fact_1443_bits__div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% bits_div_by_0
thf(fact_1444_mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mod_0
thf(fact_1445_mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mod_0
thf(fact_1446_mod__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% mod_0
thf(fact_1447_mod__by__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ zero_zero_nat )
      = A ) ).

% mod_by_0
thf(fact_1448_mod__by__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ zero_zero_int )
      = A ) ).

% mod_by_0
thf(fact_1449_mod__by__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% mod_by_0
thf(fact_1450_mod__self,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ A )
      = zero_zero_nat ) ).

% mod_self
thf(fact_1451_mod__self,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ A )
      = zero_zero_int ) ).

% mod_self
thf(fact_1452_mod__self,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ A )
      = zero_z3403309356797280102nteger ) ).

% mod_self
thf(fact_1453_bits__mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_mod_0
thf(fact_1454_bits__mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_mod_0
thf(fact_1455_bits__mod__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_0
thf(fact_1456_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_1457_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_1458_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_1459_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_1460_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_1461_mod__add__self1,axiom,
    ! [B2: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
      = ( modulo_modulo_nat @ A @ B2 ) ) ).

% mod_add_self1
thf(fact_1462_mod__add__self1,axiom,
    ! [B2: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% mod_add_self1
thf(fact_1463_mod__add__self1,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B2 @ A ) @ B2 )
      = ( modulo364778990260209775nteger @ A @ B2 ) ) ).

% mod_add_self1
thf(fact_1464_mod__add__self2,axiom,
    ! [A: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
      = ( modulo_modulo_nat @ A @ B2 ) ) ).

% mod_add_self2
thf(fact_1465_mod__add__self2,axiom,
    ! [A: int,B2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% mod_add_self2
thf(fact_1466_mod__add__self2,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ B2 )
      = ( modulo364778990260209775nteger @ A @ B2 ) ) ).

% mod_add_self2
thf(fact_1467_minus__mod__self2,axiom,
    ! [A: int,B2: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ B2 )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% minus_mod_self2
thf(fact_1468_minus__mod__self2,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ B2 )
      = ( modulo364778990260209775nteger @ A @ B2 ) ) ).

% minus_mod_self2
thf(fact_1469_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_1470_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_1471_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_1472_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_1473_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_1474_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_1475_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_1476_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_1477_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( modulo_modulo_nat @ M @ N )
        = M ) ) ).

% mod_less
thf(fact_1478_add__le__same__cancel1,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ B2 @ A ) @ B2 )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% add_le_same_cancel1
thf(fact_1479_add__le__same__cancel1,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ B2 @ A ) @ B2 )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% add_le_same_cancel1
thf(fact_1480_add__le__same__cancel1,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_1481_add__le__same__cancel1,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel1
thf(fact_1482_add__le__same__cancel2,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% add_le_same_cancel2
thf(fact_1483_add__le__same__cancel2,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B2 ) @ B2 )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% add_le_same_cancel2
thf(fact_1484_add__le__same__cancel2,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_1485_add__le__same__cancel2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel2
thf(fact_1486_le__add__same__cancel1,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_1487_le__add__same__cancel1,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B2 ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_1488_le__add__same__cancel1,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_1489_le__add__same__cancel1,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_1490_le__add__same__cancel2,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B2 @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_1491_le__add__same__cancel2,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B2 @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_1492_le__add__same__cancel2,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_1493_le__add__same__cancel2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B2 @ A ) )
      = ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_1494_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_1495_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_1496_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_1497_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_1498_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_1499_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_1500_diff__ge__0__iff__ge,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B2 ) )
      = ( ord_less_eq_real @ B2 @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_1501_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B2 ) )
      = ( ord_less_eq_rat @ B2 @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_1502_diff__ge__0__iff__ge,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B2 ) )
      = ( ord_less_eq_int @ B2 @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_1503_add__less__same__cancel1,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ ( plus_plus_real @ B2 @ A ) @ B2 )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% add_less_same_cancel1
thf(fact_1504_add__less__same__cancel1,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ B2 @ A ) @ B2 )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% add_less_same_cancel1
thf(fact_1505_add__less__same__cancel1,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_1506_add__less__same__cancel1,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel1
thf(fact_1507_add__less__same__cancel2,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% add_less_same_cancel2
thf(fact_1508_add__less__same__cancel2,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ B2 )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% add_less_same_cancel2
thf(fact_1509_add__less__same__cancel2,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_1510_add__less__same__cancel2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel2
thf(fact_1511_less__add__same__cancel1,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B2 ) )
      = ( ord_less_real @ zero_zero_real @ B2 ) ) ).

% less_add_same_cancel1
thf(fact_1512_less__add__same__cancel1,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B2 ) )
      = ( ord_less_rat @ zero_zero_rat @ B2 ) ) ).

% less_add_same_cancel1
thf(fact_1513_less__add__same__cancel1,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
      = ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).

% less_add_same_cancel1
thf(fact_1514_less__add__same__cancel1,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B2 ) )
      = ( ord_less_int @ zero_zero_int @ B2 ) ) ).

% less_add_same_cancel1
thf(fact_1515_less__add__same__cancel2,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ ( plus_plus_real @ B2 @ A ) )
      = ( ord_less_real @ zero_zero_real @ B2 ) ) ).

% less_add_same_cancel2
thf(fact_1516_less__add__same__cancel2,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ ( plus_plus_rat @ B2 @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ B2 ) ) ).

% less_add_same_cancel2
thf(fact_1517_less__add__same__cancel2,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
      = ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).

% less_add_same_cancel2
thf(fact_1518_less__add__same__cancel2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ B2 @ A ) )
      = ( ord_less_int @ zero_zero_int @ B2 ) ) ).

% less_add_same_cancel2
thf(fact_1519_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_1520_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_1521_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_1522_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_1523_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_1524_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_1525_diff__gt__0__iff__gt,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B2 ) )
      = ( ord_less_real @ B2 @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_1526_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B2 ) )
      = ( ord_less_rat @ B2 @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_1527_diff__gt__0__iff__gt,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B2 ) )
      = ( ord_less_int @ B2 @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_1528_mult__cancel__left1,axiom,
    ! [C: complex,B2: complex] :
      ( ( C
        = ( times_times_complex @ C @ B2 ) )
      = ( ( C = zero_zero_complex )
        | ( B2 = one_one_complex ) ) ) ).

% mult_cancel_left1
thf(fact_1529_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_1530_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_1531_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_1532_mult__cancel__left2,axiom,
    ! [C: complex,A: complex] :
      ( ( ( times_times_complex @ C @ A )
        = C )
      = ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_left2
thf(fact_1533_mult__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ( times_times_real @ C @ A )
        = C )
      = ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
thf(fact_1534_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ( times_times_rat @ C @ A )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
thf(fact_1535_mult__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ( times_times_int @ C @ A )
        = C )
      = ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
thf(fact_1536_mult__cancel__right1,axiom,
    ! [C: complex,B2: complex] :
      ( ( C
        = ( times_times_complex @ B2 @ C ) )
      = ( ( C = zero_zero_complex )
        | ( B2 = one_one_complex ) ) ) ).

% mult_cancel_right1
thf(fact_1537_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_1538_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_1539_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_1540_mult__cancel__right2,axiom,
    ! [A: complex,C: complex] :
      ( ( ( times_times_complex @ A @ C )
        = C )
      = ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_right2
thf(fact_1541_mult__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ( times_times_real @ A @ C )
        = C )
      = ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
thf(fact_1542_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ( times_times_rat @ A @ C )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
thf(fact_1543_mult__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ( times_times_int @ A @ C )
        = C )
      = ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
thf(fact_1544_sum__squares__eq__zero__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) )
        = zero_zero_real )
      = ( ( X3 = zero_zero_real )
        & ( Y3 = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_1545_sum__squares__eq__zero__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) )
        = zero_zero_rat )
      = ( ( X3 = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_1546_sum__squares__eq__zero__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) )
        = zero_zero_int )
      = ( ( X3 = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_1547_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
    = zero_zero_complex ) ).

% diff_numeral_special(9)
thf(fact_1548_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_real @ one_one_real @ one_one_real )
    = zero_zero_real ) ).

% diff_numeral_special(9)
thf(fact_1549_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
    = zero_zero_rat ) ).

% diff_numeral_special(9)
thf(fact_1550_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_1551_mult__divide__mult__cancel__left__if,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( ( C = zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B2 ) )
          = zero_zero_complex ) )
      & ( ( C != zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B2 ) )
          = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_1552_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ( C = zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
          = zero_zero_real ) )
      & ( ( C != zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
          = ( divide_divide_real @ A @ B2 ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_1553_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ( C = zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
          = zero_zero_rat ) )
      & ( ( C != zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
          = ( divide_divide_rat @ A @ B2 ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_1554_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B2 ) )
        = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_1555_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
        = ( divide_divide_real @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_1556_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
        = ( divide_divide_rat @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_1557_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B2 @ C ) )
        = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_1558_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B2 @ C ) )
        = ( divide_divide_real @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_1559_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B2 @ C ) )
        = ( divide_divide_rat @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_1560_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B2 @ C ) )
        = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_1561_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
        = ( divide_divide_real @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_1562_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
        = ( divide_divide_rat @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_1563_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B2 ) )
        = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_1564_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B2 ) )
        = ( divide_divide_real @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_1565_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B2 ) )
        = ( divide_divide_rat @ A @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_1566_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ( C = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
          = zero_zero_nat ) )
      & ( ( C != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
          = ( divide_divide_nat @ A @ B2 ) ) ) ) ).

% div_mult_mult1_if
thf(fact_1567_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ( C = zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
          = zero_zero_int ) )
      & ( ( C != zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
          = ( divide_divide_int @ A @ B2 ) ) ) ) ).

% div_mult_mult1_if
thf(fact_1568_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
        = ( divide_divide_nat @ A @ B2 ) ) ) ).

% div_mult_mult2
thf(fact_1569_div__mult__mult2,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
        = ( divide_divide_int @ A @ B2 ) ) ) ).

% div_mult_mult2
thf(fact_1570_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
        = ( divide_divide_nat @ A @ B2 ) ) ) ).

% div_mult_mult1
thf(fact_1571_div__mult__mult1,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
        = ( divide_divide_int @ A @ B2 ) ) ) ).

% div_mult_mult1
thf(fact_1572_nonzero__mult__div__cancel__left,axiom,
    ! [A: complex,B2: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B2 ) @ A )
        = B2 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1573_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ B2 ) @ A )
        = B2 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1574_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ B2 ) @ A )
        = B2 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1575_nonzero__mult__div__cancel__left,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ A )
        = B2 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1576_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B2: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ A )
        = B2 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1577_nonzero__mult__div__cancel__right,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B2 ) @ B2 )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1578_nonzero__mult__div__cancel__right,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ B2 ) @ B2 )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1579_nonzero__mult__div__cancel__right,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ B2 ) @ B2 )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1580_nonzero__mult__div__cancel__right,axiom,
    ! [B2: nat,A: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ B2 )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1581_nonzero__mult__div__cancel__right,axiom,
    ! [B2: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ B2 )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1582_diff__add__zero,axiom,
    ! [A: nat,B2: nat] :
      ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_1583_divide__eq__1__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ B2 )
        = one_one_complex )
      = ( ( B2 != zero_zero_complex )
        & ( A = B2 ) ) ) ).

% divide_eq_1_iff
thf(fact_1584_divide__eq__1__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ( divide_divide_real @ A @ B2 )
        = one_one_real )
      = ( ( B2 != zero_zero_real )
        & ( A = B2 ) ) ) ).

% divide_eq_1_iff
thf(fact_1585_divide__eq__1__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( divide_divide_rat @ A @ B2 )
        = one_one_rat )
      = ( ( B2 != zero_zero_rat )
        & ( A = B2 ) ) ) ).

% divide_eq_1_iff
thf(fact_1586_one__eq__divide__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( one_one_complex
        = ( divide1717551699836669952omplex @ A @ B2 ) )
      = ( ( B2 != zero_zero_complex )
        & ( A = B2 ) ) ) ).

% one_eq_divide_iff
thf(fact_1587_one__eq__divide__iff,axiom,
    ! [A: real,B2: real] :
      ( ( one_one_real
        = ( divide_divide_real @ A @ B2 ) )
      = ( ( B2 != zero_zero_real )
        & ( A = B2 ) ) ) ).

% one_eq_divide_iff
thf(fact_1588_one__eq__divide__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ A @ B2 ) )
      = ( ( B2 != zero_zero_rat )
        & ( A = B2 ) ) ) ).

% one_eq_divide_iff
thf(fact_1589_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% divide_self
thf(fact_1590_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% divide_self
thf(fact_1591_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% divide_self
thf(fact_1592_divide__self__if,axiom,
    ! [A: complex] :
      ( ( ( A = zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ A @ A )
          = zero_zero_complex ) )
      & ( ( A != zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ A @ A )
          = one_one_complex ) ) ) ).

% divide_self_if
thf(fact_1593_divide__self__if,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( ( divide_divide_real @ A @ A )
          = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( ( divide_divide_real @ A @ A )
          = one_one_real ) ) ) ).

% divide_self_if
thf(fact_1594_divide__self__if,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( ( divide_divide_rat @ A @ A )
          = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( ( divide_divide_rat @ A @ A )
          = one_one_rat ) ) ) ).

% divide_self_if
thf(fact_1595_divide__eq__eq__1,axiom,
    ! [B2: real,A: real] :
      ( ( ( divide_divide_real @ B2 @ A )
        = one_one_real )
      = ( ( A != zero_zero_real )
        & ( A = B2 ) ) ) ).

% divide_eq_eq_1
thf(fact_1596_divide__eq__eq__1,axiom,
    ! [B2: rat,A: rat] :
      ( ( ( divide_divide_rat @ B2 @ A )
        = one_one_rat )
      = ( ( A != zero_zero_rat )
        & ( A = B2 ) ) ) ).

% divide_eq_eq_1
thf(fact_1597_eq__divide__eq__1,axiom,
    ! [B2: real,A: real] :
      ( ( one_one_real
        = ( divide_divide_real @ B2 @ A ) )
      = ( ( A != zero_zero_real )
        & ( A = B2 ) ) ) ).

% eq_divide_eq_1
thf(fact_1598_eq__divide__eq__1,axiom,
    ! [B2: rat,A: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ B2 @ A ) )
      = ( ( A != zero_zero_rat )
        & ( A = B2 ) ) ) ).

% eq_divide_eq_1
thf(fact_1599_one__divide__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( divide_divide_real @ one_one_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% one_divide_eq_0_iff
thf(fact_1600_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( divide_divide_rat @ one_one_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
thf(fact_1601_zero__eq__1__divide__iff,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( divide_divide_real @ one_one_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
thf(fact_1602_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( divide_divide_rat @ one_one_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
thf(fact_1603_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% div_self
thf(fact_1604_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% div_self
thf(fact_1605_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% div_self
thf(fact_1606_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ A @ A )
        = one_one_nat ) ) ).

% div_self
thf(fact_1607_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ A @ A )
        = one_one_int ) ) ).

% div_self
thf(fact_1608_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
      = zero_zero_rat ) ).

% power_0_Suc
thf(fact_1609_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_1610_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
      = zero_zero_real ) ).

% power_0_Suc
thf(fact_1611_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
      = zero_zero_int ) ).

% power_0_Suc
thf(fact_1612_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
      = zero_zero_complex ) ).

% power_0_Suc
thf(fact_1613_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_rat ) ).

% power_zero_numeral
thf(fact_1614_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_nat ) ).

% power_zero_numeral
thf(fact_1615_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
      = zero_zero_real ) ).

% power_zero_numeral
thf(fact_1616_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
      = zero_zero_int ) ).

% power_zero_numeral
thf(fact_1617_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
      = zero_zero_complex ) ).

% power_zero_numeral
thf(fact_1618_mod__mult__self1__is__0,axiom,
    ! [B2: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ B2 @ A ) @ B2 )
      = zero_zero_nat ) ).

% mod_mult_self1_is_0
thf(fact_1619_mod__mult__self1__is__0,axiom,
    ! [B2: int,A: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ B2 @ A ) @ B2 )
      = zero_zero_int ) ).

% mod_mult_self1_is_0
thf(fact_1620_mod__mult__self1__is__0,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B2 @ A ) @ B2 )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self1_is_0
thf(fact_1621_mod__mult__self2__is__0,axiom,
    ! [A: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ B2 )
      = zero_zero_nat ) ).

% mod_mult_self2_is_0
thf(fact_1622_mod__mult__self2__is__0,axiom,
    ! [A: int,B2: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ B2 )
      = zero_zero_int ) ).

% mod_mult_self2_is_0
thf(fact_1623_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ B2 )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self2_is_0
thf(fact_1624_mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% mod_by_1
thf(fact_1625_mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% mod_by_1
thf(fact_1626_mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% mod_by_1
thf(fact_1627_bits__mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% bits_mod_by_1
thf(fact_1628_bits__mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% bits_mod_by_1
thf(fact_1629_bits__mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_by_1
thf(fact_1630_mod__div__trivial,axiom,
    ! [A: nat,B2: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
      = zero_zero_nat ) ).

% mod_div_trivial
thf(fact_1631_mod__div__trivial,axiom,
    ! [A: int,B2: int] :
      ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 )
      = zero_zero_int ) ).

% mod_div_trivial
thf(fact_1632_mod__div__trivial,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ B2 )
      = zero_z3403309356797280102nteger ) ).

% mod_div_trivial
thf(fact_1633_bits__mod__div__trivial,axiom,
    ! [A: nat,B2: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
      = zero_zero_nat ) ).

% bits_mod_div_trivial
thf(fact_1634_bits__mod__div__trivial,axiom,
    ! [A: int,B2: int] :
      ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 )
      = zero_zero_int ) ).

% bits_mod_div_trivial
thf(fact_1635_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ B2 )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_div_trivial
thf(fact_1636_power__Suc0__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1637_power__Suc0__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1638_power__Suc0__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1639_power__Suc0__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_1640_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B2 ) ) @ B2 )
      = ( modulo_modulo_nat @ A @ B2 ) ) ).

% mod_mult_self1
thf(fact_1641_mod__mult__self1,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B2 ) ) @ B2 )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% mod_mult_self1
thf(fact_1642_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B2 ) ) @ B2 )
      = ( modulo364778990260209775nteger @ A @ B2 ) ) ).

% mod_mult_self1
thf(fact_1643_mod__mult__self2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B2 @ C ) ) @ B2 )
      = ( modulo_modulo_nat @ A @ B2 ) ) ).

% mod_mult_self2
thf(fact_1644_mod__mult__self2,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B2 @ C ) ) @ B2 )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% mod_mult_self2
thf(fact_1645_mod__mult__self2,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) ) @ B2 )
      = ( modulo364778990260209775nteger @ A @ B2 ) ) ).

% mod_mult_self2
thf(fact_1646_mod__mult__self3,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B2 ) @ A ) @ B2 )
      = ( modulo_modulo_nat @ A @ B2 ) ) ).

% mod_mult_self3
thf(fact_1647_mod__mult__self3,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B2 ) @ A ) @ B2 )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% mod_mult_self3
thf(fact_1648_mod__mult__self3,axiom,
    ! [C: code_integer,B2: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B2 ) @ A ) @ B2 )
      = ( modulo364778990260209775nteger @ A @ B2 ) ) ).

% mod_mult_self3
thf(fact_1649_mod__mult__self4,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ C ) @ A ) @ B2 )
      = ( modulo_modulo_nat @ A @ B2 ) ) ).

% mod_mult_self4
thf(fact_1650_mod__mult__self4,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B2 @ C ) @ A ) @ B2 )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% mod_mult_self4
thf(fact_1651_mod__mult__self4,axiom,
    ! [B2: code_integer,C: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ C ) @ A ) @ B2 )
      = ( modulo364778990260209775nteger @ A @ B2 ) ) ).

% mod_mult_self4
thf(fact_1652_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_1653_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_1654_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_1655_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_1656_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_1657_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_1658_div__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
      = M ) ).

% div_by_Suc_0
thf(fact_1659_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_1660_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_1661_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_1662_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_1663_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_1664_div__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( divide_divide_nat @ M @ N )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_1665_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_1666_power__Suc__0,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_1667_nat__power__eq__Suc__0__iff,axiom,
    ! [X3: nat,M: nat] :
      ( ( ( power_power_nat @ X3 @ M )
        = ( suc @ zero_zero_nat ) )
      = ( ( M = zero_zero_nat )
        | ( X3
          = ( suc @ zero_zero_nat ) ) ) ) ).

% nat_power_eq_Suc_0_iff
thf(fact_1668_nat__zero__less__power__iff,axiom,
    ! [X3: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X3 @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X3 )
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
thf(fact_1669_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_1670_mod__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% mod_by_Suc_0
thf(fact_1671_zero__le__divide__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% zero_le_divide_1_iff
thf(fact_1672_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% zero_le_divide_1_iff
thf(fact_1673_divide__le__0__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% divide_le_0_1_iff
thf(fact_1674_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% divide_le_0_1_iff
thf(fact_1675_zero__less__divide__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% zero_less_divide_1_iff
thf(fact_1676_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% zero_less_divide_1_iff
thf(fact_1677_less__divide__eq__1__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
        = ( ord_less_real @ A @ B2 ) ) ) ).

% less_divide_eq_1_pos
thf(fact_1678_less__divide__eq__1__pos,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
        = ( ord_less_rat @ A @ B2 ) ) ) ).

% less_divide_eq_1_pos
thf(fact_1679_less__divide__eq__1__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
        = ( ord_less_real @ B2 @ A ) ) ) ).

% less_divide_eq_1_neg
thf(fact_1680_less__divide__eq__1__neg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
        = ( ord_less_rat @ B2 @ A ) ) ) ).

% less_divide_eq_1_neg
thf(fact_1681_divide__less__eq__1__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
        = ( ord_less_real @ B2 @ A ) ) ) ).

% divide_less_eq_1_pos
thf(fact_1682_divide__less__eq__1__pos,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
        = ( ord_less_rat @ B2 @ A ) ) ) ).

% divide_less_eq_1_pos
thf(fact_1683_divide__less__eq__1__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
        = ( ord_less_real @ A @ B2 ) ) ) ).

% divide_less_eq_1_neg
thf(fact_1684_divide__less__eq__1__neg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
        = ( ord_less_rat @ A @ B2 ) ) ) ).

% divide_less_eq_1_neg
thf(fact_1685_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% divide_less_0_1_iff
thf(fact_1686_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% divide_less_0_1_iff
thf(fact_1687_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: complex,W: num,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B2 @ ( numera6690914467698888265omplex @ W ) )
        = A )
      = ( ( ( ( numera6690914467698888265omplex @ W )
           != zero_zero_complex )
         => ( B2
            = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) ) )
        & ( ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_1688_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: real,W: num,A: real] :
      ( ( ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) )
        = A )
      = ( ( ( ( numeral_numeral_real @ W )
           != zero_zero_real )
         => ( B2
            = ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
        & ( ( ( numeral_numeral_real @ W )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_1689_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W: num,A: rat] :
      ( ( ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) )
        = A )
      = ( ( ( ( numeral_numeral_rat @ W )
           != zero_zero_rat )
         => ( B2
            = ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
        & ( ( ( numeral_numeral_rat @ W )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_1690_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B2: complex,W: num] :
      ( ( A
        = ( divide1717551699836669952omplex @ B2 @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( ( ( numera6690914467698888265omplex @ W )
           != zero_zero_complex )
         => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) )
            = B2 ) )
        & ( ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_1691_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B2: real,W: num] :
      ( ( A
        = ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ( numeral_numeral_real @ W )
           != zero_zero_real )
         => ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
            = B2 ) )
        & ( ( ( numeral_numeral_real @ W )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_1692_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B2: rat,W: num] :
      ( ( A
        = ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ( numeral_numeral_rat @ W )
           != zero_zero_rat )
         => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
            = B2 ) )
        & ( ( ( numeral_numeral_rat @ W )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_1693_nonzero__divide__mult__cancel__left,axiom,
    ! [A: complex,B2: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B2 ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ B2 ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_1694_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B2 ) )
        = ( divide_divide_real @ one_one_real @ B2 ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_1695_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B2 ) )
        = ( divide_divide_rat @ one_one_rat @ B2 ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_1696_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ B2 @ ( times_times_complex @ A @ B2 ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_1697_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( divide_divide_real @ B2 @ ( times_times_real @ A @ B2 ) )
        = ( divide_divide_real @ one_one_real @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_1698_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( divide_divide_rat @ B2 @ ( times_times_rat @ A @ B2 ) )
        = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_1699_div__mult__self1,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B2 ) ) @ B2 )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).

% div_mult_self1
thf(fact_1700_div__mult__self1,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B2 ) ) @ B2 )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).

% div_mult_self1
thf(fact_1701_div__mult__self2,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B2 @ C ) ) @ B2 )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).

% div_mult_self2
thf(fact_1702_div__mult__self2,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B2 @ C ) ) @ B2 )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).

% div_mult_self2
thf(fact_1703_div__mult__self3,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B2 ) @ A ) @ B2 )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).

% div_mult_self3
thf(fact_1704_div__mult__self3,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B2 ) @ A ) @ B2 )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).

% div_mult_self3
thf(fact_1705_div__mult__self4,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ C ) @ A ) @ B2 )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).

% div_mult_self4
thf(fact_1706_div__mult__self4,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B2 @ C ) @ A ) @ B2 )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).

% div_mult_self4
thf(fact_1707_power__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( ( power_power_rat @ A @ N )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_1708_power__eq__0__iff,axiom,
    ! [A: nat,N: nat] :
      ( ( ( power_power_nat @ A @ N )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_1709_power__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( ( power_power_real @ A @ N )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_1710_power__eq__0__iff,axiom,
    ! [A: int,N: nat] :
      ( ( ( power_power_int @ A @ N )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_1711_power__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( ( power_power_complex @ A @ N )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_1712_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_1713_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_1714_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_1715_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_1716_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_1717_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_1718_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% Suc_mod_mult_self1
thf(fact_1719_Suc__mod__mult__self2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% Suc_mod_mult_self2
thf(fact_1720_Suc__mod__mult__self3,axiom,
    ! [K: nat,N: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% Suc_mod_mult_self3
thf(fact_1721_Suc__mod__mult__self4,axiom,
    ! [N: nat,K: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% Suc_mod_mult_self4
thf(fact_1722_le__divide__eq__1__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
        = ( ord_less_eq_real @ A @ B2 ) ) ) ).

% le_divide_eq_1_pos
thf(fact_1723_le__divide__eq__1__pos,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
        = ( ord_less_eq_rat @ A @ B2 ) ) ) ).

% le_divide_eq_1_pos
thf(fact_1724_le__divide__eq__1__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
        = ( ord_less_eq_real @ B2 @ A ) ) ) ).

% le_divide_eq_1_neg
thf(fact_1725_le__divide__eq__1__neg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
        = ( ord_less_eq_rat @ B2 @ A ) ) ) ).

% le_divide_eq_1_neg
thf(fact_1726_divide__le__eq__1__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
        = ( ord_less_eq_real @ B2 @ A ) ) ) ).

% divide_le_eq_1_pos
thf(fact_1727_divide__le__eq__1__pos,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
        = ( ord_less_eq_rat @ B2 @ A ) ) ) ).

% divide_le_eq_1_pos
thf(fact_1728_divide__le__eq__1__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
        = ( ord_less_eq_real @ A @ B2 ) ) ) ).

% divide_le_eq_1_neg
thf(fact_1729_divide__le__eq__1__neg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
        = ( ord_less_eq_rat @ A @ B2 ) ) ) ).

% divide_le_eq_1_neg
thf(fact_1730_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_1731_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_1732_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_1733_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_1734_power__mono__iff,axiom,
    ! [A: real,B2: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) )
            = ( ord_less_eq_real @ A @ B2 ) ) ) ) ) ).

% power_mono_iff
thf(fact_1735_power__mono__iff,axiom,
    ! [A: rat,B2: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) )
            = ( ord_less_eq_rat @ A @ B2 ) ) ) ) ) ).

% power_mono_iff
thf(fact_1736_power__mono__iff,axiom,
    ! [A: nat,B2: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
            = ( ord_less_eq_nat @ A @ B2 ) ) ) ) ) ).

% power_mono_iff
thf(fact_1737_power__mono__iff,axiom,
    ! [A: int,B2: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
            = ( ord_less_eq_int @ A @ B2 ) ) ) ) ) ).

% power_mono_iff
thf(fact_1738_zero__eq__power2,axiom,
    ! [A: rat] :
      ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% zero_eq_power2
thf(fact_1739_zero__eq__power2,axiom,
    ! [A: nat] :
      ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% zero_eq_power2
thf(fact_1740_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% zero_eq_power2
thf(fact_1741_zero__eq__power2,axiom,
    ! [A: int] :
      ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% zero_eq_power2
thf(fact_1742_zero__eq__power2,axiom,
    ! [A: complex] :
      ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% zero_eq_power2
thf(fact_1743_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_1744_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_1745_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_1746_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_1747_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_1748_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_1749_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_1750_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_1751_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_1752_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_1753_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_1754_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_1755_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_1756_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_1757_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_1758_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_1759_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_1760_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
thf(fact_1761_power2__less__eq__zero__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
thf(fact_1762_power2__less__eq__zero__iff,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
thf(fact_1763_power2__eq__iff__nonneg,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X3 = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_1764_power2__eq__iff__nonneg,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ( ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X3 = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_1765_power2__eq__iff__nonneg,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
       => ( ( ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X3 = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_1766_power2__eq__iff__nonneg,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X3 = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_1767_zero__less__power2,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A != zero_zero_real ) ) ).

% zero_less_power2
thf(fact_1768_zero__less__power2,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A != zero_zero_rat ) ) ).

% zero_less_power2
thf(fact_1769_zero__less__power2,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A != zero_zero_int ) ) ).

% zero_less_power2
thf(fact_1770_sum__power2__eq__zero__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_rat )
      = ( ( X3 = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_1771_sum__power2__eq__zero__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_real )
      = ( ( X3 = zero_zero_real )
        & ( Y3 = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_1772_sum__power2__eq__zero__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_int )
      = ( ( X3 = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_1773_not__mod__2__eq__0__eq__1,axiom,
    ! [A: nat] :
      ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != zero_zero_nat )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_1774_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != zero_zero_int )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_1775_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_1776_not__mod__2__eq__1__eq__0,axiom,
    ! [A: nat] :
      ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != one_one_nat )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_1777_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != one_one_int )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_1778_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != one_one_Code_integer )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_1779_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_1780_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_1781_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_1782_zero__reorient,axiom,
    ! [X3: complex] :
      ( ( zero_zero_complex = X3 )
      = ( X3 = zero_zero_complex ) ) ).

% zero_reorient
thf(fact_1783_zero__reorient,axiom,
    ! [X3: real] :
      ( ( zero_zero_real = X3 )
      = ( X3 = zero_zero_real ) ) ).

% zero_reorient
thf(fact_1784_zero__reorient,axiom,
    ! [X3: rat] :
      ( ( zero_zero_rat = X3 )
      = ( X3 = zero_zero_rat ) ) ).

% zero_reorient
thf(fact_1785_zero__reorient,axiom,
    ! [X3: nat] :
      ( ( zero_zero_nat = X3 )
      = ( X3 = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_1786_zero__reorient,axiom,
    ! [X3: int] :
      ( ( zero_zero_int = X3 )
      = ( X3 = zero_zero_int ) ) ).

% zero_reorient
thf(fact_1787_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_1788_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B2 ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_1789_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B2 ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_1790_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_1791_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ord_less_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_1792_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
     => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ B2 ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_1793_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( modulo_modulo_nat @ A @ B2 )
        = A )
      = ( ( divide_divide_nat @ A @ B2 )
        = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_1794_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B2: int] :
      ( ( ( modulo_modulo_int @ A @ B2 )
        = A )
      = ( ( divide_divide_int @ A @ B2 )
        = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_1795_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B2 )
        = A )
      = ( ( divide6298287555418463151nteger @ A @ B2 )
        = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_1796_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_1797_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_1798_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( ( modulo_modulo_nat @ M @ D )
        = zero_zero_nat )
     => ? [Q3: nat] :
          ( M
          = ( times_times_nat @ D @ Q3 ) ) ) ).

% mod_eq_0D
thf(fact_1799_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).

% VEBT_internal.naive_member.simps(2)
thf(fact_1800_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).

% VEBT_internal.minNull.simps(1)
thf(fact_1801_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).

% VEBT_internal.minNull.simps(2)
thf(fact_1802_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).

% VEBT_internal.minNull.simps(3)
thf(fact_1803_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B2: $o,X3: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B2 ) @ X3 )
      = ( ( ( X3 = zero_zero_nat )
         => A )
        & ( ( X3 != zero_zero_nat )
         => ( ( ( X3 = one_one_nat )
             => B2 )
            & ( X3 = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
thf(fact_1804_vebt__insert_Osimps_I1_J,axiom,
    ! [X3: nat,A: $o,B2: $o] :
      ( ( ( X3 = zero_zero_nat )
       => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B2 ) @ X3 )
          = ( vEBT_Leaf @ $true @ B2 ) ) )
      & ( ( X3 != zero_zero_nat )
       => ( ( ( X3 = one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B2 ) @ X3 )
              = ( vEBT_Leaf @ A @ $true ) ) )
          & ( ( X3 != one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B2 ) @ X3 )
              = ( vEBT_Leaf @ A @ B2 ) ) ) ) ) ) ).

% vebt_insert.simps(1)
thf(fact_1805_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_1806_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_1807_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_1808_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ B2 )
       => ( ( modulo364778990260209775nteger @ A @ B2 )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_1809_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ B2 )
       => ( ( modulo_modulo_nat @ A @ B2 )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_1810_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ B2 )
       => ( ( modulo_modulo_int @ A @ B2 )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_1811_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = zero_zero_nat )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
thf(fact_1812_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = zero_zero_int )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
thf(fact_1813_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
thf(fact_1814_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_1815_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_1816_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
      = zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(1)
thf(fact_1817_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X5: nat] :
          ( X3
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X5 ) )
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
            ( X3
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT,X5: nat] :
              ( X3
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ X5 ) ) ) ) ).

% VEBT_internal.naive_member.cases
thf(fact_1818_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_1819_div__less__mono,axiom,
    ! [A4: nat,B5: nat,N: nat] :
      ( ( ord_less_nat @ A4 @ B5 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( modulo_modulo_nat @ A4 @ N )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B5 @ N )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A4 @ N ) @ ( divide_divide_nat @ B5 @ N ) ) ) ) ) ) ).

% div_less_mono
thf(fact_1820_invar__vebt_Ointros_I1_J,axiom,
    ! [A: $o,B2: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B2 ) @ ( suc @ zero_zero_nat ) ) ).

% invar_vebt.intros(1)
thf(fact_1821_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_1822_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_1823_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_1824_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_1825_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_1826_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_1827_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_1828_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_1829_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_1830_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_1831_mod__mult__eq,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ).

% mod_mult_eq
thf(fact_1832_mod__mult__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C ) ) ).

% mod_mult_eq
thf(fact_1833_mod__mult__eq,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ).

% mod_mult_eq
thf(fact_1834_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A2: nat,B2: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A2 @ C ) )
     => ( ( ( modulo_modulo_nat @ B2 @ C )
          = ( modulo_modulo_nat @ B @ C ) )
       => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C )
          = ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_1835_mod__mult__cong,axiom,
    ! [A: int,C: int,A2: int,B2: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A2 @ C ) )
     => ( ( ( modulo_modulo_int @ B2 @ C )
          = ( modulo_modulo_int @ B @ C ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C )
          = ( modulo_modulo_int @ ( times_times_int @ A2 @ B ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_1836_mod__mult__cong,axiom,
    ! [A: code_integer,C: code_integer,A2: code_integer,B2: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A2 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B2 @ C )
          = ( modulo364778990260209775nteger @ B @ C ) )
       => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
          = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_1837_mod__mult__mult2,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
      = ( times_times_nat @ ( modulo_modulo_nat @ A @ B2 ) @ C ) ) ).

% mod_mult_mult2
thf(fact_1838_mod__mult__mult2,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
      = ( times_times_int @ ( modulo_modulo_int @ A @ B2 ) @ C ) ) ).

% mod_mult_mult2
thf(fact_1839_mod__mult__mult2,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B2 @ C ) )
      = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ C ) ) ).

% mod_mult_mult2
thf(fact_1840_mult__mod__right,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B2 ) )
      = ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ).

% mult_mod_right
thf(fact_1841_mult__mod__right,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B2 ) )
      = ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ).

% mult_mod_right
thf(fact_1842_mult__mod__right,axiom,
    ! [C: code_integer,A: code_integer,B2: code_integer] :
      ( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B2 ) )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B2 ) ) ) ).

% mult_mod_right
thf(fact_1843_mod__mult__left__eq,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B2 ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_1844_mod__mult__left__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B2 ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_1845_mod__mult__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B2 ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_1846_mod__mult__right__eq,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_1847_mod__mult__right__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_1848_mod__mult__right__eq,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_1849_mod__add__eq,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).

% mod_add_eq
thf(fact_1850_mod__add__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).

% mod_add_eq
thf(fact_1851_mod__add__eq,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C ) ) ).

% mod_add_eq
thf(fact_1852_mod__add__cong,axiom,
    ! [A: nat,C: nat,A2: nat,B2: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A2 @ C ) )
     => ( ( ( modulo_modulo_nat @ B2 @ C )
          = ( modulo_modulo_nat @ B @ C ) )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
          = ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ B ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_1853_mod__add__cong,axiom,
    ! [A: int,C: int,A2: int,B2: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A2 @ C ) )
     => ( ( ( modulo_modulo_int @ B2 @ C )
          = ( modulo_modulo_int @ B @ C ) )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C )
          = ( modulo_modulo_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_1854_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A2: code_integer,B2: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A2 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B2 @ C )
          = ( modulo364778990260209775nteger @ B @ C ) )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
          = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_1855_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B2 ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).

% mod_add_left_eq
thf(fact_1856_mod__add__left__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B2 ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).

% mod_add_left_eq
thf(fact_1857_mod__add__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B2 ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C ) ) ).

% mod_add_left_eq
thf(fact_1858_mod__add__right__eq,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).

% mod_add_right_eq
thf(fact_1859_mod__add__right__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).

% mod_add_right_eq
thf(fact_1860_mod__add__right__eq,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C ) ) ).

% mod_add_right_eq
thf(fact_1861_mod__diff__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).

% mod_diff_eq
thf(fact_1862_mod__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C ) ) ).

% mod_diff_eq
thf(fact_1863_mod__diff__cong,axiom,
    ! [A: int,C: int,A2: int,B2: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A2 @ C ) )
     => ( ( ( modulo_modulo_int @ B2 @ C )
          = ( modulo_modulo_int @ B @ C ) )
       => ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C )
          = ( modulo_modulo_int @ ( minus_minus_int @ A2 @ B ) @ C ) ) ) ) ).

% mod_diff_cong
thf(fact_1864_mod__diff__cong,axiom,
    ! [A: code_integer,C: code_integer,A2: code_integer,B2: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A2 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B2 @ C )
          = ( modulo364778990260209775nteger @ B @ C ) )
       => ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C )
          = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ C ) ) ) ) ).

% mod_diff_cong
thf(fact_1865_mod__diff__left__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B2 ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).

% mod_diff_left_eq
thf(fact_1866_mod__diff__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B2 ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C ) ) ).

% mod_diff_left_eq
thf(fact_1867_mod__diff__right__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).

% mod_diff_right_eq
thf(fact_1868_mod__diff__right__eq,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C ) ) ).

% mod_diff_right_eq
thf(fact_1869_power__mod,axiom,
    ! [A: nat,B2: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B2 ) @ N ) @ B2 )
      = ( modulo_modulo_nat @ ( power_power_nat @ A @ N ) @ B2 ) ) ).

% power_mod
thf(fact_1870_power__mod,axiom,
    ! [A: int,B2: int,N: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B2 ) @ N ) @ B2 )
      = ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ B2 ) ) ).

% power_mod
thf(fact_1871_power__mod,axiom,
    ! [A: code_integer,B2: code_integer,N: nat] :
      ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ N ) @ B2 )
      = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N ) @ B2 ) ) ).

% power_mod
thf(fact_1872_VEBT_Oexhaust,axiom,
    ! [Y3: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
          ( Y3
         != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
     => ~ ! [X21: $o,X222: $o] :
            ( Y3
           != ( vEBT_Leaf @ X21 @ X222 ) ) ) ).

% VEBT.exhaust
thf(fact_1873_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X212: $o,X223: $o] :
      ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
     != ( vEBT_Leaf @ X212 @ X223 ) ) ).

% VEBT.distinct(1)
thf(fact_1874_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,D3: nat] :
          ( X3
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D3 ) )
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
            ( X3
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Deg3 ) ) ) ).

% VEBT_internal.valid'.cases
thf(fact_1875_mod__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% mod_Suc_eq
thf(fact_1876_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).

% mod_Suc_Suc_eq
thf(fact_1877_mod__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).

% mod_less_eq_dividend
thf(fact_1878_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] :
      ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat )
      = none_nat ) ).

% vebt_pred.simps(1)
thf(fact_1879_le__numeral__extra_I3_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% le_numeral_extra(3)
thf(fact_1880_le__numeral__extra_I3_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).

% le_numeral_extra(3)
thf(fact_1881_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_1882_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_1883_zero__le,axiom,
    ! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).

% zero_le
thf(fact_1884_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).

% less_numeral_extra(3)
thf(fact_1885_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).

% less_numeral_extra(3)
thf(fact_1886_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_1887_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_1888_field__lbound__gt__zero,axiom,
    ! [D1: real,D22: real] :
      ( ( ord_less_real @ zero_zero_real @ D1 )
     => ( ( ord_less_real @ zero_zero_real @ D22 )
       => ? [E2: real] :
            ( ( ord_less_real @ zero_zero_real @ E2 )
            & ( ord_less_real @ E2 @ D1 )
            & ( ord_less_real @ E2 @ D22 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_1889_field__lbound__gt__zero,axiom,
    ! [D1: rat,D22: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ D1 )
     => ( ( ord_less_rat @ zero_zero_rat @ D22 )
       => ? [E2: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ E2 )
            & ( ord_less_rat @ E2 @ D1 )
            & ( ord_less_rat @ E2 @ D22 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_1890_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_1891_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_1892_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_1893_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_1894_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_complex
     != ( numera6690914467698888265omplex @ N ) ) ).

% zero_neq_numeral
thf(fact_1895_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_real
     != ( numeral_numeral_real @ N ) ) ).

% zero_neq_numeral
thf(fact_1896_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_rat
     != ( numeral_numeral_rat @ N ) ) ).

% zero_neq_numeral
thf(fact_1897_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_nat
     != ( numeral_numeral_nat @ N ) ) ).

% zero_neq_numeral
thf(fact_1898_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_int
     != ( numeral_numeral_int @ N ) ) ).

% zero_neq_numeral
thf(fact_1899_mult__not__zero,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( times_times_complex @ A @ B2 )
       != zero_zero_complex )
     => ( ( A != zero_zero_complex )
        & ( B2 != zero_zero_complex ) ) ) ).

% mult_not_zero
thf(fact_1900_mult__not__zero,axiom,
    ! [A: real,B2: real] :
      ( ( ( times_times_real @ A @ B2 )
       != zero_zero_real )
     => ( ( A != zero_zero_real )
        & ( B2 != zero_zero_real ) ) ) ).

% mult_not_zero
thf(fact_1901_mult__not__zero,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( times_times_rat @ A @ B2 )
       != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B2 != zero_zero_rat ) ) ) ).

% mult_not_zero
thf(fact_1902_mult__not__zero,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( times_times_nat @ A @ B2 )
       != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B2 != zero_zero_nat ) ) ) ).

% mult_not_zero
thf(fact_1903_mult__not__zero,axiom,
    ! [A: int,B2: int] :
      ( ( ( times_times_int @ A @ B2 )
       != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B2 != zero_zero_int ) ) ) ).

% mult_not_zero
thf(fact_1904_divisors__zero,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( times_times_complex @ A @ B2 )
        = zero_zero_complex )
     => ( ( A = zero_zero_complex )
        | ( B2 = zero_zero_complex ) ) ) ).

% divisors_zero
thf(fact_1905_divisors__zero,axiom,
    ! [A: real,B2: real] :
      ( ( ( times_times_real @ A @ B2 )
        = zero_zero_real )
     => ( ( A = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% divisors_zero
thf(fact_1906_divisors__zero,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( times_times_rat @ A @ B2 )
        = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% divisors_zero
thf(fact_1907_divisors__zero,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( times_times_nat @ A @ B2 )
        = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B2 = zero_zero_nat ) ) ) ).

% divisors_zero
thf(fact_1908_divisors__zero,axiom,
    ! [A: int,B2: int] :
      ( ( ( times_times_int @ A @ B2 )
        = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B2 = zero_zero_int ) ) ) ).

% divisors_zero
thf(fact_1909_no__zero__divisors,axiom,
    ! [A: complex,B2: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B2 != zero_zero_complex )
       => ( ( times_times_complex @ A @ B2 )
         != zero_zero_complex ) ) ) ).

% no_zero_divisors
thf(fact_1910_no__zero__divisors,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( B2 != zero_zero_real )
       => ( ( times_times_real @ A @ B2 )
         != zero_zero_real ) ) ) ).

% no_zero_divisors
thf(fact_1911_no__zero__divisors,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( ( times_times_rat @ A @ B2 )
         != zero_zero_rat ) ) ) ).

% no_zero_divisors
thf(fact_1912_no__zero__divisors,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B2 != zero_zero_nat )
       => ( ( times_times_nat @ A @ B2 )
         != zero_zero_nat ) ) ) ).

% no_zero_divisors
thf(fact_1913_no__zero__divisors,axiom,
    ! [A: int,B2: int] :
      ( ( A != zero_zero_int )
     => ( ( B2 != zero_zero_int )
       => ( ( times_times_int @ A @ B2 )
         != zero_zero_int ) ) ) ).

% no_zero_divisors
thf(fact_1914_mult__left__cancel,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( times_times_complex @ C @ A )
          = ( times_times_complex @ C @ B2 ) )
        = ( A = B2 ) ) ) ).

% mult_left_cancel
thf(fact_1915_mult__left__cancel,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ C @ A )
          = ( times_times_real @ C @ B2 ) )
        = ( A = B2 ) ) ) ).

% mult_left_cancel
thf(fact_1916_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ C @ A )
          = ( times_times_rat @ C @ B2 ) )
        = ( A = B2 ) ) ) ).

% mult_left_cancel
thf(fact_1917_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ C @ A )
          = ( times_times_nat @ C @ B2 ) )
        = ( A = B2 ) ) ) ).

% mult_left_cancel
thf(fact_1918_mult__left__cancel,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ C @ A )
          = ( times_times_int @ C @ B2 ) )
        = ( A = B2 ) ) ) ).

% mult_left_cancel
thf(fact_1919_mult__right__cancel,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( times_times_complex @ A @ C )
          = ( times_times_complex @ B2 @ C ) )
        = ( A = B2 ) ) ) ).

% mult_right_cancel
thf(fact_1920_mult__right__cancel,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A @ C )
          = ( times_times_real @ B2 @ C ) )
        = ( A = B2 ) ) ) ).

% mult_right_cancel
thf(fact_1921_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A @ C )
          = ( times_times_rat @ B2 @ C ) )
        = ( A = B2 ) ) ) ).

% mult_right_cancel
thf(fact_1922_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ A @ C )
          = ( times_times_nat @ B2 @ C ) )
        = ( A = B2 ) ) ) ).

% mult_right_cancel
thf(fact_1923_mult__right__cancel,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ A @ C )
          = ( times_times_int @ B2 @ C ) )
        = ( A = B2 ) ) ) ).

% mult_right_cancel
thf(fact_1924_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
thf(fact_1925_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
thf(fact_1926_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
thf(fact_1927_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_1928_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_1929_verit__sum__simplify,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% verit_sum_simplify
thf(fact_1930_verit__sum__simplify,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% verit_sum_simplify
thf(fact_1931_verit__sum__simplify,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% verit_sum_simplify
thf(fact_1932_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_1933_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_1934_add_Ogroup__left__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_1935_add_Ogroup__left__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_1936_add_Ogroup__left__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_1937_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_1938_add_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% add.comm_neutral
thf(fact_1939_add_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.comm_neutral
thf(fact_1940_add_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.comm_neutral
thf(fact_1941_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_1942_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_1943_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_1944_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_1945_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_1946_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_1947_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_1948_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: complex,Z5: complex] : Y6 = Z5 )
    = ( ^ [A5: complex,B4: complex] :
          ( ( minus_minus_complex @ A5 @ B4 )
          = zero_zero_complex ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1949_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: real,Z5: real] : Y6 = Z5 )
    = ( ^ [A5: real,B4: real] :
          ( ( minus_minus_real @ A5 @ B4 )
          = zero_zero_real ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1950_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: rat,Z5: rat] : Y6 = Z5 )
    = ( ^ [A5: rat,B4: rat] :
          ( ( minus_minus_rat @ A5 @ B4 )
          = zero_zero_rat ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1951_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: int,Z5: int] : Y6 = Z5 )
    = ( ^ [A5: int,B4: int] :
          ( ( minus_minus_int @ A5 @ B4 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1952_power__not__zero,axiom,
    ! [A: rat,N: nat] :
      ( ( A != zero_zero_rat )
     => ( ( power_power_rat @ A @ N )
       != zero_zero_rat ) ) ).

% power_not_zero
thf(fact_1953_power__not__zero,axiom,
    ! [A: nat,N: nat] :
      ( ( A != zero_zero_nat )
     => ( ( power_power_nat @ A @ N )
       != zero_zero_nat ) ) ).

% power_not_zero
thf(fact_1954_power__not__zero,axiom,
    ! [A: real,N: nat] :
      ( ( A != zero_zero_real )
     => ( ( power_power_real @ A @ N )
       != zero_zero_real ) ) ).

% power_not_zero
thf(fact_1955_power__not__zero,axiom,
    ! [A: int,N: nat] :
      ( ( A != zero_zero_int )
     => ( ( power_power_int @ A @ N )
       != zero_zero_int ) ) ).

% power_not_zero
thf(fact_1956_power__not__zero,axiom,
    ! [A: complex,N: nat] :
      ( ( A != zero_zero_complex )
     => ( ( power_power_complex @ A @ N )
       != zero_zero_complex ) ) ).

% power_not_zero
thf(fact_1957_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) ).

% VEBT_internal.membermima.simps(2)
thf(fact_1958_num_Osize_I4_J,axiom,
    ( ( size_size_num @ one )
    = zero_zero_nat ) ).

% num.size(4)
thf(fact_1959_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] :
      ( zero_zero_nat
     != ( suc @ X2 ) ) ).

% nat.distinct(1)
thf(fact_1960_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_1961_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_1962_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat
        = ( suc @ X2 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_1963_old_Onat_Oexhaust,axiom,
    ! [Y3: nat] :
      ( ( Y3 != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y3
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_1964_nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( P @ N3 )
           => ( P @ ( suc @ N3 ) ) )
       => ( P @ N ) ) ) ).

% nat_induct
thf(fact_1965_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [X5: nat] : ( P @ X5 @ zero_zero_nat )
     => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
       => ( ! [X5: nat,Y4: nat] :
              ( ( P @ X5 @ Y4 )
             => ( P @ ( suc @ X5 ) @ ( suc @ Y4 ) ) )
         => ( P @ M @ N ) ) ) ) ).

% diff_induct
thf(fact_1966_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N3: nat] :
            ( ( P @ ( suc @ N3 ) )
           => ( P @ N3 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_1967_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_1968_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_1969_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_1970_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M4: nat] :
          ( N
          = ( suc @ M4 ) ) ) ).

% not0_implies_Suc
thf(fact_1971_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( ~ ( P @ N3 )
             => ? [M2: nat] :
                  ( ( ord_less_nat @ M2 @ N3 )
                  & ~ ( P @ M2 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_1972_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_1973_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_1974_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_1975_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_1976_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_1977_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_1978_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_1979_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_1980_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_1981_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_1982_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_1983_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_1984_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_1985_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_1986_mult__0,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% mult_0
thf(fact_1987_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_1988_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_1989_power__eq__imp__eq__base,axiom,
    ! [A: real,N: nat,B2: real] :
      ( ( ( power_power_real @ A @ N )
        = ( power_power_real @ B2 @ N ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_1990_power__eq__imp__eq__base,axiom,
    ! [A: rat,N: nat,B2: rat] :
      ( ( ( power_power_rat @ A @ N )
        = ( power_power_rat @ B2 @ N ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_1991_power__eq__imp__eq__base,axiom,
    ! [A: nat,N: nat,B2: nat] :
      ( ( ( power_power_nat @ A @ N )
        = ( power_power_nat @ B2 @ N ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_1992_power__eq__imp__eq__base,axiom,
    ! [A: int,N: nat,B2: int] :
      ( ( ( power_power_int @ A @ N )
        = ( power_power_int @ B2 @ N ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_1993_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: real,B2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
         => ( ( ( power_power_real @ A @ N )
              = ( power_power_real @ B2 @ N ) )
            = ( A = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_1994_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: rat,B2: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
         => ( ( ( power_power_rat @ A @ N )
              = ( power_power_rat @ B2 @ N ) )
            = ( A = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_1995_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
         => ( ( ( power_power_nat @ A @ N )
              = ( power_power_nat @ B2 @ N ) )
            = ( A = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_1996_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: int,B2: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
         => ( ( ( power_power_int @ A @ N )
              = ( power_power_int @ B2 @ N ) )
            = ( A = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_1997_lambda__zero,axiom,
    ( ( ^ [H: complex] : zero_zero_complex )
    = ( times_times_complex @ zero_zero_complex ) ) ).

% lambda_zero
thf(fact_1998_lambda__zero,axiom,
    ( ( ^ [H: real] : zero_zero_real )
    = ( times_times_real @ zero_zero_real ) ) ).

% lambda_zero
thf(fact_1999_lambda__zero,axiom,
    ( ( ^ [H: rat] : zero_zero_rat )
    = ( times_times_rat @ zero_zero_rat ) ) ).

% lambda_zero
thf(fact_2000_lambda__zero,axiom,
    ( ( ^ [H: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_2001_lambda__zero,axiom,
    ( ( ^ [H: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_2002_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) @ X3 )
      = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) ) ).

% vebt_insert.simps(2)
thf(fact_2003_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X3: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb ) @ X3 )
      = ( ( X3 = Mi )
        | ( X3 = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
thf(fact_2004_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_2005_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B2 @ C ) )
        = ( plus_plus_nat @ ( times_times_nat @ B2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) @ ( modulo_modulo_nat @ A @ B2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_2006_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B2 @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B2 @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) @ ( modulo_modulo_int @ A @ B2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_2007_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_2008_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_2009_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_2010_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_2011_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_2012_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_2013_mod__eqE,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ B2 @ C ) )
     => ~ ! [D3: int] :
            ( B2
           != ( plus_plus_int @ A @ ( times_times_int @ C @ D3 ) ) ) ) ).

% mod_eqE
thf(fact_2014_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ B2 @ C ) )
     => ~ ! [D3: code_integer] :
            ( B2
           != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D3 ) ) ) ) ).

% mod_eqE
thf(fact_2015_div__add1__eq,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B2 @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_2016_div__add1__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
      = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_2017_div__add1__eq,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_2018_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_2019_mod__induct,axiom,
    ! [P: nat > $o,N: nat,P4: nat,M: nat] :
      ( ( P @ N )
     => ( ( ord_less_nat @ N @ P4 )
       => ( ( ord_less_nat @ M @ P4 )
         => ( ! [N3: nat] :
                ( ( ord_less_nat @ N3 @ P4 )
               => ( ( P @ N3 )
                 => ( P @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P4 ) ) ) )
           => ( P @ M ) ) ) ) ) ).

% mod_induct
thf(fact_2020_mod__Suc__le__divisor,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).

% mod_Suc_le_divisor
thf(fact_2021_power__strict__mono,axiom,
    ! [A: real,B2: real,N: nat] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_2022_power__strict__mono,axiom,
    ! [A: rat,B2: rat,N: nat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_2023_power__strict__mono,axiom,
    ! [A: nat,B2: nat,N: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_2024_power__strict__mono,axiom,
    ! [A: int,B2: int,N: nat] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_2025_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( ord_less_nat @ M3 @ N2 ) @ M3 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M3 @ N2 ) @ N2 ) ) ) ) ).

% mod_if
thf(fact_2026_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_2027_vebt__insert_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X5: nat] :
          ( X3
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X5 ) )
     => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X5: nat] :
            ( X3
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) @ X5 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X5: nat] :
              ( X3
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ X5 ) )
         => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
                ( X3
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ X5 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
                  ( X3
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X5 ) ) ) ) ) ) ).

% vebt_insert.cases
thf(fact_2028_vebt__member_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X5: nat] :
          ( X3
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X5 ) )
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X5: nat] :
            ( X3
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X5 ) )
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X5: nat] :
              ( X3
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X5 ) )
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
                ( X3
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X5 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
                  ( X3
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X5 ) ) ) ) ) ) ).

% vebt_member.cases
thf(fact_2029_le__mod__geq,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( modulo_modulo_nat @ M @ N )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).

% le_mod_geq
thf(fact_2030_nat__mod__eq__iff,axiom,
    ! [X3: nat,N: nat,Y3: nat] :
      ( ( ( modulo_modulo_nat @ X3 @ N )
        = ( modulo_modulo_nat @ Y3 @ N ) )
      = ( ? [Q1: nat,Q22: nat] :
            ( ( plus_plus_nat @ X3 @ ( times_times_nat @ N @ Q1 ) )
            = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).

% nat_mod_eq_iff
thf(fact_2031_VEBT__internal_OminNull_Ocases,axiom,
    ! [X3: vEBT_VEBT] :
      ( ( X3
       != ( vEBT_Leaf @ $false @ $false ) )
     => ( ! [Uv2: $o] :
            ( X3
           != ( vEBT_Leaf @ $true @ Uv2 ) )
       => ( ! [Uu2: $o] :
              ( X3
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X3
               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X3
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.cases
thf(fact_2032_vebt__pred_Osimps_I2_J,axiom,
    ! [A: $o,Uw: $o] :
      ( ( A
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
          = none_nat ) ) ) ).

% vebt_pred.simps(2)
thf(fact_2033_vebt__mint_Osimps_I1_J,axiom,
    ! [A: $o,B2: $o] :
      ( ( A
       => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B2 ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A
       => ( ( B2
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B2 ) )
              = ( some_nat @ one_one_nat ) ) )
          & ( ~ B2
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B2 ) )
              = none_nat ) ) ) ) ) ).

% vebt_mint.simps(1)
thf(fact_2034_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).

% not_numeral_le_zero
thf(fact_2035_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).

% not_numeral_le_zero
thf(fact_2036_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).

% not_numeral_le_zero
thf(fact_2037_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).

% not_numeral_le_zero
thf(fact_2038_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).

% zero_le_numeral
thf(fact_2039_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).

% zero_le_numeral
thf(fact_2040_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).

% zero_le_numeral
thf(fact_2041_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).

% zero_le_numeral
thf(fact_2042_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2043_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2044_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2045_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2046_zero__le__mult__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) ) ) ) ).

% zero_le_mult_iff
thf(fact_2047_zero__le__mult__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) ) ) ).

% zero_le_mult_iff
thf(fact_2048_zero__le__mult__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ zero_zero_int @ B2 ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ B2 @ zero_zero_int ) ) ) ) ).

% zero_le_mult_iff
thf(fact_2049_mult__nonneg__nonpos2,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ B2 @ A ) @ zero_zero_real ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2050_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ B2 @ A ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2051_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ B2 @ A ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2052_mult__nonneg__nonpos2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ B2 @ A ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2053_mult__nonpos__nonneg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2054_mult__nonpos__nonneg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2055_mult__nonpos__nonneg,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2056_mult__nonpos__nonneg,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2057_mult__nonneg__nonpos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2058_mult__nonneg__nonpos,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2059_mult__nonneg__nonpos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2060_mult__nonneg__nonpos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2061_mult__nonneg__nonneg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2062_mult__nonneg__nonneg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2063_mult__nonneg__nonneg,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2064_mult__nonneg__nonneg,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2065_split__mult__neg__le,axiom,
    ! [A: real,B2: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) ) )
     => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ).

% split_mult_neg_le
thf(fact_2066_split__mult__neg__le,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ).

% split_mult_neg_le
thf(fact_2067_split__mult__neg__le,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
          & ( ord_less_eq_nat @ B2 @ zero_zero_nat ) )
        | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
          & ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ).

% split_mult_neg_le
thf(fact_2068_split__mult__neg__le,axiom,
    ! [A: int,B2: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ B2 @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B2 ) ) )
     => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ).

% split_mult_neg_le
thf(fact_2069_mult__le__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ) ) ).

% mult_le_0_iff
thf(fact_2070_mult__le__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ) ) ).

% mult_le_0_iff
thf(fact_2071_mult__le__0__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ B2 @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ) ) ).

% mult_le_0_iff
thf(fact_2072_mult__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2073_mult__right__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2074_mult__right__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2075_mult__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2076_mult__right__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2077_mult__right__mono__neg,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B2 @ A )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2078_mult__right__mono__neg,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2079_mult__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_2080_mult__left__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_2081_mult__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_2082_mult__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_2083_mult__nonpos__nonpos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2084_mult__nonpos__nonpos,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2085_mult__nonpos__nonpos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2086_mult__left__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2087_mult__left__mono__neg,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B2 @ A )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2088_mult__left__mono__neg,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2089_split__mult__pos__le,axiom,
    ! [A: real,B2: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ).

% split_mult_pos_le
thf(fact_2090_split__mult__pos__le,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ).

% split_mult_pos_le
thf(fact_2091_split__mult__pos__le,axiom,
    ! [A: int,B2: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ zero_zero_int @ B2 ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ).

% split_mult_pos_le
thf(fact_2092_zero__le__square,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).

% zero_le_square
thf(fact_2093_zero__le__square,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).

% zero_le_square
thf(fact_2094_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_2095_mult__mono_H,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2096_mult__mono_H,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2097_mult__mono_H,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2098_mult__mono_H,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2099_mult__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ 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 @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2100_mult__mono,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ 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 @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2101_mult__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ 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 @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2102_mult__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ 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 @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2103_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).

% not_numeral_less_zero
thf(fact_2104_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).

% not_numeral_less_zero
thf(fact_2105_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_2106_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).

% not_numeral_less_zero
thf(fact_2107_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).

% zero_less_numeral
thf(fact_2108_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).

% zero_less_numeral
thf(fact_2109_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).

% zero_less_numeral
thf(fact_2110_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).

% zero_less_numeral
thf(fact_2111_not__one__le__zero,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% not_one_le_zero
thf(fact_2112_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_2113_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_2114_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_2115_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_2116_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_2117_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_2118_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_2119_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_2120_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_2121_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_2122_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_2123_add__decreasing,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ C @ B2 )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_2124_add__decreasing,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ C @ B2 )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_2125_add__decreasing,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_2126_add__decreasing,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ C @ B2 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_2127_add__increasing,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_2128_add__increasing,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ord_less_eq_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_2129_add__increasing,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_2130_add__increasing,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_2131_add__decreasing2,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_eq_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ B2 )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_2132_add__decreasing2,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ B2 )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_2133_add__decreasing2,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A @ B2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_2134_add__decreasing2,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ A @ B2 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_2135_add__increasing2,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ B2 @ A )
       => ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_2136_add__increasing2,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ B2 @ A )
       => ( ord_less_eq_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_2137_add__increasing2,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B2 @ A )
       => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_2138_add__increasing2,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ B2 @ A )
       => ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_2139_add__nonneg__nonneg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_2140_add__nonneg__nonneg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_2141_add__nonneg__nonneg,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_2142_add__nonneg__nonneg,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_2143_add__nonpos__nonpos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).

% add_nonpos_nonpos
thf(fact_2144_add__nonpos__nonpos,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).

% add_nonpos_nonpos
thf(fact_2145_add__nonpos__nonpos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_2146_add__nonpos__nonpos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% add_nonpos_nonpos
thf(fact_2147_add__nonneg__eq__0__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ( plus_plus_real @ X3 @ Y3 )
            = zero_zero_real )
          = ( ( X3 = zero_zero_real )
            & ( Y3 = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2148_add__nonneg__eq__0__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ( ( plus_plus_rat @ X3 @ Y3 )
            = zero_zero_rat )
          = ( ( X3 = zero_zero_rat )
            & ( Y3 = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2149_add__nonneg__eq__0__iff,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
       => ( ( ( plus_plus_nat @ X3 @ Y3 )
            = zero_zero_nat )
          = ( ( X3 = zero_zero_nat )
            & ( Y3 = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2150_add__nonneg__eq__0__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( ( plus_plus_int @ X3 @ Y3 )
            = zero_zero_int )
          = ( ( X3 = zero_zero_int )
            & ( Y3 = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2151_add__nonpos__eq__0__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ X3 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
       => ( ( ( plus_plus_real @ X3 @ Y3 )
            = zero_zero_real )
          = ( ( X3 = zero_zero_real )
            & ( Y3 = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2152_add__nonpos__eq__0__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
       => ( ( ( plus_plus_rat @ X3 @ Y3 )
            = zero_zero_rat )
          = ( ( X3 = zero_zero_rat )
            & ( Y3 = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2153_add__nonpos__eq__0__iff,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X3 @ Y3 )
            = zero_zero_nat )
          = ( ( X3 = zero_zero_nat )
            & ( Y3 = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2154_add__nonpos__eq__0__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ X3 @ zero_zero_int )
     => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
       => ( ( ( plus_plus_int @ X3 @ Y3 )
            = zero_zero_int )
          = ( ( X3 = zero_zero_int )
            & ( Y3 = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2155_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2156_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2157_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2158_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2159_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A @ B2 ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B2 @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2160_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A @ B2 ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B2 @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2161_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A @ B2 ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B2 @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2162_mult__strict__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2163_mult__strict__right__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2164_mult__strict__right__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2165_mult__strict__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2166_mult__strict__right__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2167_mult__strict__right__mono__neg,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B2 @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2168_mult__strict__right__mono__neg,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_int @ B2 @ A )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2169_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A @ B2 ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B2 @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2170_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A @ B2 ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B2 @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2171_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A @ B2 ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B2 @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2172_mult__strict__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2173_mult__strict__left__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2174_mult__strict__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2175_mult__strict__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2176_mult__strict__left__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2177_mult__strict__left__mono__neg,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B2 @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2178_mult__strict__left__mono__neg,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_int @ B2 @ A )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2179_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
        = ( ord_less_real @ A @ B2 ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2180_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
        = ( ord_less_rat @ A @ B2 ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2181_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
        = ( ord_less_int @ A @ B2 ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2182_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
        = ( ord_less_real @ B2 @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2183_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
        = ( ord_less_rat @ B2 @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2184_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
        = ( ord_less_int @ B2 @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2185_zero__less__mult__pos2,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B2 @ A ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).

% zero_less_mult_pos2
thf(fact_2186_zero__less__mult__pos2,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B2 @ A ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ).

% zero_less_mult_pos2
thf(fact_2187_zero__less__mult__pos2,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B2 @ A ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).

% zero_less_mult_pos2
thf(fact_2188_zero__less__mult__pos2,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B2 @ A ) )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).

% zero_less_mult_pos2
thf(fact_2189_zero__less__mult__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).

% zero_less_mult_pos
thf(fact_2190_zero__less__mult__pos,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ).

% zero_less_mult_pos
thf(fact_2191_zero__less__mult__pos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).

% zero_less_mult_pos
thf(fact_2192_zero__less__mult__pos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).

% zero_less_mult_pos
thf(fact_2193_zero__less__mult__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ zero_zero_real @ B2 ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B2 @ zero_zero_real ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2194_zero__less__mult__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ zero_zero_rat @ B2 ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B2 @ zero_zero_rat ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2195_zero__less__mult__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ A )
          & ( ord_less_int @ zero_zero_int @ B2 ) )
        | ( ( ord_less_int @ A @ zero_zero_int )
          & ( ord_less_int @ B2 @ zero_zero_int ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2196_mult__pos__neg2,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ B2 @ A ) @ zero_zero_real ) ) ) ).

% mult_pos_neg2
thf(fact_2197_mult__pos__neg2,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ B2 @ A ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg2
thf(fact_2198_mult__pos__neg2,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ B2 @ A ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg2
thf(fact_2199_mult__pos__neg2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ B2 @ A ) @ zero_zero_int ) ) ) ).

% mult_pos_neg2
thf(fact_2200_mult__pos__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).

% mult_pos_pos
thf(fact_2201_mult__pos__pos,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).

% mult_pos_pos
thf(fact_2202_mult__pos__pos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) ) ) ) ).

% mult_pos_pos
thf(fact_2203_mult__pos__pos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).

% mult_pos_pos
thf(fact_2204_mult__pos__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).

% mult_pos_neg
thf(fact_2205_mult__pos__neg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg
thf(fact_2206_mult__pos__neg,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg
thf(fact_2207_mult__pos__neg,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% mult_pos_neg
thf(fact_2208_mult__neg__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).

% mult_neg_pos
thf(fact_2209_mult__neg__pos,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).

% mult_neg_pos
thf(fact_2210_mult__neg__pos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% mult_neg_pos
thf(fact_2211_mult__neg__pos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% mult_neg_pos
thf(fact_2212_mult__less__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B2 @ zero_zero_real ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ zero_zero_real @ B2 ) ) ) ) ).

% mult_less_0_iff
thf(fact_2213_mult__less__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B2 @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ) ).

% mult_less_0_iff
thf(fact_2214_mult__less__0__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int )
      = ( ( ( ord_less_int @ zero_zero_int @ A )
          & ( ord_less_int @ B2 @ zero_zero_int ) )
        | ( ( ord_less_int @ A @ zero_zero_int )
          & ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).

% mult_less_0_iff
thf(fact_2215_not__square__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).

% not_square_less_zero
thf(fact_2216_not__square__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_2217_not__square__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_2218_mult__neg__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).

% mult_neg_neg
thf(fact_2219_mult__neg__neg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).

% mult_neg_neg
thf(fact_2220_mult__neg__neg,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).

% mult_neg_neg
thf(fact_2221_vebt__maxt_Osimps_I1_J,axiom,
    ! [B2: $o,A: $o] :
      ( ( B2
       => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B2 ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B2
       => ( ( A
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B2 ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B2 ) )
              = none_nat ) ) ) ) ) ).

% vebt_maxt.simps(1)
thf(fact_2222_le__iff__diff__le__0,axiom,
    ( ord_less_eq_real
    = ( ^ [A5: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A5 @ B4 ) @ zero_zero_real ) ) ) ).

% le_iff_diff_le_0
thf(fact_2223_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A5: rat,B4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A5 @ B4 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_2224_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_2225_less__numeral__extra_I1_J,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% less_numeral_extra(1)
thf(fact_2226_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_2227_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_2228_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_2229_not__one__less__zero,axiom,
    ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).

% not_one_less_zero
thf(fact_2230_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_2231_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_2232_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_2233_zero__less__one,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% zero_less_one
thf(fact_2234_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_2235_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_2236_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_2237_add__less__zeroD,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X3 @ Y3 ) @ zero_zero_real )
     => ( ( ord_less_real @ X3 @ zero_zero_real )
        | ( ord_less_real @ Y3 @ zero_zero_real ) ) ) ).

% add_less_zeroD
thf(fact_2238_add__less__zeroD,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ zero_zero_rat )
     => ( ( ord_less_rat @ X3 @ zero_zero_rat )
        | ( ord_less_rat @ Y3 @ zero_zero_rat ) ) ) ).

% add_less_zeroD
thf(fact_2239_add__less__zeroD,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_int @ ( plus_plus_int @ X3 @ Y3 ) @ zero_zero_int )
     => ( ( ord_less_int @ X3 @ zero_zero_int )
        | ( ord_less_int @ Y3 @ zero_zero_int ) ) ) ).

% add_less_zeroD
thf(fact_2240_add__neg__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).

% add_neg_neg
thf(fact_2241_add__neg__neg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).

% add_neg_neg
thf(fact_2242_add__neg__neg,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_2243_add__neg__neg,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% add_neg_neg
thf(fact_2244_add__pos__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_2245_add__pos__pos,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_2246_add__pos__pos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_2247_add__pos__pos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_2248_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ~ ! [C2: nat] :
            ( ( B2
              = ( plus_plus_nat @ A @ C2 ) )
           => ( C2 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_2249_pos__add__strict,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B2 @ C )
       => ( ord_less_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_2250_pos__add__strict,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_2251_pos__add__strict,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_2252_pos__add__strict,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B2 @ C )
       => ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_2253_divide__right__mono__neg,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_2254_divide__right__mono__neg,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_2255_divide__nonpos__nonpos,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ X3 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_2256_divide__nonpos__nonpos,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_2257_divide__nonpos__nonneg,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ X3 @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).

% divide_nonpos_nonneg
thf(fact_2258_divide__nonpos__nonneg,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_nonneg
thf(fact_2259_divide__nonneg__nonpos,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).

% divide_nonneg_nonpos
thf(fact_2260_divide__nonneg__nonpos,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_nonpos
thf(fact_2261_divide__nonneg__nonneg,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_2262_divide__nonneg__nonneg,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_2263_zero__le__divide__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B2 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) ) ) ) ).

% zero_le_divide_iff
thf(fact_2264_zero__le__divide__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) ) ) ).

% zero_le_divide_iff
thf(fact_2265_divide__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).

% divide_right_mono
thf(fact_2266_divide__right__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ) ).

% divide_right_mono
thf(fact_2267_divide__le__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B2 ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ) ) ).

% divide_le_0_iff
thf(fact_2268_divide__le__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B2 ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ) ) ).

% divide_le_0_iff
thf(fact_2269_less__iff__diff__less__0,axiom,
    ( ord_less_real
    = ( ^ [A5: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A5 @ B4 ) @ zero_zero_real ) ) ) ).

% less_iff_diff_less_0
thf(fact_2270_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A5: rat,B4: rat] : ( ord_less_rat @ ( minus_minus_rat @ A5 @ B4 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_2271_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A5: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_2272_divide__strict__right__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_2273_divide__strict__right__mono__neg,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B2 @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_2274_divide__strict__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).

% divide_strict_right_mono
thf(fact_2275_divide__strict__right__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ) ).

% divide_strict_right_mono
thf(fact_2276_zero__less__divide__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ zero_zero_real @ B2 ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B2 @ zero_zero_real ) ) ) ) ).

% zero_less_divide_iff
thf(fact_2277_zero__less__divide__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ zero_zero_rat @ B2 ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B2 @ zero_zero_rat ) ) ) ) ).

% zero_less_divide_iff
thf(fact_2278_divide__less__cancel,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B2 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_real @ B2 @ A ) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
thf(fact_2279_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B2 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B2 @ A ) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
thf(fact_2280_divide__less__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A @ B2 ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B2 @ zero_zero_real ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ zero_zero_real @ B2 ) ) ) ) ).

% divide_less_0_iff
thf(fact_2281_divide__less__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A @ B2 ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B2 @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ) ).

% divide_less_0_iff
thf(fact_2282_divide__pos__pos,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).

% divide_pos_pos
thf(fact_2283_divide__pos__pos,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).

% divide_pos_pos
thf(fact_2284_divide__pos__neg,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ Y3 @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).

% divide_pos_neg
thf(fact_2285_divide__pos__neg,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_pos_neg
thf(fact_2286_divide__neg__pos,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ord_less_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).

% divide_neg_pos
thf(fact_2287_divide__neg__pos,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ X3 @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_neg_pos
thf(fact_2288_divide__neg__neg,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ zero_zero_real )
     => ( ( ord_less_real @ Y3 @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).

% divide_neg_neg
thf(fact_2289_divide__neg__neg,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ X3 @ zero_zero_rat )
     => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).

% divide_neg_neg
thf(fact_2290_zero__le__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).

% zero_le_power
thf(fact_2291_zero__le__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).

% zero_le_power
thf(fact_2292_zero__le__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).

% zero_le_power
thf(fact_2293_zero__le__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).

% zero_le_power
thf(fact_2294_power__mono,axiom,
    ! [A: real,B2: real,N: nat] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ).

% power_mono
thf(fact_2295_power__mono,axiom,
    ! [A: rat,B2: rat,N: nat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ).

% power_mono
thf(fact_2296_power__mono,axiom,
    ! [A: nat,B2: nat,N: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ).

% power_mono
thf(fact_2297_power__mono,axiom,
    ! [A: int,B2: int,N: nat] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).

% power_mono
thf(fact_2298_zero__less__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).

% zero_less_power
thf(fact_2299_zero__less__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).

% zero_less_power
thf(fact_2300_zero__less__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).

% zero_less_power
thf(fact_2301_zero__less__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).

% zero_less_power
thf(fact_2302_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X3 )
     => ( ! [Uv2: $o] :
            ( X3
           != ( vEBT_Leaf @ $true @ Uv2 ) )
       => ( ! [Uu2: $o] :
              ( X3
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( X3
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).

% VEBT_internal.minNull.elims(3)
thf(fact_2303_frac__eq__eq,axiom,
    ! [Y3: complex,Z: complex,X3: complex,W: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( ( divide1717551699836669952omplex @ X3 @ Y3 )
            = ( divide1717551699836669952omplex @ W @ Z ) )
          = ( ( times_times_complex @ X3 @ Z )
            = ( times_times_complex @ W @ Y3 ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2304_frac__eq__eq,axiom,
    ! [Y3: real,Z: real,X3: real,W: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ( divide_divide_real @ X3 @ Y3 )
            = ( divide_divide_real @ W @ Z ) )
          = ( ( times_times_real @ X3 @ Z )
            = ( times_times_real @ W @ Y3 ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2305_frac__eq__eq,axiom,
    ! [Y3: rat,Z: rat,X3: rat,W: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ( divide_divide_rat @ X3 @ Y3 )
            = ( divide_divide_rat @ W @ Z ) )
          = ( ( times_times_rat @ X3 @ Z )
            = ( times_times_rat @ W @ Y3 ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2306_divide__eq__eq,axiom,
    ! [B2: complex,C: complex,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B2 @ C )
        = A )
      = ( ( ( C != zero_zero_complex )
         => ( B2
            = ( times_times_complex @ A @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq
thf(fact_2307_divide__eq__eq,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( ( divide_divide_real @ B2 @ C )
        = A )
      = ( ( ( C != zero_zero_real )
         => ( B2
            = ( times_times_real @ A @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq
thf(fact_2308_divide__eq__eq,axiom,
    ! [B2: rat,C: rat,A: rat] :
      ( ( ( divide_divide_rat @ B2 @ C )
        = A )
      = ( ( ( C != zero_zero_rat )
         => ( B2
            = ( times_times_rat @ A @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
thf(fact_2309_eq__divide__eq,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( A
        = ( divide1717551699836669952omplex @ B2 @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ A @ C )
            = B2 ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq
thf(fact_2310_eq__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( A
        = ( divide_divide_real @ B2 @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ A @ C )
            = B2 ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq
thf(fact_2311_eq__divide__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( A
        = ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A @ C )
            = B2 ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
thf(fact_2312_divide__eq__imp,axiom,
    ! [C: complex,B2: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( B2
          = ( times_times_complex @ A @ C ) )
       => ( ( divide1717551699836669952omplex @ B2 @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_2313_divide__eq__imp,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( B2
          = ( times_times_real @ A @ C ) )
       => ( ( divide_divide_real @ B2 @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_2314_divide__eq__imp,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B2
          = ( times_times_rat @ A @ C ) )
       => ( ( divide_divide_rat @ B2 @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_2315_eq__divide__imp,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( times_times_complex @ A @ C )
          = B2 )
       => ( A
          = ( divide1717551699836669952omplex @ B2 @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_2316_eq__divide__imp,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A @ C )
          = B2 )
       => ( A
          = ( divide_divide_real @ B2 @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_2317_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A @ C )
          = B2 )
       => ( A
          = ( divide_divide_rat @ B2 @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_2318_nonzero__divide__eq__eq,axiom,
    ! [C: complex,B2: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( divide1717551699836669952omplex @ B2 @ C )
          = A )
        = ( B2
          = ( times_times_complex @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_2319_nonzero__divide__eq__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( ( divide_divide_real @ B2 @ C )
          = A )
        = ( B2
          = ( times_times_real @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_2320_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( divide_divide_rat @ B2 @ C )
          = A )
        = ( B2
          = ( times_times_rat @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_2321_nonzero__eq__divide__eq,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( A
          = ( divide1717551699836669952omplex @ B2 @ C ) )
        = ( ( times_times_complex @ A @ C )
          = B2 ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_2322_nonzero__eq__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( A
          = ( divide_divide_real @ B2 @ C ) )
        = ( ( times_times_real @ A @ C )
          = B2 ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_2323_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A
          = ( divide_divide_rat @ B2 @ C ) )
        = ( ( times_times_rat @ A @ C )
          = B2 ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_2324_right__inverse__eq,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( ( divide1717551699836669952omplex @ A @ B2 )
          = one_one_complex )
        = ( A = B2 ) ) ) ).

% right_inverse_eq
thf(fact_2325_right__inverse__eq,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( ( divide_divide_real @ A @ B2 )
          = one_one_real )
        = ( A = B2 ) ) ) ).

% right_inverse_eq
thf(fact_2326_right__inverse__eq,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( ( divide_divide_rat @ A @ B2 )
          = one_one_rat )
        = ( A = B2 ) ) ) ).

% right_inverse_eq
thf(fact_2327_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X3 )
     => ( ( X3
         != ( vEBT_Leaf @ $false @ $false ) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X3
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).

% VEBT_internal.minNull.elims(2)
thf(fact_2328_power__0,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% power_0
thf(fact_2329_power__0,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_2330_power__0,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% power_0
thf(fact_2331_power__0,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_2332_power__0,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% power_0
thf(fact_2333_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_2334_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M4: nat] :
          ( N
          = ( suc @ M4 ) ) ) ).

% gr0_implies_Suc
thf(fact_2335_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_2336_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M3: nat] :
            ( N
            = ( suc @ M3 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_2337_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_2338_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_2339_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_2340_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K3 )
               => ~ ( P @ I2 ) )
            & ( P @ K3 ) ) ) ) ).

% ex_least_nat_le
thf(fact_2341_option_Osize_I4_J,axiom,
    ! [X2: nat] :
      ( ( size_size_option_nat @ ( some_nat @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_2342_option_Osize_I4_J,axiom,
    ! [X2: product_prod_nat_nat] :
      ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_2343_option_Osize_I4_J,axiom,
    ! [X2: num] :
      ( ( size_size_option_num @ ( some_num @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_2344_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K3: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K3 )
          & ( ( plus_plus_nat @ I @ K3 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_2345_option_Osize_I3_J,axiom,
    ( ( size_size_option_nat @ none_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2346_option_Osize_I3_J,axiom,
    ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2347_option_Osize_I3_J,axiom,
    ( ( size_size_option_num @ none_num )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2348_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_2349_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_2350_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_2351_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_2352_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_2353_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_2354_divmod__digit__0_I2_J,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) )
          = ( modulo_modulo_nat @ A @ B2 ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_2355_divmod__digit__0_I2_J,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) )
          = ( modulo_modulo_int @ A @ B2 ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_2356_divmod__digit__0_I2_J,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) )
          = ( modulo364778990260209775nteger @ A @ B2 ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_2357_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_2358_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_2359_bits__stable__imp__add__self,axiom,
    ! [A: nat] :
      ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
thf(fact_2360_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
        = zero_zero_int ) ) ).

% bits_stable_imp_add_self
thf(fact_2361_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
thf(fact_2362_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_2363_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_2364_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X3 )
      = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) ) ).

% vebt_insert.simps(3)
thf(fact_2365_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X3: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X3 ) ).

% vebt_member.simps(3)
thf(fact_2366_verit__le__mono__div,axiom,
    ! [A4: nat,B5: nat,N: nat] :
      ( ( ord_less_nat @ A4 @ B5 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A4 @ N )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B5 @ N )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B5 @ N ) ) ) ) ).

% verit_le_mono_div
thf(fact_2367_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
          ( X3
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
            ( X3
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X5: nat] :
              ( X3
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X5 ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
                ( X3
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ X5 ) )
           => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT,X5: nat] :
                  ( X3
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ X5 ) ) ) ) ) ) ).

% VEBT_internal.membermima.cases
thf(fact_2368_vebt__pred_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o] :
          ( X3
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) )
     => ( ! [A3: $o,Uw2: $o] :
            ( X3
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A3: $o,B3: $o,Va2: nat] :
              ( X3
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va2 ) ) ) )
         => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] :
                ( X3
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Vb2 ) )
           => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
                  ( X3
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) @ Vf ) )
             => ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
                    ( X3
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
                      ( X3
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X5 ) ) ) ) ) ) ) ) ).

% vebt_pred.cases
thf(fact_2369_vebt__pred_Osimps_I3_J,axiom,
    ! [B2: $o,A: $o,Va: nat] :
      ( ( B2
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B2 ) @ ( suc @ ( suc @ Va ) ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B2
       => ( ( A
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B2 ) @ ( suc @ ( suc @ Va ) ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B2 ) @ ( suc @ ( suc @ Va ) ) )
              = none_nat ) ) ) ) ) ).

% vebt_pred.simps(3)
thf(fact_2370_divmod__digit__0_I1_J,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
          = ( divide_divide_nat @ A @ B2 ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_2371_divmod__digit__0_I1_J,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
          = ( divide_divide_int @ A @ B2 ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_2372_divmod__digit__0_I1_J,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) )
          = ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_2373_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_2374_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_2375_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_2376_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_2377_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_2378_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_2379_div__mult1__eq,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B2 @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_2380_div__mult1__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B2 @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_2381_div__mult1__eq,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B2 @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_2382_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) @ ( modulo_modulo_nat @ A @ B2 ) ) @ C )
      = ( plus_plus_nat @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_2383_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) @ ( modulo_modulo_int @ A @ B2 ) ) @ C )
      = ( plus_plus_int @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_2384_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) @ ( modulo364778990260209775nteger @ A @ B2 ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_2385_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A @ B2 ) ) @ C )
      = ( plus_plus_nat @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_2386_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A @ B2 ) ) @ C )
      = ( plus_plus_int @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_2387_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) @ ( modulo364778990260209775nteger @ A @ B2 ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_2388_mod__div__decomp,axiom,
    ! [A: nat,B2: nat] :
      ( A
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A @ B2 ) ) ) ).

% mod_div_decomp
thf(fact_2389_mod__div__decomp,axiom,
    ! [A: int,B2: int] :
      ( A
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A @ B2 ) ) ) ).

% mod_div_decomp
thf(fact_2390_mod__div__decomp,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( A
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ).

% mod_div_decomp
thf(fact_2391_div__mult__mod__eq,axiom,
    ! [A: nat,B2: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A @ B2 ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_2392_div__mult__mod__eq,axiom,
    ! [A: int,B2: int] :
      ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A @ B2 ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_2393_div__mult__mod__eq,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) @ ( modulo364778990260209775nteger @ A @ B2 ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_2394_mod__div__mult__eq,axiom,
    ! [A: nat,B2: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B2 ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_2395_mod__div__mult__eq,axiom,
    ! [A: int,B2: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B2 ) @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_2396_mod__div__mult__eq,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_2397_mod__mult__div__eq,axiom,
    ! [A: nat,B2: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B2 ) @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_2398_mod__mult__div__eq,axiom,
    ! [A: int,B2: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B2 ) @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_2399_mod__mult__div__eq,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_2400_mult__div__mod__eq,axiom,
    ! [B2: nat,A: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) @ ( modulo_modulo_nat @ A @ B2 ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_2401_mult__div__mod__eq,axiom,
    ! [B2: int,A: int] :
      ( ( plus_plus_int @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) @ ( modulo_modulo_int @ A @ B2 ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_2402_mult__div__mod__eq,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) @ ( modulo364778990260209775nteger @ A @ B2 ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_2403_minus__div__mult__eq__mod,axiom,
    ! [A: nat,B2: nat] :
      ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) )
      = ( modulo_modulo_nat @ A @ B2 ) ) ).

% minus_div_mult_eq_mod
thf(fact_2404_minus__div__mult__eq__mod,axiom,
    ! [A: int,B2: int] :
      ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% minus_div_mult_eq_mod
thf(fact_2405_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) )
      = ( modulo364778990260209775nteger @ A @ B2 ) ) ).

% minus_div_mult_eq_mod
thf(fact_2406_minus__mod__eq__div__mult,axiom,
    ! [A: nat,B2: nat] :
      ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B2 ) )
      = ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) ) ).

% minus_mod_eq_div_mult
thf(fact_2407_minus__mod__eq__div__mult,axiom,
    ! [A: int,B2: int] :
      ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B2 ) )
      = ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) ) ).

% minus_mod_eq_div_mult
thf(fact_2408_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B2 ) )
      = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) ) ).

% minus_mod_eq_div_mult
thf(fact_2409_minus__mod__eq__mult__div,axiom,
    ! [A: nat,B2: nat] :
      ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B2 ) )
      = ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_2410_minus__mod__eq__mult__div,axiom,
    ! [A: int,B2: int] :
      ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B2 ) )
      = ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_2411_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B2 ) )
      = ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_2412_minus__mult__div__eq__mod,axiom,
    ! [A: nat,B2: nat] :
      ( ( minus_minus_nat @ A @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) )
      = ( modulo_modulo_nat @ A @ B2 ) ) ).

% minus_mult_div_eq_mod
thf(fact_2413_minus__mult__div__eq__mod,axiom,
    ! [A: int,B2: int] :
      ( ( minus_minus_int @ A @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) )
      = ( modulo_modulo_int @ A @ B2 ) ) ).

% minus_mult_div_eq_mod
thf(fact_2414_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) )
      = ( modulo364778990260209775nteger @ A @ B2 ) ) ).

% minus_mult_div_eq_mod
thf(fact_2415_mod__eq__nat1E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q2 )
        = ( modulo_modulo_nat @ N @ Q2 ) )
     => ( ( ord_less_eq_nat @ N @ M )
       => ~ ! [S3: nat] :
              ( M
             != ( plus_plus_nat @ N @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_2416_mod__eq__nat2E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q2 )
        = ( modulo_modulo_nat @ N @ Q2 ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ~ ! [S3: nat] :
              ( N
             != ( plus_plus_nat @ M @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_2417_nat__mod__eq__lemma,axiom,
    ! [X3: nat,N: nat,Y3: nat] :
      ( ( ( modulo_modulo_nat @ X3 @ N )
        = ( modulo_modulo_nat @ Y3 @ N ) )
     => ( ( ord_less_eq_nat @ Y3 @ X3 )
       => ? [Q3: nat] :
            ( X3
            = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N @ Q3 ) ) ) ) ) ).

% nat_mod_eq_lemma
thf(fact_2418_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Y3: $o] :
      ( ( ( vEBT_VEBT_minNull @ X3 )
        = Y3 )
     => ( ( ( X3
            = ( vEBT_Leaf @ $false @ $false ) )
         => ~ Y3 )
       => ( ( ? [Uv2: $o] :
                ( X3
                = ( vEBT_Leaf @ $true @ Uv2 ) )
           => Y3 )
         => ( ( ? [Uu2: $o] :
                  ( X3
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
             => Y3 )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( X3
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
               => ~ Y3 )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( X3
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                 => Y3 ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
thf(fact_2419_div__mod__decomp,axiom,
    ! [A4: nat,N: nat] :
      ( A4
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A4 @ N ) @ N ) @ ( modulo_modulo_nat @ A4 @ N ) ) ) ).

% div_mod_decomp
thf(fact_2420_mod__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).

% mod_mult2_eq
thf(fact_2421_modulo__nat__def,axiom,
    ( modulo_modulo_nat
    = ( ^ [M3: nat,N2: nat] : ( minus_minus_nat @ M3 @ ( times_times_nat @ ( divide_divide_nat @ M3 @ N2 ) @ N2 ) ) ) ) ).

% modulo_nat_def
thf(fact_2422_mult__less__le__imp__less,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2423_mult__less__le__imp__less,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2424_mult__less__le__imp__less,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2425_mult__less__le__imp__less,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2426_mult__le__less__imp__less,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2427_mult__le__less__imp__less,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2428_mult__le__less__imp__less,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2429_mult__le__less__imp__less,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2430_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_2431_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_2432_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_2433_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_2434_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_2435_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_2436_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_2437_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_2438_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
        = ( ord_less_eq_real @ A @ B2 ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2439_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
        = ( ord_less_eq_rat @ A @ B2 ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2440_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
        = ( ord_less_eq_int @ A @ B2 ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2441_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
        = ( ord_less_eq_real @ B2 @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2442_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
        = ( ord_less_eq_rat @ B2 @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2443_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
        = ( ord_less_eq_int @ B2 @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2444_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B2 ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B2 @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2445_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B2 ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B2 @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2446_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B2 ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B2 @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2447_mult__strict__mono_H,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2448_mult__strict__mono_H,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2449_mult__strict__mono_H,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2450_mult__strict__mono_H,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2451_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_2452_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_2453_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_2454_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_2455_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B2 ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B2 @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2456_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B2 ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B2 @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2457_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B2 ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B2 @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2458_mult__strict__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_real @ A @ 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 @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2459_mult__strict__mono,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ 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 @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2460_mult__strict__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ 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 @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2461_mult__strict__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_int @ A @ 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 @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2462_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_2463_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_2464_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_2465_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_2466_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B2 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B2 @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2467_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B2 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2468_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B2 ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B2 @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2469_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B2 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B2 @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2470_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B2 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2471_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B2 ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B2 @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2472_field__le__epsilon,axiom,
    ! [X3: real,Y3: real] :
      ( ! [E2: real] :
          ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( ord_less_eq_real @ X3 @ ( plus_plus_real @ Y3 @ E2 ) ) )
     => ( ord_less_eq_real @ X3 @ Y3 ) ) ).

% field_le_epsilon
thf(fact_2473_field__le__epsilon,axiom,
    ! [X3: rat,Y3: rat] :
      ( ! [E2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E2 )
         => ( ord_less_eq_rat @ X3 @ ( plus_plus_rat @ Y3 @ E2 ) ) )
     => ( ord_less_eq_rat @ X3 @ Y3 ) ) ).

% field_le_epsilon
thf(fact_2474_add__strict__increasing2,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B2 @ C )
       => ( ord_less_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_2475_add__strict__increasing2,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_2476_add__strict__increasing2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_2477_add__strict__increasing2,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B2 @ C )
       => ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_2478_add__strict__increasing,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ord_less_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_2479_add__strict__increasing,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_2480_add__strict__increasing,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_2481_add__strict__increasing,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_2482_add__pos__nonneg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_2483_add__pos__nonneg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_2484_add__pos__nonneg,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_2485_add__pos__nonneg,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_2486_add__nonpos__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).

% add_nonpos_neg
thf(fact_2487_add__nonpos__neg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).

% add_nonpos_neg
thf(fact_2488_add__nonpos__neg,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_2489_add__nonpos__neg,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% add_nonpos_neg
thf(fact_2490_add__nonneg__pos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_2491_add__nonneg__pos,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_2492_add__nonneg__pos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_2493_add__nonneg__pos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_2494_add__neg__nonpos,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).

% add_neg_nonpos
thf(fact_2495_add__neg__nonpos,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).

% add_neg_nonpos
thf(fact_2496_add__neg__nonpos,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_2497_add__neg__nonpos,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% add_neg_nonpos
thf(fact_2498_divide__nonpos__pos,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ X3 @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).

% divide_nonpos_pos
thf(fact_2499_divide__nonpos__pos,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_pos
thf(fact_2500_divide__nonpos__neg,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ X3 @ zero_zero_real )
     => ( ( ord_less_real @ Y3 @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).

% divide_nonpos_neg
thf(fact_2501_divide__nonpos__neg,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
     => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).

% divide_nonpos_neg
thf(fact_2502_divide__nonneg__pos,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).

% divide_nonneg_pos
thf(fact_2503_divide__nonneg__pos,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).

% divide_nonneg_pos
thf(fact_2504_divide__nonneg__neg,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ Y3 @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).

% divide_nonneg_neg
thf(fact_2505_divide__nonneg__neg,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_neg
thf(fact_2506_divide__le__cancel,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B2 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B2 @ A ) ) ) ) ).

% divide_le_cancel
thf(fact_2507_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B2 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).

% divide_le_cancel
thf(fact_2508_frac__less2,axiom,
    ! [X3: real,Y3: real,W: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ X3 @ Y3 )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_real @ W @ Z )
           => ( ord_less_real @ ( divide_divide_real @ X3 @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).

% frac_less2
thf(fact_2509_frac__less2,axiom,
    ! [X3: rat,Y3: rat,W: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_eq_rat @ X3 @ Y3 )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_rat @ W @ Z )
           => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).

% frac_less2
thf(fact_2510_frac__less,axiom,
    ! [X3: real,Y3: real,W: real,Z: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ Y3 )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_eq_real @ W @ Z )
           => ( ord_less_real @ ( divide_divide_real @ X3 @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).

% frac_less
thf(fact_2511_frac__less,axiom,
    ! [X3: rat,Y3: rat,W: rat,Z: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_rat @ X3 @ Y3 )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_eq_rat @ W @ Z )
           => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).

% frac_less
thf(fact_2512_frac__le,axiom,
    ! [Y3: real,X3: real,W: real,Z: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_eq_real @ X3 @ Y3 )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_eq_real @ W @ Z )
           => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).

% frac_le
thf(fact_2513_frac__le,axiom,
    ! [Y3: rat,X3: rat,W: rat,Z: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_eq_rat @ X3 @ Y3 )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_eq_rat @ W @ Z )
           => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).

% frac_le
thf(fact_2514_div__positive,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ A )
       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).

% div_positive
thf(fact_2515_div__positive,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_eq_int @ B2 @ A )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) ) ) ) ).

% div_positive
thf(fact_2516_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ B2 )
       => ( ( divide_divide_nat @ A @ B2 )
          = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_2517_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ B2 )
       => ( ( divide_divide_int @ A @ B2 )
          = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_2518_mult__left__le__one__le,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ Y3 @ X3 ) @ X3 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_2519_mult__left__le__one__le,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ Y3 @ X3 ) @ X3 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_2520_mult__left__le__one__le,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( ord_less_eq_int @ Y3 @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ Y3 @ X3 ) @ X3 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_2521_mult__right__le__one__le,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ X3 @ Y3 ) @ X3 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_2522_mult__right__le__one__le,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ X3 @ Y3 ) @ X3 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_2523_mult__right__le__one__le,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( ord_less_eq_int @ Y3 @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ X3 @ Y3 ) @ X3 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_2524_mult__le__one,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ 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 @ A @ B2 ) @ one_one_real ) ) ) ) ).

% mult_le_one
thf(fact_2525_mult__le__one,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ 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 @ A @ B2 ) @ one_one_rat ) ) ) ) ).

% mult_le_one
thf(fact_2526_mult__le__one,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ 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 @ A @ B2 ) @ one_one_nat ) ) ) ) ).

% mult_le_one
thf(fact_2527_mult__le__one,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ 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 @ A @ B2 ) @ one_one_int ) ) ) ) ).

% mult_le_one
thf(fact_2528_mult__left__le,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_eq_real @ C @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_2529_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_eq_rat @ C @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_2530_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_2531_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_eq_int @ C @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_2532_sum__squares__le__zero__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real )
      = ( ( X3 = zero_zero_real )
        & ( Y3 = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_2533_sum__squares__le__zero__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat )
      = ( ( X3 = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_2534_sum__squares__le__zero__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int )
      = ( ( X3 = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_2535_sum__squares__ge__zero,axiom,
    ! [X3: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ).

% sum_squares_ge_zero
thf(fact_2536_sum__squares__ge__zero,axiom,
    ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) ) ) ).

% sum_squares_ge_zero
thf(fact_2537_sum__squares__ge__zero,axiom,
    ! [X3: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) ) ) ).

% sum_squares_ge_zero
thf(fact_2538_power__less__imp__less__base,axiom,
    ! [A: real,N: nat,B2: real] :
      ( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_real @ A @ B2 ) ) ) ).

% power_less_imp_less_base
thf(fact_2539_power__less__imp__less__base,axiom,
    ! [A: rat,N: nat,B2: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ A @ B2 ) ) ) ).

% power_less_imp_less_base
thf(fact_2540_power__less__imp__less__base,axiom,
    ! [A: nat,N: nat,B2: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ A @ B2 ) ) ) ).

% power_less_imp_less_base
thf(fact_2541_power__less__imp__less__base,axiom,
    ! [A: int,N: nat,B2: int] :
      ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ A @ B2 ) ) ) ).

% power_less_imp_less_base
thf(fact_2542_sum__squares__gt__zero__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) )
      = ( ( X3 != zero_zero_real )
        | ( Y3 != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_2543_sum__squares__gt__zero__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) ) )
      = ( ( X3 != zero_zero_rat )
        | ( Y3 != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_2544_sum__squares__gt__zero__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) ) )
      = ( ( X3 != zero_zero_int )
        | ( Y3 != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_2545_not__sum__squares__lt__zero,axiom,
    ! [X3: real,Y3: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real ) ).

% not_sum_squares_lt_zero
thf(fact_2546_not__sum__squares__lt__zero,axiom,
    ! [X3: rat,Y3: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat ) ).

% not_sum_squares_lt_zero
thf(fact_2547_not__sum__squares__lt__zero,axiom,
    ! [X3: int,Y3: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int ) ).

% not_sum_squares_lt_zero
thf(fact_2548_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2549_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2550_zero__less__two,axiom,
    ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).

% zero_less_two
thf(fact_2551_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_2552_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_2553_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_2554_divide__strict__left__mono__neg,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
         => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_2555_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_2556_divide__strict__left__mono,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
         => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_2557_divide__strict__left__mono,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B2 @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_2558_mult__imp__less__div__pos,axiom,
    ! [Y3: real,Z: real,X3: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_real @ ( times_times_real @ Z @ Y3 ) @ X3 )
       => ( ord_less_real @ Z @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_2559_mult__imp__less__div__pos,axiom,
    ! [Y3: rat,Z: rat,X3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y3 ) @ X3 )
       => ( ord_less_rat @ Z @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_2560_mult__imp__div__pos__less,axiom,
    ! [Y3: real,X3: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_real @ X3 @ ( times_times_real @ Z @ Y3 ) )
       => ( ord_less_real @ ( divide_divide_real @ X3 @ Y3 ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_2561_mult__imp__div__pos__less,axiom,
    ! [Y3: rat,X3: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_rat @ X3 @ ( times_times_rat @ Z @ Y3 ) )
       => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_2562_pos__less__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ C ) )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).

% pos_less_divide_eq
thf(fact_2563_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ) ).

% pos_less_divide_eq
thf(fact_2564_pos__divide__less__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ A )
        = ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_2565_pos__divide__less__eq,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
        = ( ord_less_rat @ B2 @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_2566_neg__less__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ C ) )
        = ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_2567_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
        = ( ord_less_rat @ B2 @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_2568_neg__divide__less__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ A )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).

% neg_divide_less_eq
thf(fact_2569_neg__divide__less__eq,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ) ).

% neg_divide_less_eq
thf(fact_2570_less__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_2571_less__divide__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B2 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B2 @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_2572_divide__less__eq,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_2573_divide__less__eq,axiom,
    ! [B2: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B2 @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B2 ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_2574_less__divide__eq__1,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ A @ B2 ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B2 @ A ) ) ) ) ).

% less_divide_eq_1
thf(fact_2575_less__divide__eq__1,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ A @ B2 ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B2 @ A ) ) ) ) ).

% less_divide_eq_1
thf(fact_2576_divide__less__eq__1,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B2 @ A ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ A @ B2 ) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
thf(fact_2577_divide__less__eq__1,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B2 @ A ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ A @ B2 ) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
thf(fact_2578_power__le__one,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).

% power_le_one
thf(fact_2579_power__le__one,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).

% power_le_one
thf(fact_2580_power__le__one,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ A @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).

% power_le_one
thf(fact_2581_power__le__one,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ A @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).

% power_le_one
thf(fact_2582_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: complex,C: complex,W: num] :
      ( ( ( divide1717551699836669952omplex @ B2 @ C )
        = ( numera6690914467698888265omplex @ W ) )
      = ( ( ( C != zero_zero_complex )
         => ( B2
            = ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_2583_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_2584_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_2585_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: complex,C: complex] :
      ( ( ( numera6690914467698888265omplex @ W )
        = ( divide1717551699836669952omplex @ B2 @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C )
            = B2 ) )
        & ( ( C = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_2586_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_2587_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_2588_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: complex,A: complex,B2: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B2 )
          = B2 ) )
      & ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B2 )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_2589_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: real,A: real,B2: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
          = B2 ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
          = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_2590_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: rat,A: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B2 )
          = B2 ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B2 )
          = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_2591_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: complex,A: complex,B2: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_2592_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: real,A: real,B2: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_2593_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: rat,A: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B2 @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B2 @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_2594_add__frac__eq,axiom,
    ! [Y3: complex,Z: complex,X3: complex,W: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X3 @ Z ) @ ( times_times_complex @ W @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2595_add__frac__eq,axiom,
    ! [Y3: real,Z: real,X3: real,W: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X3 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2596_add__frac__eq,axiom,
    ! [Y3: rat,Z: rat,X3: rat,W: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2597_add__frac__num,axiom,
    ! [Y3: complex,X3: complex,Z: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ Z )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Z @ Y3 ) ) @ Y3 ) ) ) ).

% add_frac_num
thf(fact_2598_add__frac__num,axiom,
    ! [Y3: real,X3: real,Z: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Y3 ) @ Z )
        = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Z @ Y3 ) ) @ Y3 ) ) ) ).

% add_frac_num
thf(fact_2599_add__frac__num,axiom,
    ! [Y3: rat,X3: rat,Z: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ Z )
        = ( divide_divide_rat @ ( plus_plus_rat @ X3 @ ( times_times_rat @ Z @ Y3 ) ) @ Y3 ) ) ) ).

% add_frac_num
thf(fact_2600_add__num__frac,axiom,
    ! [Y3: complex,Z: complex,X3: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X3 @ Y3 ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Z @ Y3 ) ) @ Y3 ) ) ) ).

% add_num_frac
thf(fact_2601_add__num__frac,axiom,
    ! [Y3: real,Z: real,X3: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X3 @ Y3 ) )
        = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Z @ Y3 ) ) @ Y3 ) ) ) ).

% add_num_frac
thf(fact_2602_add__num__frac,axiom,
    ! [Y3: rat,Z: rat,X3: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X3 @ Y3 ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ X3 @ ( times_times_rat @ Z @ Y3 ) ) @ Y3 ) ) ) ).

% add_num_frac
thf(fact_2603_add__divide__eq__iff,axiom,
    ! [Z: complex,X3: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ X3 @ ( divide1717551699836669952omplex @ Y3 @ Z ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X3 @ Z ) @ Y3 ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_2604_add__divide__eq__iff,axiom,
    ! [Z: real,X3: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ X3 @ ( divide_divide_real @ Y3 @ Z ) )
        = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X3 @ Z ) @ Y3 ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_2605_add__divide__eq__iff,axiom,
    ! [Z: rat,X3: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ X3 @ ( divide_divide_rat @ Y3 @ Z ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ Z ) @ Y3 ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_2606_divide__add__eq__iff,axiom,
    ! [Z: complex,X3: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Z ) @ Y3 )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_2607_divide__add__eq__iff,axiom,
    ! [Z: real,X3: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Z ) @ Y3 )
        = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_2608_divide__add__eq__iff,axiom,
    ! [Z: rat,X3: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ Z ) @ Y3 )
        = ( divide_divide_rat @ ( plus_plus_rat @ X3 @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_2609_power__le__imp__le__base,axiom,
    ! [A: real,N: nat,B2: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_eq_real @ A @ B2 ) ) ) ).

% power_le_imp_le_base
thf(fact_2610_power__le__imp__le__base,axiom,
    ! [A: rat,N: nat,B2: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_eq_rat @ A @ B2 ) ) ) ).

% power_le_imp_le_base
thf(fact_2611_power__le__imp__le__base,axiom,
    ! [A: nat,N: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_eq_nat @ A @ B2 ) ) ) ).

% power_le_imp_le_base
thf(fact_2612_power__le__imp__le__base,axiom,
    ! [A: int,N: nat,B2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ A @ B2 ) ) ) ).

% power_le_imp_le_base
thf(fact_2613_power__inject__base,axiom,
    ! [A: real,N: nat,B2: real] :
      ( ( ( power_power_real @ A @ ( suc @ N ) )
        = ( power_power_real @ B2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
         => ( A = B2 ) ) ) ) ).

% power_inject_base
thf(fact_2614_power__inject__base,axiom,
    ! [A: rat,N: nat,B2: rat] :
      ( ( ( power_power_rat @ A @ ( suc @ N ) )
        = ( power_power_rat @ B2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
         => ( A = B2 ) ) ) ) ).

% power_inject_base
thf(fact_2615_power__inject__base,axiom,
    ! [A: nat,N: nat,B2: nat] :
      ( ( ( power_power_nat @ A @ ( suc @ N ) )
        = ( power_power_nat @ B2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
         => ( A = B2 ) ) ) ) ).

% power_inject_base
thf(fact_2616_power__inject__base,axiom,
    ! [A: int,N: nat,B2: int] :
      ( ( ( power_power_int @ A @ ( suc @ N ) )
        = ( power_power_int @ B2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
         => ( A = B2 ) ) ) ) ).

% power_inject_base
thf(fact_2617_div__add__self1,axiom,
    ! [B2: nat,A: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).

% div_add_self1
thf(fact_2618_div__add__self1,axiom,
    ! [B2: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
        = ( plus_plus_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).

% div_add_self1
thf(fact_2619_div__add__self2,axiom,
    ! [B2: nat,A: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).

% div_add_self2
thf(fact_2620_div__add__self2,axiom,
    ! [B2: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
        = ( plus_plus_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).

% div_add_self2
thf(fact_2621_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: complex,A: complex,B2: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_2622_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: real,A: real,B2: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_2623_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: rat,A: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B2 @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B2 @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_2624_diff__frac__eq,axiom,
    ! [Y3: complex,Z: complex,X3: complex,W: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X3 @ Z ) @ ( times_times_complex @ W @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_2625_diff__frac__eq,axiom,
    ! [Y3: real,Z: real,X3: real,W: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_2626_diff__frac__eq,axiom,
    ! [Y3: rat,Z: rat,X3: rat,W: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_2627_diff__divide__eq__iff,axiom,
    ! [Z: complex,X3: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ X3 @ ( divide1717551699836669952omplex @ Y3 @ Z ) )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X3 @ Z ) @ Y3 ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_2628_diff__divide__eq__iff,axiom,
    ! [Z: real,X3: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ X3 @ ( divide_divide_real @ Y3 @ Z ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z ) @ Y3 ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_2629_diff__divide__eq__iff,axiom,
    ! [Z: rat,X3: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ X3 @ ( divide_divide_rat @ Y3 @ Z ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z ) @ Y3 ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_2630_divide__diff__eq__iff,axiom,
    ! [Z: complex,X3: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X3 @ Z ) @ Y3 )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X3 @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_2631_divide__diff__eq__iff,axiom,
    ! [Z: real,X3: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ X3 @ Z ) @ Y3 )
        = ( divide_divide_real @ ( minus_minus_real @ X3 @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_2632_divide__diff__eq__iff,axiom,
    ! [Z: rat,X3: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ X3 @ Z ) @ Y3 )
        = ( divide_divide_rat @ ( minus_minus_rat @ X3 @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_2633_mod__double__modulus,axiom,
    ! [M: code_integer,X3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
       => ( ( ( modulo364778990260209775nteger @ X3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( modulo364778990260209775nteger @ X3 @ M ) )
          | ( ( modulo364778990260209775nteger @ X3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X3 @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_2634_mod__double__modulus,axiom,
    ! [M: nat,X3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
       => ( ( ( modulo_modulo_nat @ X3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_nat @ X3 @ M ) )
          | ( ( modulo_modulo_nat @ X3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_nat @ ( modulo_modulo_nat @ X3 @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_2635_mod__double__modulus,axiom,
    ! [M: int,X3: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ X3 )
       => ( ( ( modulo_modulo_int @ X3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_int @ X3 @ M ) )
          | ( ( modulo_modulo_int @ X3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_int @ ( modulo_modulo_int @ X3 @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_2636_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ( ord_le3102999989581377725nteger @ B2 @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
            = ( modulo364778990260209775nteger @ A @ B2 ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_2637_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ( ord_less_eq_nat @ B2 @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
            = ( modulo_modulo_nat @ A @ B2 ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_2638_divmod__digit__1_I2_J,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ( ord_less_eq_int @ B2 @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
            = ( modulo_modulo_int @ A @ B2 ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_2639_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X3 ) ).

% vebt_member.simps(4)
thf(fact_2640_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_2641_num_Osize_I5_J,axiom,
    ! [X2: num] :
      ( ( size_size_num @ ( bit0 @ X2 ) )
      = ( plus_plus_nat @ ( size_size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(5)
thf(fact_2642_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N )
            & ! [I2: nat] :
                ( ( ord_less_eq_nat @ I2 @ K3 )
               => ~ ( P @ I2 ) )
            & ( P @ ( suc @ K3 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_2643_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_2644_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_2645_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_2646_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_2647_length__pos__if__in__set,axiom,
    ! [X3: real,Xs2: list_real] :
      ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2648_length__pos__if__in__set,axiom,
    ! [X3: complex,Xs2: list_complex] :
      ( ( member_complex @ X3 @ ( set_complex2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2649_length__pos__if__in__set,axiom,
    ! [X3: int,Xs2: list_int] :
      ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2650_length__pos__if__in__set,axiom,
    ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2651_length__pos__if__in__set,axiom,
    ! [X3: $o,Xs2: list_o] :
      ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2652_length__pos__if__in__set,axiom,
    ! [X3: nat,Xs2: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_2653_nat__induct__non__zero,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( P @ one_one_nat )
       => ( ! [N3: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N3 )
             => ( ( P @ N3 )
               => ( P @ ( suc @ N3 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_non_zero
thf(fact_2654_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_2655_nat__diff__split,axiom,
    ! [P: nat > $o,A: nat,B2: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B2 ) )
      = ( ( ( ord_less_nat @ A @ B2 )
         => ( P @ zero_zero_nat ) )
        & ! [D2: nat] :
            ( ( A
              = ( plus_plus_nat @ B2 @ D2 ) )
           => ( P @ D2 ) ) ) ) ).

% nat_diff_split
thf(fact_2656_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A: nat,B2: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B2 ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B2 )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D2: nat] :
                ( ( A
                  = ( plus_plus_nat @ B2 @ D2 ) )
                & ~ ( P @ D2 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_2657_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_2658_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_2659_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_2660_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_2661_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q2 )
     => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N )
        = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q2 ) ) ) ) ).

% div_less_iff_less_mult
thf(fact_2662_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_2663_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_2664_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_2665_nat__induct2,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ( P @ one_one_nat )
       => ( ! [N3: nat] :
              ( ( P @ N3 )
             => ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct2
thf(fact_2666_vebt__mint_Ocases,axiom,
    ! [X3: vEBT_VEBT] :
      ( ! [A3: $o,B3: $o] :
          ( X3
         != ( vEBT_Leaf @ A3 @ B3 ) )
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
            ( X3
           != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
              ( X3
             != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).

% vebt_mint.cases
thf(fact_2667_field__le__mult__one__interval,axiom,
    ! [X3: real,Y3: real] :
      ( ! [Z4: real] :
          ( ( ord_less_real @ zero_zero_real @ Z4 )
         => ( ( ord_less_real @ Z4 @ one_one_real )
           => ( ord_less_eq_real @ ( times_times_real @ Z4 @ X3 ) @ Y3 ) ) )
     => ( ord_less_eq_real @ X3 @ Y3 ) ) ).

% field_le_mult_one_interval
thf(fact_2668_field__le__mult__one__interval,axiom,
    ! [X3: rat,Y3: rat] :
      ( ! [Z4: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ Z4 )
         => ( ( ord_less_rat @ Z4 @ one_one_rat )
           => ( ord_less_eq_rat @ ( times_times_rat @ Z4 @ X3 ) @ Y3 ) ) )
     => ( ord_less_eq_rat @ X3 @ Y3 ) ) ).

% field_le_mult_one_interval
thf(fact_2669_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ one_one_real ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2670_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2671_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2672_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_2673_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_2674_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_2675_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ one_one_real ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2676_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2677_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2678_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_2679_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_2680_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_2681_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ one_one_real ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2682_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2683_mult__le__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2684_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_2685_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_2686_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_2687_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ one_one_real ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_2688_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_2689_mult__le__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_2690_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_2691_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_2692_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_2693_divide__left__mono__neg,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_2694_divide__left__mono__neg,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_2695_mult__imp__le__div__pos,axiom,
    ! [Y3: real,Z: real,X3: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y3 ) @ X3 )
       => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_2696_mult__imp__le__div__pos,axiom,
    ! [Y3: rat,Z: rat,X3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y3 ) @ X3 )
       => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_2697_mult__imp__div__pos__le,axiom,
    ! [Y3: real,X3: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_eq_real @ X3 @ ( times_times_real @ Z @ Y3 ) )
       => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_2698_mult__imp__div__pos__le,axiom,
    ! [Y3: rat,X3: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_eq_rat @ X3 @ ( times_times_rat @ Z @ Y3 ) )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_2699_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ C ) )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).

% pos_le_divide_eq
thf(fact_2700_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ) ).

% pos_le_divide_eq
thf(fact_2701_pos__divide__le__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ A )
        = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_2702_pos__divide__le__eq,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
        = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_2703_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ C ) )
        = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_2704_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
        = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_2705_neg__divide__le__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ A )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).

% neg_divide_le_eq
thf(fact_2706_neg__divide__le__eq,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ) ).

% neg_divide_le_eq
thf(fact_2707_divide__left__mono,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).

% divide_left_mono
thf(fact_2708_divide__left__mono,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B2 @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).

% divide_left_mono
thf(fact_2709_le__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_2710_le__divide__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B2 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_2711_divide__le__eq,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_2712_divide__le__eq,axiom,
    ! [B2: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B2 ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_2713_le__divide__eq__1,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ A @ B2 ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B2 @ A ) ) ) ) ).

% le_divide_eq_1
thf(fact_2714_le__divide__eq__1,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ A @ B2 ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).

% le_divide_eq_1
thf(fact_2715_divide__le__eq__1,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B2 @ A ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ A @ B2 ) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
thf(fact_2716_divide__le__eq__1,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B2 @ A ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ A @ B2 ) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
thf(fact_2717_convex__bound__le,axiom,
    ! [X3: real,A: real,Y3: real,U: real,V: real] :
      ( ( ord_less_eq_real @ X3 @ A )
     => ( ( ord_less_eq_real @ Y3 @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X3 ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2718_convex__bound__le,axiom,
    ! [X3: rat,A: rat,Y3: rat,U: rat,V: rat] :
      ( ( ord_less_eq_rat @ X3 @ A )
     => ( ( ord_less_eq_rat @ Y3 @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X3 ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2719_convex__bound__le,axiom,
    ! [X3: int,A: int,Y3: int,U: int,V: int] :
      ( ( ord_less_eq_int @ X3 @ A )
     => ( ( ord_less_eq_int @ Y3 @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X3 ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2720_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_2721_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_2722_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_2723_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_2724_frac__le__eq,axiom,
    ! [Y3: real,Z: real,X3: real,W: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
          = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) @ zero_zero_real ) ) ) ) ).

% frac_le_eq
thf(fact_2725_frac__le__eq,axiom,
    ! [Y3: rat,Z: rat,X3: rat,W: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
          = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_le_eq
thf(fact_2726_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ( ord_le3102999989581377725nteger @ B2 @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) @ one_one_Code_integer )
            = ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_2727_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ( ord_less_eq_nat @ B2 @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) @ one_one_nat )
            = ( divide_divide_nat @ A @ B2 ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_2728_divmod__digit__1_I1_J,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ( ord_less_eq_int @ B2 @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) @ one_one_int )
            = ( divide_divide_int @ A @ B2 ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_2729_frac__less__eq,axiom,
    ! [Y3: real,Z: real,X3: real,W: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ord_less_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
          = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) @ zero_zero_real ) ) ) ) ).

% frac_less_eq
thf(fact_2730_frac__less__eq,axiom,
    ! [Y3: rat,Z: rat,X3: rat,W: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
          = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_less_eq
thf(fact_2731_power__Suc__less,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_2732_power__Suc__less,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_2733_power__Suc__less,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_2734_power__Suc__less,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_2735_power__Suc__le__self,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2736_power__Suc__le__self,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2737_power__Suc__le__self,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ A @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2738_power__Suc__le__self,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ A @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2739_power__Suc__less__one,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).

% power_Suc_less_one
thf(fact_2740_power__Suc__less__one,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).

% power_Suc_less_one
thf(fact_2741_power__Suc__less__one,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).

% power_Suc_less_one
thf(fact_2742_power__Suc__less__one,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).

% power_Suc_less_one
thf(fact_2743_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: real] :
      ( ( ord_less_nat @ N @ N4 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( ord_less_real @ A @ one_one_real )
         => ( ord_less_real @ ( power_power_real @ A @ N4 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2744_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: rat] :
      ( ( ord_less_nat @ N @ N4 )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ( ord_less_rat @ A @ one_one_rat )
         => ( ord_less_rat @ ( power_power_rat @ A @ N4 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2745_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( ( ord_less_nat @ N @ N4 )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ A @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2746_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( ( ord_less_nat @ N @ N4 )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ( ord_less_int @ A @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2747_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: real] :
      ( ( ord_less_eq_nat @ N @ N4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ A @ one_one_real )
         => ( ord_less_eq_real @ ( power_power_real @ A @ N4 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2748_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: rat] :
      ( ( ord_less_eq_nat @ N @ N4 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ A @ one_one_rat )
         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N4 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2749_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ N4 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ A @ one_one_nat )
         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2750_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( ( ord_less_eq_nat @ N @ N4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ A @ one_one_int )
         => ( ord_less_eq_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2751_vebt__mint_Oelims,axiom,
    ! [X3: vEBT_VEBT,Y3: option_nat] :
      ( ( ( vEBT_vebt_mint @ X3 )
        = Y3 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X3
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( A3
                 => ( Y3
                    = ( some_nat @ zero_zero_nat ) ) )
                & ( ~ A3
                 => ( ( B3
                     => ( Y3
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B3
                     => ( Y3 = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X3
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y3 != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y3
                 != ( some_nat @ Mi2 ) ) ) ) ) ) ).

% vebt_mint.elims
thf(fact_2752_zero__power2,axiom,
    ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_rat ) ).

% zero_power2
thf(fact_2753_zero__power2,axiom,
    ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% zero_power2
thf(fact_2754_zero__power2,axiom,
    ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_real ) ).

% zero_power2
thf(fact_2755_zero__power2,axiom,
    ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% zero_power2
thf(fact_2756_zero__power2,axiom,
    ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_complex ) ).

% zero_power2
thf(fact_2757_vebt__maxt_Oelims,axiom,
    ! [X3: vEBT_VEBT,Y3: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X3 )
        = Y3 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X3
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( B3
                 => ( Y3
                    = ( some_nat @ one_one_nat ) ) )
                & ( ~ B3
                 => ( ( A3
                     => ( Y3
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A3
                     => ( Y3 = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X3
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y3 != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y3
                 != ( some_nat @ Ma2 ) ) ) ) ) ) ).

% vebt_maxt.elims
thf(fact_2758_self__le__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_2759_self__le__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_2760_self__le__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_2761_self__le__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_2762_one__less__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_2763_one__less__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_2764_one__less__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_2765_one__less__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_2766_numeral__2__eq__2,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ one ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% numeral_2_eq_2
thf(fact_2767_pos2,axiom,
    ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).

% pos2
thf(fact_2768_power__diff,axiom,
    ! [A: complex,N: nat,M: nat] :
      ( ( A != zero_zero_complex )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_2769_power__diff,axiom,
    ! [A: real,N: nat,M: nat] :
      ( ( A != zero_zero_real )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_2770_power__diff,axiom,
    ! [A: rat,N: nat,M: nat] :
      ( ( A != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_2771_power__diff,axiom,
    ! [A: nat,N: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_2772_power__diff,axiom,
    ! [A: int,N: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_2773_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M3: nat,N2: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M3 @ N2 )
            | ( N2 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M3 @ N2 ) @ N2 ) ) ) ) ) ).

% div_if
thf(fact_2774_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_2775_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_2776_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_2777_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ) ) ) ).

% add_eq_if
thf(fact_2778_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q2 )
     => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q2 ) )
        = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_2779_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_2780_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_2781_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_2782_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
        = Y3 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X3
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( Y3
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X3
                = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
           => Y3 )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] :
                    ( X3
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
               => ( Y3
                  = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
thf(fact_2783_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X3
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => A3 )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => B3 )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [S3: vEBT_VEBT] :
                  ( X3
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
             => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
thf(fact_2784_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X3
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( ( ( Xa2 = zero_zero_nat )
               => A3 )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => B3 )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X3
             != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] :
                    ( X3
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
thf(fact_2785_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ) ) ) ).

% mult_eq_if
thf(fact_2786_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve2 ) @ Vf2 )
      = none_nat ) ).

% vebt_pred.simps(5)
thf(fact_2787_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
      ( ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
        = Y3 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X3
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => Y3 )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X3
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
           => Y3 )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
               => ( Y3
                  = ( ~ ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X3
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
                 => ( Y3
                    = ( ~ ( ( Xa2 = Mi2 )
                          | ( Xa2 = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X3
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
                   => ( Y3
                      = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
thf(fact_2788_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X3 @ Xa2 )
     => ( ! [Uu2: $o,Uv2: $o] :
            ( X3
           != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X3
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
               => ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X3
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
                 => ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 )
                    | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X3
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
thf(fact_2789_convex__bound__lt,axiom,
    ! [X3: real,A: real,Y3: real,U: real,V: real] :
      ( ( ord_less_real @ X3 @ A )
     => ( ( ord_less_real @ Y3 @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X3 ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2790_convex__bound__lt,axiom,
    ! [X3: rat,A: rat,Y3: rat,U: rat,V: rat] :
      ( ( ord_less_rat @ X3 @ A )
     => ( ( ord_less_rat @ Y3 @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X3 ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2791_convex__bound__lt,axiom,
    ! [X3: int,A: int,Y3: int,U: int,V: int] :
      ( ( ord_less_int @ X3 @ A )
     => ( ( ord_less_int @ Y3 @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X3 ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2792_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_2793_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_2794_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_2795_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_2796_half__gt__zero,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% half_gt_zero
thf(fact_2797_half__gt__zero,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% half_gt_zero
thf(fact_2798_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% half_gt_zero_iff
thf(fact_2799_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% half_gt_zero_iff
thf(fact_2800_scaling__mono,axiom,
    ! [U: real,V: real,R: real,S: real] :
      ( ( ord_less_eq_real @ U @ V )
     => ( ( ord_less_eq_real @ zero_zero_real @ R )
       => ( ( ord_less_eq_real @ R @ S )
         => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_2801_scaling__mono,axiom,
    ! [U: rat,V: rat,R: rat,S: rat] :
      ( ( ord_less_eq_rat @ U @ V )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ R )
       => ( ( ord_less_eq_rat @ R @ S )
         => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_2802_zero__le__power2,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_2803_zero__le__power2,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_2804_zero__le__power2,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_2805_power2__eq__imp__eq,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
         => ( X3 = Y3 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_2806_power2__eq__imp__eq,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
         => ( X3 = Y3 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_2807_power2__eq__imp__eq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
         => ( X3 = Y3 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_2808_power2__eq__imp__eq,axiom,
    ! [X3: int,Y3: int] :
      ( ( ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ X3 )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
         => ( X3 = Y3 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_2809_power2__le__imp__le,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ X3 @ Y3 ) ) ) ).

% power2_le_imp_le
thf(fact_2810_power2__le__imp__le,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_eq_rat @ X3 @ Y3 ) ) ) ).

% power2_le_imp_le
thf(fact_2811_power2__le__imp__le,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
       => ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).

% power2_le_imp_le
thf(fact_2812_power2__le__imp__le,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ord_less_eq_int @ X3 @ Y3 ) ) ) ).

% power2_le_imp_le
thf(fact_2813_power2__less__0,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).

% power2_less_0
thf(fact_2814_power2__less__0,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).

% power2_less_0
thf(fact_2815_power2__less__0,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).

% power2_less_0
thf(fact_2816_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_2817_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_2818_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_2819_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_2820_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_2821_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_2822_power__diff__power__eq,axiom,
    ! [A: nat,N: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
            = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
            = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_2823_power__diff__power__eq,axiom,
    ! [A: int,N: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
            = ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
            = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_2824_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_2825_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_2826_power__eq__if,axiom,
    ( power_power_complex
    = ( ^ [P5: complex,M3: nat] : ( if_complex @ ( M3 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P5 @ ( power_power_complex @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2827_power__eq__if,axiom,
    ( power_power_real
    = ( ^ [P5: real,M3: nat] : ( if_real @ ( M3 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P5 @ ( power_power_real @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2828_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P5: rat,M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P5 @ ( power_power_rat @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2829_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P5: nat,M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P5 @ ( power_power_nat @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2830_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P5: int,M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P5 @ ( power_power_int @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2831_power__minus__mult,axiom,
    ! [N: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_complex @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_2832_power__minus__mult,axiom,
    ! [N: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_real @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_2833_power__minus__mult,axiom,
    ! [N: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_rat @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_2834_power__minus__mult,axiom,
    ! [N: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_nat @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_2835_power__minus__mult,axiom,
    ! [N: nat,A: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_int @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_2836_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_2837_split__div_H,axiom,
    ! [P: nat > $o,M: nat,N: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N ) )
      = ( ( ( N = zero_zero_nat )
          & ( P @ zero_zero_nat ) )
        | ? [Q4: nat] :
            ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M )
            & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
            & ( P @ Q4 ) ) ) ) ).

% split_div'
thf(fact_2838_div__exp__mod__exp__eq,axiom,
    ! [A: nat,N: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
      = ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_2839_div__exp__mod__exp__eq,axiom,
    ! [A: int,N: nat,M: nat] :
      ( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
      = ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_2840_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,N: nat,M: nat] :
      ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
      = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_2841_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 )
      = none_nat ) ).

% vebt_pred.simps(6)
thf(fact_2842_power2__less__imp__less,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ord_less_real @ X3 @ Y3 ) ) ) ).

% power2_less_imp_less
thf(fact_2843_power2__less__imp__less,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_rat @ X3 @ Y3 ) ) ) ).

% power2_less_imp_less
thf(fact_2844_power2__less__imp__less,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
       => ( ord_less_nat @ X3 @ Y3 ) ) ) ).

% power2_less_imp_less
thf(fact_2845_power2__less__imp__less,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ord_less_int @ X3 @ Y3 ) ) ) ).

% power2_less_imp_less
thf(fact_2846_sum__power2__ge__zero,axiom,
    ! [X3: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_2847_sum__power2__ge__zero,axiom,
    ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_2848_sum__power2__ge__zero,axiom,
    ! [X3: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_2849_sum__power2__le__zero__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
      = ( ( X3 = zero_zero_real )
        & ( Y3 = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_2850_sum__power2__le__zero__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
      = ( ( X3 = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_2851_sum__power2__le__zero__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
      = ( ( X3 = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_2852_not__sum__power2__lt__zero,axiom,
    ! [X3: real,Y3: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).

% not_sum_power2_lt_zero
thf(fact_2853_not__sum__power2__lt__zero,axiom,
    ! [X3: rat,Y3: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).

% not_sum_power2_lt_zero
thf(fact_2854_not__sum__power2__lt__zero,axiom,
    ! [X3: int,Y3: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).

% not_sum_power2_lt_zero
thf(fact_2855_sum__power2__gt__zero__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X3 != zero_zero_real )
        | ( Y3 != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_2856_sum__power2__gt__zero__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X3 != zero_zero_rat )
        | ( Y3 != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_2857_sum__power2__gt__zero__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X3 != zero_zero_int )
        | ( Y3 != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_2858_zero__le__even__power_H,axiom,
    ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% zero_le_even_power'
thf(fact_2859_zero__le__even__power_H,axiom,
    ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% zero_le_even_power'
thf(fact_2860_zero__le__even__power_H,axiom,
    ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% zero_le_even_power'
thf(fact_2861_nat__bit__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( P @ N3 )
           => ( ( ord_less_nat @ zero_zero_nat @ N3 )
             => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
       => ( ! [N3: nat] :
              ( ( P @ N3 )
             => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_bit_induct
thf(fact_2862_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_2863_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_2864_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
        = ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_2865_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
        = ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_2866_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
        = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_2867_vebt__member_Oelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_vebt_member @ X3 @ Xa2 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X3
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => A3 )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => B3 )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] :
                  ( X3
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
             => ~ ( ( Xa2 != Mi2 )
                 => ( ( Xa2 != Ma2 )
                   => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                      & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                       => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                          & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                           => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                               => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
thf(fact_2868_vebt__member_Oelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
      ( ( ( vEBT_vebt_member @ X3 @ Xa2 )
        = Y3 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X3
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( Y3
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X3
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => Y3 )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X3
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
             => Y3 )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
               => Y3 )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                   => ( Y3
                      = ( ~ ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
thf(fact_2869_vebt__member_Oelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_vebt_member @ X3 @ Xa2 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X3
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( ( ( Xa2 = zero_zero_nat )
               => A3 )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => B3 )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X3
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                ( X3
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X3
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                   => ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
thf(fact_2870_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_2871_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_2872_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_2873_odd__power__less__zero,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).

% odd_power_less_zero
thf(fact_2874_odd__power__less__zero,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).

% odd_power_less_zero
thf(fact_2875_odd__power__less__zero,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).

% odd_power_less_zero
thf(fact_2876_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X3: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
thf(fact_2877_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X3: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_low @ X3 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
thf(fact_2878_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                ( X3
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
           => ~ ( ( Xa2 = Mi2 )
                | ( Xa2 = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] :
                  ( X3
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
             => ~ ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 )
                  | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
         => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] :
                    ( X3
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
thf(fact_2879_arith__geo__mean,axiom,
    ! [U: real,X3: real,Y3: real] :
      ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_real @ X3 @ Y3 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
         => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_2880_arith__geo__mean,axiom,
    ! [U: rat,X3: rat,Y3: rat] :
      ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_rat @ X3 @ Y3 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
         => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_2881_invar__vebt_Osimps,axiom,
    ( vEBT_invar_vebt
    = ( ^ [A1: vEBT_VEBT,A22: nat] :
          ( ( ? [A5: $o,B4: $o] :
                ( A1
                = ( vEBT_Leaf @ A5 @ B4 ) )
            & ( A22
              = ( suc @ zero_zero_nat ) ) )
          | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X @ N2 ) )
              & ( vEBT_invar_vebt @ Summary3 @ N2 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
              & ( A22
                = ( plus_plus_nat @ N2 @ N2 ) )
              & ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X7 )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
          | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X @ N2 ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
              & ( A22
                = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
              & ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X7 )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
          | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A1
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X @ N2 ) )
              & ( vEBT_invar_vebt @ Summary3 @ N2 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
              & ( A22
                = ( plus_plus_nat @ N2 @ N2 ) )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
                 => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X7 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                   => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
                          = I4 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
                      & ! [X: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X @ N2 )
                              = I4 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X @ N2 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X )
                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) )
          | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A1
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X @ N2 ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
              & ( A22
                = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
                 => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X7 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                   => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
                          = I4 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
                      & ! [X: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X @ N2 )
                              = I4 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X @ N2 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X )
                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
thf(fact_2882_invar__vebt_Ocases,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( ( vEBT_invar_vebt @ A12 @ A23 )
     => ( ( ? [A3: $o,B3: $o] :
              ( A12
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( A23
           != ( suc @ zero_zero_nat ) ) )
       => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( ( A23 = Deg2 )
               => ( ! [X6: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                     => ( vEBT_invar_vebt @ X6 @ N3 ) )
                 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                   => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                     => ( ( M4 = N3 )
                       => ( ( Deg2
                            = ( plus_plus_nat @ N3 @ M4 ) )
                         => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
                           => ~ ! [X6: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_12 ) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
                ( ( A12
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( A23 = Deg2 )
                 => ( ! [X6: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ( vEBT_invar_vebt @ X6 @ N3 ) )
                   => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                       => ( ( M4
                            = ( suc @ N3 ) )
                         => ( ( Deg2
                              = ( plus_plus_nat @ N3 @ M4 ) )
                           => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
                             => ~ ! [X6: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_12 ) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A12
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
                 => ( ( A23 = Deg2 )
                   => ( ! [X6: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_invar_vebt @ X6 @ N3 ) )
                     => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                         => ( ( M4 = N3 )
                           => ( ( Deg2
                                = ( plus_plus_nat @ N3 @ M4 ) )
                             => ( ! [I2: nat] :
                                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                   => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X7 ) )
                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X6: vEBT_VEBT] :
                                        ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_12 ) ) )
                                 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                   => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I2: nat] :
                                              ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                             => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
                                                    = I2 )
                                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
                                                & ! [X6: nat] :
                                                    ( ( ( ( vEBT_VEBT_high @ X6 @ N3 )
                                                        = I2 )
                                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ X6 @ N3 ) ) )
                                                   => ( ( ord_less_nat @ Mi2 @ X6 )
                                                      & ( ord_less_eq_nat @ X6 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A12
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
                   => ( ( A23 = Deg2 )
                     => ( ! [X6: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                           => ( vEBT_invar_vebt @ X6 @ N3 ) )
                       => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                           => ( ( M4
                                = ( suc @ N3 ) )
                             => ( ( Deg2
                                  = ( plus_plus_nat @ N3 @ M4 ) )
                               => ( ! [I2: nat] :
                                      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                     => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X7 ) )
                                        = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X6: vEBT_VEBT] :
                                          ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_12 ) ) )
                                   => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                     => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I2: nat] :
                                                ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
                                                      = I2 )
                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
                                                  & ! [X6: nat] :
                                                      ( ( ( ( vEBT_VEBT_high @ X6 @ N3 )
                                                          = I2 )
                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ X6 @ N3 ) ) )
                                                     => ( ( ord_less_nat @ Mi2 @ X6 )
                                                        & ( ord_less_eq_nat @ X6 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
thf(fact_2883_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X3: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X3 ) ).

% vebt_member.simps(2)
thf(fact_2884_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) ) ).

% VEBT_internal.minNull.simps(5)
thf(fact_2885_inrange,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).

% inrange
thf(fact_2886_set__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_2887_set__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_2888_vebt__pred_Opelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y3: option_nat] :
      ( ( ( vEBT_vebt_pred @ X3 @ Xa2 )
        = Y3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X3
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y3 = none_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,Uw2: $o] :
                ( ( X3
                  = ( vEBT_Leaf @ A3 @ Uw2 ) )
               => ( ( Xa2
                    = ( suc @ zero_zero_nat ) )
                 => ( ( ( A3
                       => ( Y3
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A3
                       => ( Y3 = none_nat ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( X3
                    = ( vEBT_Leaf @ A3 @ B3 ) )
                 => ! [Va2: nat] :
                      ( ( Xa2
                        = ( suc @ ( suc @ Va2 ) ) )
                     => ( ( ( B3
                           => ( Y3
                              = ( some_nat @ one_one_nat ) ) )
                          & ( ~ B3
                           => ( ( A3
                               => ( Y3
                                  = ( some_nat @ zero_zero_nat ) ) )
                              & ( ~ A3
                               => ( Y3 = none_nat ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X3
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
                   => ( ( Y3 = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Xa2 ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( ( X3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) )
                     => ( ( Y3 = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) @ Xa2 ) ) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
                        ( ( X3
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) )
                       => ( ( Y3 = none_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Xa2 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X3
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                         => ( ( ( ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( Y3
                                  = ( some_nat @ Ma2 ) ) )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( Y3
                                  = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                    @ ( if_option_nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( if_option_nat
                                        @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                          = none_nat )
                                        @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
                                        @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                    @ none_nat ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
thf(fact_2889_Diff__eq__empty__iff,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( ( minus_minus_set_nat @ A4 @ B5 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A4 @ B5 ) ) ).

% Diff_eq_empty_iff
thf(fact_2890_Diff__eq__empty__iff,axiom,
    ! [A4: set_real,B5: set_real] :
      ( ( ( minus_minus_set_real @ A4 @ B5 )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ A4 @ B5 ) ) ).

% Diff_eq_empty_iff
thf(fact_2891_Diff__eq__empty__iff,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ( minus_minus_set_int @ A4 @ B5 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A4 @ B5 ) ) ).

% Diff_eq_empty_iff
thf(fact_2892_empty__subsetI,axiom,
    ! [A4: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A4 ) ).

% empty_subsetI
thf(fact_2893_empty__subsetI,axiom,
    ! [A4: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A4 ) ).

% empty_subsetI
thf(fact_2894_empty__subsetI,axiom,
    ! [A4: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A4 ) ).

% empty_subsetI
thf(fact_2895_subset__empty,axiom,
    ! [A4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat )
      = ( A4 = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_2896_subset__empty,axiom,
    ! [A4: set_real] :
      ( ( ord_less_eq_set_real @ A4 @ bot_bot_set_real )
      = ( A4 = bot_bot_set_real ) ) ).

% subset_empty
thf(fact_2897_subset__empty,axiom,
    ! [A4: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ bot_bot_set_int )
      = ( A4 = bot_bot_set_int ) ) ).

% subset_empty
thf(fact_2898_double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( plus_plus_real @ A @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% double_eq_0_iff
thf(fact_2899_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( plus_plus_rat @ A @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
thf(fact_2900_double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( plus_plus_int @ A @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% double_eq_0_iff
thf(fact_2901_unset__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_2902_unset__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_2903_set__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_2904_set__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_2905_set__bit__Suc,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_2906_flip__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_2907_flip__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_2908_flip__bit__Suc,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_2909_subsetI,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( member_nat @ X5 @ B5 ) )
     => ( ord_less_eq_set_nat @ A4 @ B5 ) ) ).

% subsetI
thf(fact_2910_subsetI,axiom,
    ! [A4: set_real,B5: set_real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( member_real @ X5 @ B5 ) )
     => ( ord_less_eq_set_real @ A4 @ B5 ) ) ).

% subsetI
thf(fact_2911_subsetI,axiom,
    ! [A4: set_complex,B5: set_complex] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A4 )
         => ( member_complex @ X5 @ B5 ) )
     => ( ord_le211207098394363844omplex @ A4 @ B5 ) ) ).

% subsetI
thf(fact_2912_subsetI,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( member_int @ X5 @ B5 ) )
     => ( ord_less_eq_set_int @ A4 @ B5 ) ) ).

% subsetI
thf(fact_2913_subset__antisym,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B5 )
     => ( ( ord_less_eq_set_int @ B5 @ A4 )
       => ( A4 = B5 ) ) ) ).

% subset_antisym
thf(fact_2914_psubsetI,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B5 )
     => ( ( A4 != B5 )
       => ( ord_less_set_int @ A4 @ B5 ) ) ) ).

% psubsetI
thf(fact_2915_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_2916_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_2917_i0__less,axiom,
    ! [N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
      = ( N != zero_z5237406670263579293d_enat ) ) ).

% i0_less
thf(fact_2918_idiff__0__right,axiom,
    ! [N: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
      = N ) ).

% idiff_0_right
thf(fact_2919_idiff__0,axiom,
    ! [N: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
      = zero_z5237406670263579293d_enat ) ).

% idiff_0
thf(fact_2920_not__real__square__gt__zero,axiom,
    ! [X3: real] :
      ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X3 @ X3 ) ) )
      = ( X3 = zero_zero_real ) ) ).

% not_real_square_gt_zero
thf(fact_2921_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_2922_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_2923_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_2924_psubset__imp__ex__mem,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( ord_less_set_nat @ A4 @ B5 )
     => ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_2925_psubset__imp__ex__mem,axiom,
    ! [A4: set_real,B5: set_real] :
      ( ( ord_less_set_real @ A4 @ B5 )
     => ? [B3: real] : ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_2926_psubset__imp__ex__mem,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ord_less_set_int @ A4 @ B5 )
     => ? [B3: int] : ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_2927_psubset__imp__ex__mem,axiom,
    ! [A4: set_complex,B5: set_complex] :
      ( ( ord_less_set_complex @ A4 @ B5 )
     => ? [B3: complex] : ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_2928_zdiv__mono__strict,axiom,
    ! [A4: int,B5: int,N: int] :
      ( ( ord_less_int @ A4 @ B5 )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ( ( modulo_modulo_int @ A4 @ N )
            = zero_zero_int )
         => ( ( ( modulo_modulo_int @ B5 @ N )
              = zero_zero_int )
           => ( ord_less_int @ ( divide_divide_int @ A4 @ N ) @ ( divide_divide_int @ B5 @ N ) ) ) ) ) ) ).

% zdiv_mono_strict
thf(fact_2929_verit__le__mono__div__int,axiom,
    ! [A4: int,B5: int,N: int] :
      ( ( ord_less_int @ A4 @ B5 )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int
          @ ( plus_plus_int @ ( divide_divide_int @ A4 @ N )
            @ ( if_int
              @ ( ( modulo_modulo_int @ B5 @ N )
                = zero_zero_int )
              @ one_one_int
              @ zero_zero_int ) )
          @ ( divide_divide_int @ B5 @ N ) ) ) ) ).

% verit_le_mono_div_int
thf(fact_2930_zmod__zmult2__eq,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B2 @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B2 @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) @ ( modulo_modulo_int @ A @ B2 ) ) ) ) ).

% zmod_zmult2_eq
thf(fact_2931_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_2932_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_2933_div__mod__decomp__int,axiom,
    ! [A4: int,N: int] :
      ( A4
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A4 @ N ) @ N ) @ ( modulo_modulo_int @ A4 @ N ) ) ) ).

% div_mod_decomp_int
thf(fact_2934_div__neg__pos__less0,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% div_neg_pos_less0
thf(fact_2935_neg__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ B2 @ zero_zero_int )
     => ( ( ord_less_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int )
        = ( ord_less_int @ zero_zero_int @ A ) ) ) ).

% neg_imp_zdiv_neg_iff
thf(fact_2936_pos__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int )
        = ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% pos_imp_zdiv_neg_iff
thf(fact_2937_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_2938_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) )
        = ( ( ord_less_eq_int @ B2 @ A )
          & ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
thf(fact_2939_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% pos_imp_zdiv_nonneg_iff
thf(fact_2940_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ B2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) )
        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).

% neg_imp_zdiv_nonneg_iff
thf(fact_2941_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_2942_div__nonpos__pos__le0,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% div_nonpos_pos_le0
thf(fact_2943_div__nonneg__neg__le0,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int ) ) ) ).

% div_nonneg_neg_le0
thf(fact_2944_int__div__less__self,axiom,
    ! [X3: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ X3 )
     => ( ( ord_less_int @ one_one_int @ K )
       => ( ord_less_int @ ( divide_divide_int @ X3 @ K ) @ X3 ) ) ) ).

% int_div_less_self
thf(fact_2945_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_2946_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_2947_zdiv__mono2__neg,axiom,
    ! [A: int,B: int,B2: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ B2 )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ) ).

% zdiv_mono2_neg
thf(fact_2948_zdiv__mono1__neg,axiom,
    ! [A: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ A @ A2 )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B2 ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ).

% zdiv_mono1_neg
thf(fact_2949_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_2950_zdiv__mono2,axiom,
    ! [A: int,B: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ B2 )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).

% zdiv_mono2
thf(fact_2951_zdiv__mono1,axiom,
    ! [A: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ A @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).

% zdiv_mono1
thf(fact_2952_zdiv__zmult2__eq,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ).

% zdiv_zmult2_eq
thf(fact_2953_int__div__pos__eq,axiom,
    ! [A: int,B2: int,Q2: int,R: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R )
       => ( ( ord_less_int @ R @ B2 )
         => ( ( divide_divide_int @ A @ B2 )
            = Q2 ) ) ) ) ).

% int_div_pos_eq
thf(fact_2954_int__div__neg__eq,axiom,
    ! [A: int,B2: int,Q2: int,R: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R ) )
     => ( ( ord_less_eq_int @ R @ zero_zero_int )
       => ( ( ord_less_int @ B2 @ R )
         => ( ( divide_divide_int @ A @ B2 )
            = Q2 ) ) ) ) ).

% int_div_neg_eq
thf(fact_2955_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_2956_neg__zmod__mult__2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B2 @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).

% neg_zmod_mult_2
thf(fact_2957_pos__zmod__mult__2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B2 @ A ) ) ) ) ) ).

% pos_zmod_mult_2
thf(fact_2958_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_2959_not__iless0,axiom,
    ! [N: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).

% not_iless0
thf(fact_2960_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_2961_i0__lb,axiom,
    ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).

% i0_lb
thf(fact_2962_ile0__eq,axiom,
    ! [N: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
      = ( N = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
thf(fact_2963_ex__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M3: nat] :
            ( ( ord_less_eq_nat @ M3 @ N )
            & ( P @ M3 ) ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
            & ( P @ X ) ) ) ) ).

% ex_nat_less
thf(fact_2964_all__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ M3 @ N )
           => ( P @ M3 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
           => ( P @ X ) ) ) ) ).

% all_nat_less
thf(fact_2965_not__psubset__empty,axiom,
    ! [A4: set_nat] :
      ~ ( ord_less_set_nat @ A4 @ bot_bot_set_nat ) ).

% not_psubset_empty
thf(fact_2966_not__psubset__empty,axiom,
    ! [A4: set_int] :
      ~ ( ord_less_set_int @ A4 @ bot_bot_set_int ) ).

% not_psubset_empty
thf(fact_2967_not__psubset__empty,axiom,
    ! [A4: set_real] :
      ~ ( ord_less_set_real @ A4 @ bot_bot_set_real ) ).

% not_psubset_empty
thf(fact_2968_in__mono,axiom,
    ! [A4: set_nat,B5: set_nat,X3: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B5 )
     => ( ( member_nat @ X3 @ A4 )
       => ( member_nat @ X3 @ B5 ) ) ) ).

% in_mono
thf(fact_2969_in__mono,axiom,
    ! [A4: set_real,B5: set_real,X3: real] :
      ( ( ord_less_eq_set_real @ A4 @ B5 )
     => ( ( member_real @ X3 @ A4 )
       => ( member_real @ X3 @ B5 ) ) ) ).

% in_mono
thf(fact_2970_in__mono,axiom,
    ! [A4: set_complex,B5: set_complex,X3: complex] :
      ( ( ord_le211207098394363844omplex @ A4 @ B5 )
     => ( ( member_complex @ X3 @ A4 )
       => ( member_complex @ X3 @ B5 ) ) ) ).

% in_mono
thf(fact_2971_in__mono,axiom,
    ! [A4: set_int,B5: set_int,X3: int] :
      ( ( ord_less_eq_set_int @ A4 @ B5 )
     => ( ( member_int @ X3 @ A4 )
       => ( member_int @ X3 @ B5 ) ) ) ).

% in_mono
thf(fact_2972_subsetD,axiom,
    ! [A4: set_nat,B5: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B5 )
     => ( ( member_nat @ C @ A4 )
       => ( member_nat @ C @ B5 ) ) ) ).

% subsetD
thf(fact_2973_subsetD,axiom,
    ! [A4: set_real,B5: set_real,C: real] :
      ( ( ord_less_eq_set_real @ A4 @ B5 )
     => ( ( member_real @ C @ A4 )
       => ( member_real @ C @ B5 ) ) ) ).

% subsetD
thf(fact_2974_subsetD,axiom,
    ! [A4: set_complex,B5: set_complex,C: complex] :
      ( ( ord_le211207098394363844omplex @ A4 @ B5 )
     => ( ( member_complex @ C @ A4 )
       => ( member_complex @ C @ B5 ) ) ) ).

% subsetD
thf(fact_2975_subsetD,axiom,
    ! [A4: set_int,B5: set_int,C: int] :
      ( ( ord_less_eq_set_int @ A4 @ B5 )
     => ( ( member_int @ C @ A4 )
       => ( member_int @ C @ B5 ) ) ) ).

% subsetD
thf(fact_2976_equalityE,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( A4 = B5 )
     => ~ ( ( ord_less_eq_set_int @ A4 @ B5 )
         => ~ ( ord_less_eq_set_int @ B5 @ A4 ) ) ) ).

% equalityE
thf(fact_2977_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ A6 )
         => ( member_nat @ X @ B6 ) ) ) ) ).

% subset_eq
thf(fact_2978_subset__eq,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B6: set_real] :
        ! [X: real] :
          ( ( member_real @ X @ A6 )
         => ( member_real @ X @ B6 ) ) ) ) ).

% subset_eq
thf(fact_2979_subset__eq,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [A6: set_complex,B6: set_complex] :
        ! [X: complex] :
          ( ( member_complex @ X @ A6 )
         => ( member_complex @ X @ B6 ) ) ) ) ).

% subset_eq
thf(fact_2980_subset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
        ! [X: int] :
          ( ( member_int @ X @ A6 )
         => ( member_int @ X @ B6 ) ) ) ) ).

% subset_eq
thf(fact_2981_equalityD1,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( A4 = B5 )
     => ( ord_less_eq_set_int @ A4 @ B5 ) ) ).

% equalityD1
thf(fact_2982_equalityD2,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( A4 = B5 )
     => ( ord_less_eq_set_int @ B5 @ A4 ) ) ).

% equalityD2
thf(fact_2983_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [T2: nat] :
          ( ( member_nat @ T2 @ A6 )
         => ( member_nat @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_2984_subset__iff,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B6: set_real] :
        ! [T2: real] :
          ( ( member_real @ T2 @ A6 )
         => ( member_real @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_2985_subset__iff,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [A6: set_complex,B6: set_complex] :
        ! [T2: complex] :
          ( ( member_complex @ T2 @ A6 )
         => ( member_complex @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_2986_subset__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
        ! [T2: int] :
          ( ( member_int @ T2 @ A6 )
         => ( member_int @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_2987_subset__refl,axiom,
    ! [A4: set_int] : ( ord_less_eq_set_int @ A4 @ A4 ) ).

% subset_refl
thf(fact_2988_Collect__mono,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ! [X5: complex] :
          ( ( P @ X5 )
         => ( Q @ X5 ) )
     => ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).

% Collect_mono
thf(fact_2989_Collect__mono,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ! [X5: real] :
          ( ( P @ X5 )
         => ( Q @ X5 ) )
     => ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).

% Collect_mono
thf(fact_2990_Collect__mono,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ! [X5: list_nat] :
          ( ( P @ X5 )
         => ( Q @ X5 ) )
     => ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_2991_Collect__mono,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X5: nat] :
          ( ( P @ X5 )
         => ( Q @ X5 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_2992_Collect__mono,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X5: int] :
          ( ( P @ X5 )
         => ( Q @ X5 ) )
     => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).

% Collect_mono
thf(fact_2993_subset__trans,axiom,
    ! [A4: set_int,B5: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B5 )
     => ( ( ord_less_eq_set_int @ B5 @ C4 )
       => ( ord_less_eq_set_int @ A4 @ C4 ) ) ) ).

% subset_trans
thf(fact_2994_set__eq__subset,axiom,
    ( ( ^ [Y6: set_int,Z5: set_int] : Y6 = Z5 )
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_eq_set_int @ A6 @ B6 )
          & ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).

% set_eq_subset
thf(fact_2995_Collect__mono__iff,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
      = ( ! [X: complex] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_2996_Collect__mono__iff,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
      = ( ! [X: real] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_2997_Collect__mono__iff,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
      = ( ! [X: list_nat] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_2998_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X: nat] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_2999_Collect__mono__iff,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
      = ( ! [X: int] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_3000_Diff__mono,axiom,
    ! [A4: set_int,C4: set_int,D4: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ C4 )
     => ( ( ord_less_eq_set_int @ D4 @ B5 )
       => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A4 @ B5 ) @ ( minus_minus_set_int @ C4 @ D4 ) ) ) ) ).

% Diff_mono
thf(fact_3001_Diff__subset,axiom,
    ! [A4: set_int,B5: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A4 @ B5 ) @ A4 ) ).

% Diff_subset
thf(fact_3002_double__diff,axiom,
    ! [A4: set_int,B5: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B5 )
     => ( ( ord_less_eq_set_int @ B5 @ C4 )
       => ( ( minus_minus_set_int @ B5 @ ( minus_minus_set_int @ C4 @ A4 ) )
          = A4 ) ) ) ).

% double_diff
thf(fact_3003_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_set_int @ A6 @ B6 )
          | ( A6 = B6 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_3004_subset__psubset__trans,axiom,
    ! [A4: set_int,B5: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B5 )
     => ( ( ord_less_set_int @ B5 @ C4 )
       => ( ord_less_set_int @ A4 @ C4 ) ) ) ).

% subset_psubset_trans
thf(fact_3005_subset__not__subset__eq,axiom,
    ( ord_less_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_eq_set_int @ A6 @ B6 )
          & ~ ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_3006_psubset__subset__trans,axiom,
    ! [A4: set_int,B5: set_int,C4: set_int] :
      ( ( ord_less_set_int @ A4 @ B5 )
     => ( ( ord_less_eq_set_int @ B5 @ C4 )
       => ( ord_less_set_int @ A4 @ C4 ) ) ) ).

% psubset_subset_trans
thf(fact_3007_psubset__imp__subset,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ord_less_set_int @ A4 @ B5 )
     => ( ord_less_eq_set_int @ A4 @ B5 ) ) ).

% psubset_imp_subset
thf(fact_3008_psubset__eq,axiom,
    ( ord_less_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_eq_set_int @ A6 @ B6 )
          & ( A6 != B6 ) ) ) ) ).

% psubset_eq
thf(fact_3009_psubsetE,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ord_less_set_int @ A4 @ B5 )
     => ~ ( ( ord_less_eq_set_int @ A4 @ B5 )
         => ( ord_less_eq_set_int @ B5 @ A4 ) ) ) ).

% psubsetE
thf(fact_3010_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_3011_realpow__pos__nth2,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ? [R3: real] :
          ( ( ord_less_real @ zero_zero_real @ R3 )
          & ( ( power_power_real @ R3 @ ( suc @ N ) )
            = A ) ) ) ).

% realpow_pos_nth2
thf(fact_3012_real__arch__pow__inv,axiom,
    ! [Y3: real,X3: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_real @ X3 @ one_one_real )
       => ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X3 @ N3 ) @ Y3 ) ) ) ).

% real_arch_pow_inv
thf(fact_3013_realpow__pos__nth__unique,axiom,
    ! [N: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [X5: real] :
            ( ( ord_less_real @ zero_zero_real @ X5 )
            & ( ( power_power_real @ X5 @ N )
              = A )
            & ! [Y5: real] :
                ( ( ( ord_less_real @ zero_zero_real @ Y5 )
                  & ( ( power_power_real @ Y5 @ N )
                    = A ) )
               => ( Y5 = X5 ) ) ) ) ) ).

% realpow_pos_nth_unique
thf(fact_3014_realpow__pos__nth,axiom,
    ! [N: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ( ( power_power_real @ R3 @ N )
              = A ) ) ) ) ).

% realpow_pos_nth
thf(fact_3015_neg__zdiv__mult__2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( divide_divide_int @ ( plus_plus_int @ B2 @ one_one_int ) @ A ) ) ) ).

% neg_zdiv_mult_2
thf(fact_3016_pos__zdiv__mult__2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( divide_divide_int @ B2 @ A ) ) ) ).

% pos_zdiv_mult_2
thf(fact_3017_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A6 )
          @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_3018_less__eq__set__def,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( ord_less_eq_real_o
          @ ^ [X: real] : ( member_real @ X @ A6 )
          @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_3019_less__eq__set__def,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [A6: set_complex,B6: set_complex] :
          ( ord_le4573692005234683329plex_o
          @ ^ [X: complex] : ( member_complex @ X @ A6 )
          @ ^ [X: complex] : ( member_complex @ X @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_3020_less__eq__set__def,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ord_less_eq_int_o
          @ ^ [X: int] : ( member_int @ X @ A6 )
          @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_3021_Collect__subset,axiom,
    ! [A4: set_complex,P: complex > $o] :
      ( ord_le211207098394363844omplex
      @ ( collect_complex
        @ ^ [X: complex] :
            ( ( member_complex @ X @ A4 )
            & ( P @ X ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_3022_Collect__subset,axiom,
    ! [A4: set_real,P: real > $o] :
      ( ord_less_eq_set_real
      @ ( collect_real
        @ ^ [X: real] :
            ( ( member_real @ X @ A4 )
            & ( P @ X ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_3023_Collect__subset,axiom,
    ! [A4: set_list_nat,P: list_nat > $o] :
      ( ord_le6045566169113846134st_nat
      @ ( collect_list_nat
        @ ^ [X: list_nat] :
            ( ( member_list_nat @ X @ A4 )
            & ( P @ X ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_3024_Collect__subset,axiom,
    ! [A4: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ A4 )
            & ( P @ X ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_3025_Collect__subset,axiom,
    ! [A4: set_int,P: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X: int] :
            ( ( member_int @ X @ A4 )
            & ( P @ X ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_3026_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_3027_unset__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_3028_unset__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_3029_unset__bit__Suc,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_3030_vebt__member_Opelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_vebt_member @ X3 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X3
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
               => ( ( ( Xa2 = zero_zero_nat )
                   => A3 )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => B3 )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X3
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) )
                       => ( ( Xa2 != Mi2 )
                         => ( ( Xa2 != Ma2 )
                           => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                              & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                               => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                  & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                       => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
thf(fact_3031_vebt__member_Opelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
      ( ( ( vEBT_vebt_member @ X3 @ Xa2 )
        = Y3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X3
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y3
                  = ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ~ Y3
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ( ~ Y3
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X3
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ( ~ Y3
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( Y3
                          = ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
thf(fact_3032_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X3
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
               => ( ( ( Xa2 = zero_zero_nat )
                   => A3 )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => B3 )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa2 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
thf(fact_3033_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X3
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa2 ) )
                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
thf(fact_3034_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
        = Y3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X3
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y3
                  = ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
               => ( ~ Y3
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
                 => ( ( Y3
                      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
thf(fact_3035_vebt__member_Opelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_vebt_member @ X3 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X3
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) )
                 => ~ ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
thf(fact_3036_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X3 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X3
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
                   => ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X3
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
                     => ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 )
                        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X3
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) )
                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
thf(fact_3037_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
      ( ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
        = Y3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X3
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ~ Y3
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
               => ( ~ Y3
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
                 => ( ( Y3
                      = ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X3
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
                   => ( ( Y3
                        = ( ( Xa2 = Mi2 )
                          | ( Xa2 = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) ) ) )
               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X3
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
                     => ( ( Y3
                          = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
thf(fact_3038_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_3039_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_3040_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_3041_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_3042_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_3043_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_3044_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_3045_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_3046_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_3047_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_3048_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_3049_q__pos__lemma,axiom,
    ! [B: int,Q5: int,R4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) )
     => ( ( ord_less_int @ R4 @ B )
       => ( ( ord_less_int @ zero_zero_int @ B )
         => ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).

% q_pos_lemma
thf(fact_3050_int__mod__neg__eq,axiom,
    ! [A: int,B2: int,Q2: int,R: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R ) )
     => ( ( ord_less_eq_int @ R @ zero_zero_int )
       => ( ( ord_less_int @ B2 @ R )
         => ( ( modulo_modulo_int @ A @ B2 )
            = R ) ) ) ) ).

% int_mod_neg_eq
thf(fact_3051_int__mod__pos__eq,axiom,
    ! [A: int,B2: int,Q2: int,R: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R )
       => ( ( ord_less_int @ R @ B2 )
         => ( ( modulo_modulo_int @ A @ B2 )
            = R ) ) ) ) ).

% int_mod_pos_eq
thf(fact_3052_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q2: int,R: int,B: int,Q5: int,R4: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R )
        = ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) )
       => ( ( ord_less_int @ R4 @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ R )
           => ( ( ord_less_int @ zero_zero_int @ B )
             => ( ( ord_less_eq_int @ B @ B2 )
               => ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
thf(fact_3053_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q2: int,R: int,B: int,Q5: int,R4: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R )
        = ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) )
     => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ zero_zero_int )
       => ( ( ord_less_int @ R @ B2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
           => ( ( ord_less_int @ zero_zero_int @ B )
             => ( ( ord_less_eq_int @ B @ B2 )
               => ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
thf(fact_3054_unique__quotient__lemma,axiom,
    ! [B2: int,Q5: int,R4: int,Q2: int,R: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
       => ( ( ord_less_int @ R4 @ B2 )
         => ( ( ord_less_int @ R @ B2 )
           => ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ).

% unique_quotient_lemma
thf(fact_3055_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q5: int,R4: int,Q2: int,R: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R ) )
     => ( ( ord_less_eq_int @ R @ zero_zero_int )
       => ( ( ord_less_int @ B2 @ R )
         => ( ( ord_less_int @ B2 @ R4 )
           => ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ).

% unique_quotient_lemma_neg
thf(fact_3056_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( ( modulo_modulo_int @ M @ D )
        = zero_zero_int )
      = ( ? [Q4: int] :
            ( M
            = ( times_times_int @ D @ Q4 ) ) ) ) ).

% zmod_eq_0_iff
thf(fact_3057_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( ( modulo_modulo_int @ M @ D )
        = zero_zero_int )
     => ? [Q3: int] :
          ( M
          = ( times_times_int @ D @ Q3 ) ) ) ).

% zmod_eq_0D
thf(fact_3058_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_3059_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_3060_neg__mod__conj,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ B2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B2 ) @ zero_zero_int )
        & ( ord_less_int @ B2 @ ( modulo_modulo_int @ A @ B2 ) ) ) ) ).

% neg_mod_conj
thf(fact_3061_pos__mod__conj,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B2 ) )
        & ( ord_less_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 ) ) ) ).

% pos_mod_conj
thf(fact_3062_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_3063_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_3064_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_3065_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_3066_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_3067_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
thf(fact_3068_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_3069_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_3070_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_3071_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_3072_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_3073_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_3074_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_3075_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_3076_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_3077_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_3078_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_3079_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_3080_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_3081_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_3082_less__int__code_I1_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_int_code(1)
thf(fact_3083_unset__bit__less__eq,axiom,
    ! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).

% unset_bit_less_eq
thf(fact_3084_set__bit__greater__eq,axiom,
    ! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).

% set_bit_greater_eq
thf(fact_3085_verit__la__generic,axiom,
    ! [A: int,X3: int] :
      ( ( ord_less_eq_int @ A @ X3 )
      | ( A = X3 )
      | ( ord_less_eq_int @ X3 @ A ) ) ).

% verit_la_generic
thf(fact_3086_less__eq__int__code_I1_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% less_eq_int_code(1)
thf(fact_3087_times__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( times_times_int @ zero_zero_int @ L )
      = zero_zero_int ) ).

% times_int_code(2)
thf(fact_3088_times__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( times_times_int @ K @ zero_zero_int )
      = zero_zero_int ) ).

% times_int_code(1)
thf(fact_3089_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_3090_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_3091_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_3092_psubsetD,axiom,
    ! [A4: set_nat,B5: set_nat,C: nat] :
      ( ( ord_less_set_nat @ A4 @ B5 )
     => ( ( member_nat @ C @ A4 )
       => ( member_nat @ C @ B5 ) ) ) ).

% psubsetD
thf(fact_3093_psubsetD,axiom,
    ! [A4: set_real,B5: set_real,C: real] :
      ( ( ord_less_set_real @ A4 @ B5 )
     => ( ( member_real @ C @ A4 )
       => ( member_real @ C @ B5 ) ) ) ).

% psubsetD
thf(fact_3094_psubsetD,axiom,
    ! [A4: set_int,B5: set_int,C: int] :
      ( ( ord_less_set_int @ A4 @ B5 )
     => ( ( member_int @ C @ A4 )
       => ( member_int @ C @ B5 ) ) ) ).

% psubsetD
thf(fact_3095_psubsetD,axiom,
    ! [A4: set_complex,B5: set_complex,C: complex] :
      ( ( ord_less_set_complex @ A4 @ B5 )
     => ( ( member_complex @ C @ A4 )
       => ( member_complex @ C @ B5 ) ) ) ).

% psubsetD
thf(fact_3096_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( ord_less_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A6 )
          @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ).

% less_set_def
thf(fact_3097_less__set__def,axiom,
    ( ord_less_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( ord_less_real_o
          @ ^ [X: real] : ( member_real @ X @ A6 )
          @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ).

% less_set_def
thf(fact_3098_less__set__def,axiom,
    ( ord_less_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ord_less_int_o
          @ ^ [X: int] : ( member_int @ X @ A6 )
          @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ).

% less_set_def
thf(fact_3099_less__set__def,axiom,
    ( ord_less_set_complex
    = ( ^ [A6: set_complex,B6: set_complex] :
          ( ord_less_complex_o
          @ ^ [X: complex] : ( member_complex @ X @ A6 )
          @ ^ [X: complex] : ( member_complex @ X @ B6 ) ) ) ) ).

% less_set_def
thf(fact_3100_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_3101_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_3102_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X3
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
               => ~ ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
                 => ~ ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 )
                      | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
           => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) )
                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
thf(fact_3103_atLeastatMost__empty,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_rat @ B2 @ A )
     => ( ( set_or633870826150836451st_rat @ A @ B2 )
        = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
thf(fact_3104_atLeastatMost__empty,axiom,
    ! [B2: num,A: num] :
      ( ( ord_less_num @ B2 @ A )
     => ( ( set_or7049704709247886629st_num @ A @ B2 )
        = bot_bot_set_num ) ) ).

% atLeastatMost_empty
thf(fact_3105_atLeastatMost__empty,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ( set_or1269000886237332187st_nat @ A @ B2 )
        = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
thf(fact_3106_atLeastatMost__empty,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ B2 @ A )
     => ( ( set_or1266510415728281911st_int @ A @ B2 )
        = bot_bot_set_int ) ) ).

% atLeastatMost_empty
thf(fact_3107_atLeastatMost__empty,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ( set_or1222579329274155063t_real @ A @ B2 )
        = bot_bot_set_real ) ) ).

% atLeastatMost_empty
thf(fact_3108_atLeastatMost__subset__iff,axiom,
    ! [A: set_int,B2: set_int,C: set_int,D: set_int] :
      ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A @ B2 ) @ ( set_or370866239135849197et_int @ C @ D ) )
      = ( ~ ( ord_less_eq_set_int @ A @ B2 )
        | ( ( ord_less_eq_set_int @ C @ A )
          & ( ord_less_eq_set_int @ B2 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3109_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B2 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
      = ( ~ ( ord_less_eq_rat @ A @ B2 )
        | ( ( ord_less_eq_rat @ C @ A )
          & ( ord_less_eq_rat @ B2 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3110_atLeastatMost__subset__iff,axiom,
    ! [A: num,B2: num,C: num,D: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B2 ) @ ( set_or7049704709247886629st_num @ C @ D ) )
      = ( ~ ( ord_less_eq_num @ A @ B2 )
        | ( ( ord_less_eq_num @ C @ A )
          & ( ord_less_eq_num @ B2 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3111_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B2 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
      = ( ~ ( ord_less_eq_nat @ A @ B2 )
        | ( ( ord_less_eq_nat @ C @ A )
          & ( ord_less_eq_nat @ B2 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3112_atLeastatMost__subset__iff,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B2 ) @ ( set_or1266510415728281911st_int @ C @ D ) )
      = ( ~ ( ord_less_eq_int @ A @ B2 )
        | ( ( ord_less_eq_int @ C @ A )
          & ( ord_less_eq_int @ B2 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3113_atLeastatMost__subset__iff,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ~ ( ord_less_eq_real @ A @ B2 )
        | ( ( ord_less_eq_real @ C @ A )
          & ( ord_less_eq_real @ B2 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3114_atLeastatMost__empty__iff2,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( bot_bot_set_set_int
        = ( set_or370866239135849197et_int @ A @ B2 ) )
      = ( ~ ( ord_less_eq_set_int @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3115_atLeastatMost__empty__iff2,axiom,
    ! [A: rat,B2: rat] :
      ( ( bot_bot_set_rat
        = ( set_or633870826150836451st_rat @ A @ B2 ) )
      = ( ~ ( ord_less_eq_rat @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3116_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B2: num] :
      ( ( bot_bot_set_num
        = ( set_or7049704709247886629st_num @ A @ B2 ) )
      = ( ~ ( ord_less_eq_num @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3117_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B2: nat] :
      ( ( bot_bot_set_nat
        = ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( ~ ( ord_less_eq_nat @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3118_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B2: int] :
      ( ( bot_bot_set_int
        = ( set_or1266510415728281911st_int @ A @ B2 ) )
      = ( ~ ( ord_less_eq_int @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3119_atLeastatMost__empty__iff2,axiom,
    ! [A: real,B2: real] :
      ( ( bot_bot_set_real
        = ( set_or1222579329274155063t_real @ A @ B2 ) )
      = ( ~ ( ord_less_eq_real @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3120_atLeastatMost__empty__iff,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( ( set_or370866239135849197et_int @ A @ B2 )
        = bot_bot_set_set_int )
      = ( ~ ( ord_less_eq_set_int @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3121_atLeastatMost__empty__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( set_or633870826150836451st_rat @ A @ B2 )
        = bot_bot_set_rat )
      = ( ~ ( ord_less_eq_rat @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3122_atLeastatMost__empty__iff,axiom,
    ! [A: num,B2: num] :
      ( ( ( set_or7049704709247886629st_num @ A @ B2 )
        = bot_bot_set_num )
      = ( ~ ( ord_less_eq_num @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3123_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A @ B2 )
        = bot_bot_set_nat )
      = ( ~ ( ord_less_eq_nat @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3124_atLeastatMost__empty__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ( set_or1266510415728281911st_int @ A @ B2 )
        = bot_bot_set_int )
      = ( ~ ( ord_less_eq_int @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3125_atLeastatMost__empty__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ( set_or1222579329274155063t_real @ A @ B2 )
        = bot_bot_set_real )
      = ( ~ ( ord_less_eq_real @ A @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3126_decr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X5: int] :
            ( ( P @ X5 )
           => ( P @ ( minus_minus_int @ X5 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X6: int] :
              ( ( P @ X6 )
             => ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% decr_mult_lemma
thf(fact_3127_incr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X5: int] :
            ( ( P @ X5 )
           => ( P @ ( plus_plus_int @ X5 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X6: int] :
              ( ( P @ X6 )
             => ( P @ ( plus_plus_int @ X6 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% incr_mult_lemma
thf(fact_3128_atLeastAtMost__iff,axiom,
    ! [I: set_int,L: set_int,U: set_int] :
      ( ( member_set_int @ I @ ( set_or370866239135849197et_int @ L @ U ) )
      = ( ( ord_less_eq_set_int @ L @ I )
        & ( ord_less_eq_set_int @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3129_atLeastAtMost__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L @ U ) )
      = ( ( ord_less_eq_rat @ L @ I )
        & ( ord_less_eq_rat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3130_atLeastAtMost__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( ( member_num @ I @ ( set_or7049704709247886629st_num @ L @ U ) )
      = ( ( ord_less_eq_num @ L @ I )
        & ( ord_less_eq_num @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3131_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
      = ( ( ord_less_eq_nat @ L @ I )
        & ( ord_less_eq_nat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3132_atLeastAtMost__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U ) )
      = ( ( ord_less_eq_int @ L @ I )
        & ( ord_less_eq_int @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3133_atLeastAtMost__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U ) )
      = ( ( ord_less_eq_real @ L @ I )
        & ( ord_less_eq_real @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3134_Icc__eq__Icc,axiom,
    ! [L: set_int,H2: set_int,L3: set_int,H3: set_int] :
      ( ( ( set_or370866239135849197et_int @ L @ H2 )
        = ( set_or370866239135849197et_int @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_set_int @ L @ H2 )
          & ~ ( ord_less_eq_set_int @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3135_Icc__eq__Icc,axiom,
    ! [L: rat,H2: rat,L3: rat,H3: rat] :
      ( ( ( set_or633870826150836451st_rat @ L @ H2 )
        = ( set_or633870826150836451st_rat @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_rat @ L @ H2 )
          & ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3136_Icc__eq__Icc,axiom,
    ! [L: num,H2: num,L3: num,H3: num] :
      ( ( ( set_or7049704709247886629st_num @ L @ H2 )
        = ( set_or7049704709247886629st_num @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_num @ L @ H2 )
          & ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3137_Icc__eq__Icc,axiom,
    ! [L: nat,H2: nat,L3: nat,H3: nat] :
      ( ( ( set_or1269000886237332187st_nat @ L @ H2 )
        = ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_nat @ L @ H2 )
          & ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3138_Icc__eq__Icc,axiom,
    ! [L: int,H2: int,L3: int,H3: int] :
      ( ( ( set_or1266510415728281911st_int @ L @ H2 )
        = ( set_or1266510415728281911st_int @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_int @ L @ H2 )
          & ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3139_Icc__eq__Icc,axiom,
    ! [L: real,H2: real,L3: real,H3: real] :
      ( ( ( set_or1222579329274155063t_real @ L @ H2 )
        = ( set_or1222579329274155063t_real @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_real @ L @ H2 )
          & ~ ( ord_less_eq_real @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3140_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z4 )
     => ~ ( ord_less_real @ T @ X6 ) ) ).

% minf(7)
thf(fact_3141_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z4 )
     => ~ ( ord_less_rat @ T @ X6 ) ) ).

% minf(7)
thf(fact_3142_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z4 )
     => ~ ( ord_less_num @ T @ X6 ) ) ).

% minf(7)
thf(fact_3143_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z4 )
     => ~ ( ord_less_nat @ T @ X6 ) ) ).

% minf(7)
thf(fact_3144_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z4 )
     => ~ ( ord_less_int @ T @ X6 ) ) ).

% minf(7)
thf(fact_3145_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z4 )
     => ( ord_less_real @ X6 @ T ) ) ).

% minf(5)
thf(fact_3146_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z4 )
     => ( ord_less_rat @ X6 @ T ) ) ).

% minf(5)
thf(fact_3147_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z4 )
     => ( ord_less_num @ X6 @ T ) ) ).

% minf(5)
thf(fact_3148_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z4 )
     => ( ord_less_nat @ X6 @ T ) ) ).

% minf(5)
thf(fact_3149_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z4 )
     => ( ord_less_int @ X6 @ T ) ) ).

% minf(5)
thf(fact_3150_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z4 )
     => ( X6 != T ) ) ).

% minf(4)
thf(fact_3151_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z4 )
     => ( X6 != T ) ) ).

% minf(4)
thf(fact_3152_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z4 )
     => ( X6 != T ) ) ).

% minf(4)
thf(fact_3153_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z4 )
     => ( X6 != T ) ) ).

% minf(4)
thf(fact_3154_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z4 )
     => ( X6 != T ) ) ).

% minf(4)
thf(fact_3155_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z4 )
     => ( X6 != T ) ) ).

% minf(3)
thf(fact_3156_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z4 )
     => ( X6 != T ) ) ).

% minf(3)
thf(fact_3157_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z4 )
     => ( X6 != T ) ) ).

% minf(3)
thf(fact_3158_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z4 )
     => ( X6 != T ) ) ).

% minf(3)
thf(fact_3159_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z4 )
     => ( X6 != T ) ) ).

% minf(3)
thf(fact_3160_minf_I2_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z3: real] :
        ! [X5: real] :
          ( ( ord_less_real @ X5 @ Z3 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: real] :
          ! [X5: real] :
            ( ( ord_less_real @ X5 @ Z3 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: real] :
          ! [X6: real] :
            ( ( ord_less_real @ X6 @ Z4 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q6 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3161_minf_I2_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z3: rat] :
        ! [X5: rat] :
          ( ( ord_less_rat @ X5 @ Z3 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ X5 @ Z3 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: rat] :
          ! [X6: rat] :
            ( ( ord_less_rat @ X6 @ Z4 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q6 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3162_minf_I2_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z3: num] :
        ! [X5: num] :
          ( ( ord_less_num @ X5 @ Z3 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: num] :
          ! [X5: num] :
            ( ( ord_less_num @ X5 @ Z3 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: num] :
          ! [X6: num] :
            ( ( ord_less_num @ X6 @ Z4 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q6 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3163_minf_I2_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z3: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ X5 @ Z3 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ X5 @ Z3 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ X6 @ Z4 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q6 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3164_minf_I2_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z3: int] :
        ! [X5: int] :
          ( ( ord_less_int @ X5 @ Z3 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z3 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: int] :
          ! [X6: int] :
            ( ( ord_less_int @ X6 @ Z4 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q6 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3165_minf_I1_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z3: real] :
        ! [X5: real] :
          ( ( ord_less_real @ X5 @ Z3 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: real] :
          ! [X5: real] :
            ( ( ord_less_real @ X5 @ Z3 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: real] :
          ! [X6: real] :
            ( ( ord_less_real @ X6 @ Z4 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q6 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3166_minf_I1_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z3: rat] :
        ! [X5: rat] :
          ( ( ord_less_rat @ X5 @ Z3 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ X5 @ Z3 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: rat] :
          ! [X6: rat] :
            ( ( ord_less_rat @ X6 @ Z4 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q6 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3167_minf_I1_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z3: num] :
        ! [X5: num] :
          ( ( ord_less_num @ X5 @ Z3 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: num] :
          ! [X5: num] :
            ( ( ord_less_num @ X5 @ Z3 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: num] :
          ! [X6: num] :
            ( ( ord_less_num @ X6 @ Z4 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q6 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3168_minf_I1_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z3: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ X5 @ Z3 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ X5 @ Z3 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ X6 @ Z4 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q6 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3169_minf_I1_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z3: int] :
        ! [X5: int] :
          ( ( ord_less_int @ X5 @ Z3 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z3 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: int] :
          ! [X6: int] :
            ( ( ord_less_int @ X6 @ Z4 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q6 @ X6 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3170_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z4 @ X6 )
     => ( ord_less_real @ T @ X6 ) ) ).

% pinf(7)
thf(fact_3171_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z4 @ X6 )
     => ( ord_less_rat @ T @ X6 ) ) ).

% pinf(7)
thf(fact_3172_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z4 @ X6 )
     => ( ord_less_num @ T @ X6 ) ) ).

% pinf(7)
thf(fact_3173_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z4 @ X6 )
     => ( ord_less_nat @ T @ X6 ) ) ).

% pinf(7)
thf(fact_3174_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z4 @ X6 )
     => ( ord_less_int @ T @ X6 ) ) ).

% pinf(7)
thf(fact_3175_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z4 @ X6 )
     => ~ ( ord_less_real @ X6 @ T ) ) ).

% pinf(5)
thf(fact_3176_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z4 @ X6 )
     => ~ ( ord_less_rat @ X6 @ T ) ) ).

% pinf(5)
thf(fact_3177_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z4 @ X6 )
     => ~ ( ord_less_num @ X6 @ T ) ) ).

% pinf(5)
thf(fact_3178_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z4 @ X6 )
     => ~ ( ord_less_nat @ X6 @ T ) ) ).

% pinf(5)
thf(fact_3179_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z4 @ X6 )
     => ~ ( ord_less_int @ X6 @ T ) ) ).

% pinf(5)
thf(fact_3180_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z4 @ X6 )
     => ( X6 != T ) ) ).

% pinf(4)
thf(fact_3181_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z4 @ X6 )
     => ( X6 != T ) ) ).

% pinf(4)
thf(fact_3182_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z4 @ X6 )
     => ( X6 != T ) ) ).

% pinf(4)
thf(fact_3183_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z4 @ X6 )
     => ( X6 != T ) ) ).

% pinf(4)
thf(fact_3184_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z4 @ X6 )
     => ( X6 != T ) ) ).

% pinf(4)
thf(fact_3185_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z4 @ X6 )
     => ( X6 != T ) ) ).

% pinf(3)
thf(fact_3186_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z4 @ X6 )
     => ( X6 != T ) ) ).

% pinf(3)
thf(fact_3187_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z4 @ X6 )
     => ( X6 != T ) ) ).

% pinf(3)
thf(fact_3188_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z4 @ X6 )
     => ( X6 != T ) ) ).

% pinf(3)
thf(fact_3189_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z4 @ X6 )
     => ( X6 != T ) ) ).

% pinf(3)
thf(fact_3190_pinf_I2_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z3: real] :
        ! [X5: real] :
          ( ( ord_less_real @ Z3 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: real] :
          ! [X5: real] :
            ( ( ord_less_real @ Z3 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: real] :
          ! [X6: real] :
            ( ( ord_less_real @ Z4 @ X6 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q6 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3191_pinf_I2_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z3: rat] :
        ! [X5: rat] :
          ( ( ord_less_rat @ Z3 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ Z3 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: rat] :
          ! [X6: rat] :
            ( ( ord_less_rat @ Z4 @ X6 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q6 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3192_pinf_I2_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z3: num] :
        ! [X5: num] :
          ( ( ord_less_num @ Z3 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: num] :
          ! [X5: num] :
            ( ( ord_less_num @ Z3 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: num] :
          ! [X6: num] :
            ( ( ord_less_num @ Z4 @ X6 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q6 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3193_pinf_I2_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z3: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ Z3 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ Z3 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ Z4 @ X6 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q6 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3194_pinf_I2_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z3: int] :
        ! [X5: int] :
          ( ( ord_less_int @ Z3 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z3 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: int] :
          ! [X6: int] :
            ( ( ord_less_int @ Z4 @ X6 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                | ( Q6 @ X6 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3195_pinf_I1_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z3: real] :
        ! [X5: real] :
          ( ( ord_less_real @ Z3 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: real] :
          ! [X5: real] :
            ( ( ord_less_real @ Z3 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: real] :
          ! [X6: real] :
            ( ( ord_less_real @ Z4 @ X6 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q6 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3196_pinf_I1_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z3: rat] :
        ! [X5: rat] :
          ( ( ord_less_rat @ Z3 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ Z3 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: rat] :
          ! [X6: rat] :
            ( ( ord_less_rat @ Z4 @ X6 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q6 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3197_pinf_I1_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z3: num] :
        ! [X5: num] :
          ( ( ord_less_num @ Z3 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: num] :
          ! [X5: num] :
            ( ( ord_less_num @ Z3 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: num] :
          ! [X6: num] :
            ( ( ord_less_num @ Z4 @ X6 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q6 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3198_pinf_I1_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z3: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ Z3 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ Z3 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ Z4 @ X6 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q6 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3199_pinf_I1_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z3: int] :
        ! [X5: int] :
          ( ( ord_less_int @ Z3 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z3 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: int] :
          ! [X6: int] :
            ( ( ord_less_int @ Z4 @ X6 )
           => ( ( ( P @ X6 )
                & ( Q @ X6 ) )
              = ( ( P6 @ X6 )
                & ( Q6 @ X6 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3200_bounded__Max__nat,axiom,
    ! [P: nat > $o,X3: nat,M7: nat] :
      ( ( P @ X3 )
     => ( ! [X5: nat] :
            ( ( P @ X5 )
           => ( ord_less_eq_nat @ X5 @ M7 ) )
       => ~ ! [M4: nat] :
              ( ( P @ M4 )
             => ~ ! [X6: nat] :
                    ( ( P @ X6 )
                   => ( ord_less_eq_nat @ X6 @ M4 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_3201_periodic__finite__ex,axiom,
    ! [D: int,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X5: int,K3: int] :
            ( ( P @ X5 )
            = ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D ) ) ) )
       => ( ( ? [X7: int] : ( P @ X7 ) )
          = ( ? [X: int] :
                ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                & ( P @ X ) ) ) ) ) ) ).

% periodic_finite_ex
thf(fact_3202_aset_I7_J,axiom,
    ! [D4: int,A4: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A4 )
                 => ( X6
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ T @ X6 )
           => ( ord_less_int @ T @ ( plus_plus_int @ X6 @ D4 ) ) ) ) ) ).

% aset(7)
thf(fact_3203_aset_I5_J,axiom,
    ! [D4: int,T: int,A4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ A4 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A4 )
                   => ( X6
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ X6 @ T )
             => ( ord_less_int @ ( plus_plus_int @ X6 @ D4 ) @ T ) ) ) ) ) ).

% aset(5)
thf(fact_3204_aset_I4_J,axiom,
    ! [D4: int,T: int,A4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ A4 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A4 )
                   => ( X6
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X6 != T )
             => ( ( plus_plus_int @ X6 @ D4 )
               != T ) ) ) ) ) ).

% aset(4)
thf(fact_3205_aset_I3_J,axiom,
    ! [D4: int,T: int,A4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A4 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A4 )
                   => ( X6
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X6 = T )
             => ( ( plus_plus_int @ X6 @ D4 )
                = T ) ) ) ) ) ).

% aset(3)
thf(fact_3206_bset_I7_J,axiom,
    ! [D4: int,T: int,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ B5 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X6
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ T @ X6 )
             => ( ord_less_int @ T @ ( minus_minus_int @ X6 @ D4 ) ) ) ) ) ) ).

% bset(7)
thf(fact_3207_bset_I5_J,axiom,
    ! [D4: int,B5: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B5 )
                 => ( X6
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ X6 @ T )
           => ( ord_less_int @ ( minus_minus_int @ X6 @ D4 ) @ T ) ) ) ) ).

% bset(5)
thf(fact_3208_bset_I4_J,axiom,
    ! [D4: int,T: int,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ B5 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X6
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X6 != T )
             => ( ( minus_minus_int @ X6 @ D4 )
               != T ) ) ) ) ) ).

% bset(4)
thf(fact_3209_bset_I3_J,axiom,
    ! [D4: int,T: int,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X6
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X6 = T )
             => ( ( minus_minus_int @ X6 @ D4 )
                = T ) ) ) ) ) ).

% bset(3)
thf(fact_3210_aset_I8_J,axiom,
    ! [D4: int,A4: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A4 )
                 => ( X6
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ T @ X6 )
           => ( ord_less_eq_int @ T @ ( plus_plus_int @ X6 @ D4 ) ) ) ) ) ).

% aset(8)
thf(fact_3211_aset_I6_J,axiom,
    ! [D4: int,T: int,A4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A4 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A4 )
                   => ( X6
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ X6 @ T )
             => ( ord_less_eq_int @ ( plus_plus_int @ X6 @ D4 ) @ T ) ) ) ) ) ).

% aset(6)
thf(fact_3212_bset_I8_J,axiom,
    ! [D4: int,T: int,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
       => ! [X6: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X6
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ T @ X6 )
             => ( ord_less_eq_int @ T @ ( minus_minus_int @ X6 @ D4 ) ) ) ) ) ) ).

% bset(8)
thf(fact_3213_bset_I6_J,axiom,
    ! [D4: int,B5: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X6: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B5 )
                 => ( X6
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ X6 @ T )
           => ( ord_less_eq_int @ ( minus_minus_int @ X6 @ D4 ) @ T ) ) ) ) ).

% bset(6)
thf(fact_3214_cppi,axiom,
    ! [D4: int,P: int > $o,P6: int > $o,A4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z3: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z3 @ X5 )
           => ( ( P @ X5 )
              = ( P6 @ X5 ) ) )
       => ( ! [X5: int] :
              ( ! [Xa: int] :
                  ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb: int] :
                      ( ( member_int @ Xb @ A4 )
                     => ( X5
                       != ( minus_minus_int @ Xb @ Xa ) ) ) )
             => ( ( P @ X5 )
               => ( P @ ( plus_plus_int @ X5 @ D4 ) ) ) )
         => ( ! [X5: int,K3: int] :
                ( ( P6 @ X5 )
                = ( P6 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
           => ( ( ? [X7: int] : ( P @ X7 ) )
              = ( ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P6 @ X ) )
                | ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y: int] :
                        ( ( member_int @ Y @ A4 )
                        & ( P @ ( minus_minus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_3215_cpmi,axiom,
    ! [D4: int,P: int > $o,P6: int > $o,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z3: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z3 )
           => ( ( P @ X5 )
              = ( P6 @ X5 ) ) )
       => ( ! [X5: int] :
              ( ! [Xa: int] :
                  ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb: int] :
                      ( ( member_int @ Xb @ B5 )
                     => ( X5
                       != ( plus_plus_int @ Xb @ Xa ) ) ) )
             => ( ( P @ X5 )
               => ( P @ ( minus_minus_int @ X5 @ D4 ) ) ) )
         => ( ! [X5: int,K3: int] :
                ( ( P6 @ X5 )
                = ( P6 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
           => ( ( ? [X7: int] : ( P @ X7 ) )
              = ( ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P6 @ X ) )
                | ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y: int] :
                        ( ( member_int @ Y @ B5 )
                        & ( P @ ( plus_plus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_3216_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z4 )
     => ~ ( ord_less_eq_real @ T @ X6 ) ) ).

% minf(8)
thf(fact_3217_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z4 )
     => ~ ( ord_less_eq_rat @ T @ X6 ) ) ).

% minf(8)
thf(fact_3218_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z4 )
     => ~ ( ord_less_eq_num @ T @ X6 ) ) ).

% minf(8)
thf(fact_3219_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z4 )
     => ~ ( ord_less_eq_nat @ T @ X6 ) ) ).

% minf(8)
thf(fact_3220_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z4 )
     => ~ ( ord_less_eq_int @ T @ X6 ) ) ).

% minf(8)
thf(fact_3221_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z4 )
     => ( ord_less_eq_real @ X6 @ T ) ) ).

% minf(6)
thf(fact_3222_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z4 )
     => ( ord_less_eq_rat @ X6 @ T ) ) ).

% minf(6)
thf(fact_3223_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X6: num] :
      ( ( ord_less_num @ X6 @ Z4 )
     => ( ord_less_eq_num @ X6 @ T ) ) ).

% minf(6)
thf(fact_3224_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z4 )
     => ( ord_less_eq_nat @ X6 @ T ) ) ).

% minf(6)
thf(fact_3225_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z4 )
     => ( ord_less_eq_int @ X6 @ T ) ) ).

% minf(6)
thf(fact_3226_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z4 @ X6 )
     => ( ord_less_eq_real @ T @ X6 ) ) ).

% pinf(8)
thf(fact_3227_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z4 @ X6 )
     => ( ord_less_eq_rat @ T @ X6 ) ) ).

% pinf(8)
thf(fact_3228_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z4 @ X6 )
     => ( ord_less_eq_num @ T @ X6 ) ) ).

% pinf(8)
thf(fact_3229_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z4 @ X6 )
     => ( ord_less_eq_nat @ T @ X6 ) ) ).

% pinf(8)
thf(fact_3230_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z4 @ X6 )
     => ( ord_less_eq_int @ T @ X6 ) ) ).

% pinf(8)
thf(fact_3231_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z4 @ X6 )
     => ~ ( ord_less_eq_real @ X6 @ T ) ) ).

% pinf(6)
thf(fact_3232_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z4 @ X6 )
     => ~ ( ord_less_eq_rat @ X6 @ T ) ) ).

% pinf(6)
thf(fact_3233_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X6: num] :
      ( ( ord_less_num @ Z4 @ X6 )
     => ~ ( ord_less_eq_num @ X6 @ T ) ) ).

% pinf(6)
thf(fact_3234_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z4 @ X6 )
     => ~ ( ord_less_eq_nat @ X6 @ T ) ) ).

% pinf(6)
thf(fact_3235_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z4 @ X6 )
     => ~ ( ord_less_eq_int @ X6 @ T ) ) ).

% pinf(6)
thf(fact_3236_inf__period_I2_J,axiom,
    ! [P: complex > $o,D4: complex,Q: complex > $o] :
      ( ! [X5: complex,K3: complex] :
          ( ( P @ X5 )
          = ( P @ ( minus_minus_complex @ X5 @ ( times_times_complex @ K3 @ D4 ) ) ) )
     => ( ! [X5: complex,K3: complex] :
            ( ( Q @ X5 )
            = ( Q @ ( minus_minus_complex @ X5 @ ( times_times_complex @ K3 @ D4 ) ) ) )
       => ! [X6: complex,K4: complex] :
            ( ( ( P @ X6 )
              | ( Q @ X6 ) )
            = ( ( P @ ( minus_minus_complex @ X6 @ ( times_times_complex @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_complex @ X6 @ ( times_times_complex @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3237_inf__period_I2_J,axiom,
    ! [P: real > $o,D4: real,Q: real > $o] :
      ( ! [X5: real,K3: real] :
          ( ( P @ X5 )
          = ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K3 @ D4 ) ) ) )
     => ( ! [X5: real,K3: real] :
            ( ( Q @ X5 )
            = ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K3 @ D4 ) ) ) )
       => ! [X6: real,K4: real] :
            ( ( ( P @ X6 )
              | ( Q @ X6 ) )
            = ( ( P @ ( minus_minus_real @ X6 @ ( times_times_real @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_real @ X6 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3238_inf__period_I2_J,axiom,
    ! [P: rat > $o,D4: rat,Q: rat > $o] :
      ( ! [X5: rat,K3: rat] :
          ( ( P @ X5 )
          = ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K3 @ D4 ) ) ) )
     => ( ! [X5: rat,K3: rat] :
            ( ( Q @ X5 )
            = ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K3 @ D4 ) ) ) )
       => ! [X6: rat,K4: rat] :
            ( ( ( P @ X6 )
              | ( Q @ X6 ) )
            = ( ( P @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3239_inf__period_I2_J,axiom,
    ! [P: int > $o,D4: int,Q: int > $o] :
      ( ! [X5: int,K3: int] :
          ( ( P @ X5 )
          = ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
     => ( ! [X5: int,K3: int] :
            ( ( Q @ X5 )
            = ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
       => ! [X6: int,K4: int] :
            ( ( ( P @ X6 )
              | ( Q @ X6 ) )
            = ( ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3240_inf__period_I1_J,axiom,
    ! [P: complex > $o,D4: complex,Q: complex > $o] :
      ( ! [X5: complex,K3: complex] :
          ( ( P @ X5 )
          = ( P @ ( minus_minus_complex @ X5 @ ( times_times_complex @ K3 @ D4 ) ) ) )
     => ( ! [X5: complex,K3: complex] :
            ( ( Q @ X5 )
            = ( Q @ ( minus_minus_complex @ X5 @ ( times_times_complex @ K3 @ D4 ) ) ) )
       => ! [X6: complex,K4: complex] :
            ( ( ( P @ X6 )
              & ( Q @ X6 ) )
            = ( ( P @ ( minus_minus_complex @ X6 @ ( times_times_complex @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_complex @ X6 @ ( times_times_complex @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3241_inf__period_I1_J,axiom,
    ! [P: real > $o,D4: real,Q: real > $o] :
      ( ! [X5: real,K3: real] :
          ( ( P @ X5 )
          = ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K3 @ D4 ) ) ) )
     => ( ! [X5: real,K3: real] :
            ( ( Q @ X5 )
            = ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K3 @ D4 ) ) ) )
       => ! [X6: real,K4: real] :
            ( ( ( P @ X6 )
              & ( Q @ X6 ) )
            = ( ( P @ ( minus_minus_real @ X6 @ ( times_times_real @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_real @ X6 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3242_inf__period_I1_J,axiom,
    ! [P: rat > $o,D4: rat,Q: rat > $o] :
      ( ! [X5: rat,K3: rat] :
          ( ( P @ X5 )
          = ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K3 @ D4 ) ) ) )
     => ( ! [X5: rat,K3: rat] :
            ( ( Q @ X5 )
            = ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K3 @ D4 ) ) ) )
       => ! [X6: rat,K4: rat] :
            ( ( ( P @ X6 )
              & ( Q @ X6 ) )
            = ( ( P @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3243_inf__period_I1_J,axiom,
    ! [P: int > $o,D4: int,Q: int > $o] :
      ( ! [X5: int,K3: int] :
          ( ( P @ X5 )
          = ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
     => ( ! [X5: int,K3: int] :
            ( ( Q @ X5 )
            = ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
       => ! [X6: int,K4: int] :
            ( ( ( P @ X6 )
              & ( Q @ X6 ) )
            = ( ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3244_imp__le__cong,axiom,
    ! [X3: int,X8: int,P: $o,P6: $o] :
      ( ( X3 = X8 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
         => ( P = P6 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
           => P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X8 )
           => P6 ) ) ) ) ).

% imp_le_cong
thf(fact_3245_conj__le__cong,axiom,
    ! [X3: int,X8: int,P: $o,P6: $o] :
      ( ( X3 = X8 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
         => ( P = P6 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
            & P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X8 )
            & P6 ) ) ) ) ).

% conj_le_cong
thf(fact_3246_atLeastatMost__psubset__iff,axiom,
    ! [A: set_int,B2: set_int,C: set_int,D: set_int] :
      ( ( ord_less_set_set_int @ ( set_or370866239135849197et_int @ A @ B2 ) @ ( set_or370866239135849197et_int @ C @ D ) )
      = ( ( ~ ( ord_less_eq_set_int @ A @ B2 )
          | ( ( ord_less_eq_set_int @ C @ A )
            & ( ord_less_eq_set_int @ B2 @ D )
            & ( ( ord_less_set_int @ C @ A )
              | ( ord_less_set_int @ B2 @ D ) ) ) )
        & ( ord_less_eq_set_int @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3247_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B2 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_rat @ A @ B2 )
          | ( ( ord_less_eq_rat @ C @ A )
            & ( ord_less_eq_rat @ B2 @ D )
            & ( ( ord_less_rat @ C @ A )
              | ( ord_less_rat @ B2 @ D ) ) ) )
        & ( ord_less_eq_rat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3248_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B2: num,C: num,D: num] :
      ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B2 ) @ ( set_or7049704709247886629st_num @ C @ D ) )
      = ( ( ~ ( ord_less_eq_num @ A @ B2 )
          | ( ( ord_less_eq_num @ C @ A )
            & ( ord_less_eq_num @ B2 @ D )
            & ( ( ord_less_num @ C @ A )
              | ( ord_less_num @ B2 @ D ) ) ) )
        & ( ord_less_eq_num @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3249_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B2 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_nat @ A @ B2 )
          | ( ( ord_less_eq_nat @ C @ A )
            & ( ord_less_eq_nat @ B2 @ D )
            & ( ( ord_less_nat @ C @ A )
              | ( ord_less_nat @ B2 @ D ) ) ) )
        & ( ord_less_eq_nat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3250_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B2 ) @ ( set_or1266510415728281911st_int @ C @ D ) )
      = ( ( ~ ( ord_less_eq_int @ A @ B2 )
          | ( ( ord_less_eq_int @ C @ A )
            & ( ord_less_eq_int @ B2 @ D )
            & ( ( ord_less_int @ C @ A )
              | ( ord_less_int @ B2 @ D ) ) ) )
        & ( ord_less_eq_int @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3251_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ( ~ ( ord_less_eq_real @ A @ B2 )
          | ( ( ord_less_eq_real @ C @ A )
            & ( ord_less_eq_real @ B2 @ D )
            & ( ( ord_less_real @ C @ A )
              | ( ord_less_real @ B2 @ D ) ) ) )
        & ( ord_less_eq_real @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3252_minusinfinity,axiom,
    ! [D: int,P1: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X5: int,K3: int] :
            ( ( P1 @ X5 )
            = ( P1 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D ) ) ) )
       => ( ? [Z3: int] :
            ! [X5: int] :
              ( ( ord_less_int @ X5 @ Z3 )
             => ( ( P @ X5 )
                = ( P1 @ X5 ) ) )
         => ( ? [X_12: int] : ( P1 @ X_12 )
           => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).

% minusinfinity
thf(fact_3253_plusinfinity,axiom,
    ! [D: int,P6: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X5: int,K3: int] :
            ( ( P6 @ X5 )
            = ( P6 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D ) ) ) )
       => ( ? [Z3: int] :
            ! [X5: int] :
              ( ( ord_less_int @ Z3 @ X5 )
             => ( ( P @ X5 )
                = ( P6 @ X5 ) ) )
         => ( ? [X_12: int] : ( P6 @ X_12 )
           => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).

% plusinfinity
thf(fact_3254_Bolzano,axiom,
    ! [A: real,B2: real,P: real > real > $o] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ! [A3: real,B3: real,C2: real] :
            ( ( P @ A3 @ B3 )
           => ( ( P @ B3 @ C2 )
             => ( ( ord_less_eq_real @ A3 @ B3 )
               => ( ( ord_less_eq_real @ B3 @ C2 )
                 => ( P @ A3 @ C2 ) ) ) ) )
       => ( ! [X5: real] :
              ( ( ord_less_eq_real @ A @ X5 )
             => ( ( ord_less_eq_real @ X5 @ B2 )
               => ? [D5: real] :
                    ( ( ord_less_real @ zero_zero_real @ D5 )
                    & ! [A3: real,B3: real] :
                        ( ( ( ord_less_eq_real @ A3 @ X5 )
                          & ( ord_less_eq_real @ X5 @ B3 )
                          & ( ord_less_real @ ( minus_minus_real @ B3 @ A3 ) @ D5 ) )
                       => ( P @ A3 @ B3 ) ) ) ) )
         => ( P @ A @ B2 ) ) ) ) ).

% Bolzano
thf(fact_3255_mult__le__cancel__iff1,axiom,
    ! [Z: real,X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ X3 @ Z ) @ ( times_times_real @ Y3 @ Z ) )
        = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ).

% mult_le_cancel_iff1
thf(fact_3256_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X3: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ X3 @ Z ) @ ( times_times_rat @ Y3 @ Z ) )
        = ( ord_less_eq_rat @ X3 @ Y3 ) ) ) ).

% mult_le_cancel_iff1
thf(fact_3257_mult__le__cancel__iff1,axiom,
    ! [Z: int,X3: int,Y3: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ X3 @ Z ) @ ( times_times_int @ Y3 @ Z ) )
        = ( ord_less_eq_int @ X3 @ Y3 ) ) ) ).

% mult_le_cancel_iff1
thf(fact_3258_mult__le__cancel__iff2,axiom,
    ! [Z: real,X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X3 ) @ ( times_times_real @ Z @ Y3 ) )
        = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ).

% mult_le_cancel_iff2
thf(fact_3259_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X3: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X3 ) @ ( times_times_rat @ Z @ Y3 ) )
        = ( ord_less_eq_rat @ X3 @ Y3 ) ) ) ).

% mult_le_cancel_iff2
thf(fact_3260_mult__le__cancel__iff2,axiom,
    ! [Z: int,X3: int,Y3: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X3 ) @ ( times_times_int @ Z @ Y3 ) )
        = ( ord_less_eq_int @ X3 @ Y3 ) ) ) ).

% mult_le_cancel_iff2
thf(fact_3261_divides__aux__eq,axiom,
    ! [Q2: nat,R: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q2 @ R ) )
      = ( R = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_3262_divides__aux__eq,axiom,
    ! [Q2: int,R: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q2 @ R ) )
      = ( R = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_3263_product__nth,axiom,
    ! [N: nat,Xs2: list_int,Ys: list_int] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs2 ) @ ( size_size_list_int @ Ys ) ) )
     => ( ( nth_Pr4439495888332055232nt_int @ ( product_int_int @ Xs2 @ Ys ) @ N )
        = ( product_Pair_int_int @ ( nth_int @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3264_product__nth,axiom,
    ! [N: nat,Xs2: list_int,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr3474266648193625910T_VEBT @ ( produc662631939642741121T_VEBT @ Xs2 @ Ys ) @ N )
        = ( produc3329399203697025711T_VEBT @ ( nth_int @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3265_product__nth,axiom,
    ! [N: nat,Xs2: list_int,Ys: list_o] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
     => ( ( nth_Pr7514405829937366042_int_o @ ( product_int_o @ Xs2 @ Ys ) @ N )
        = ( product_Pair_int_o @ ( nth_int @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3266_product__nth,axiom,
    ! [N: nat,Xs2: list_int,Ys: list_nat] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) )
     => ( ( nth_Pr8617346907841251940nt_nat @ ( product_int_nat @ Xs2 @ Ys ) @ N )
        = ( product_Pair_int_nat @ ( nth_int @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3267_product__nth,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_int] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys ) ) )
     => ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys ) @ N )
        = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3268_product__nth,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys ) @ N )
        = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3269_product__nth,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_o] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
     => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys ) @ N )
        = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3270_product__nth,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_nat] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) )
     => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys ) @ N )
        = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3271_product__nth,axiom,
    ! [N: nat,Xs2: list_o,Ys: list_int] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys ) ) )
     => ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs2 @ Ys ) @ N )
        = ( product_Pair_o_int @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3272_product__nth,axiom,
    ! [N: nat,Xs2: list_o,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys ) @ N )
        = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3273_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A: int,Q2: int,R: int] :
      ( ( ord_less_eq_int @ B2 @ zero_zero_int )
     => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B2 @ ( product_Pair_int_int @ Q2 @ R ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ ( product_Pair_int_int @ Q2 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R ) @ one_one_int ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_3274_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% length_product
thf(fact_3275_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_o] :
      ( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_3276_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_nat] :
      ( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_3277_length__product,axiom,
    ! [Xs2: list_o,Ys: list_VEBT_VEBT] :
      ( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% length_product
thf(fact_3278_length__product,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_3279_length__product,axiom,
    ! [Xs2: list_o,Ys: list_nat] :
      ( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_3280_length__product,axiom,
    ! [Xs2: list_nat,Ys: list_VEBT_VEBT] :
      ( ( size_s4762443039079500285T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% length_product
thf(fact_3281_length__product,axiom,
    ! [Xs2: list_nat,Ys: list_o] :
      ( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_3282_length__product,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( product_nat_nat @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_3283_unique__remainder,axiom,
    ! [A: int,B2: int,Q2: int,R: int,Q5: int,R4: int] :
      ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q2 @ R ) )
     => ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q5 @ R4 ) )
       => ( R = R4 ) ) ) ).

% unique_remainder
thf(fact_3284_unique__quotient,axiom,
    ! [A: int,B2: int,Q2: int,R: int,Q5: int,R4: int] :
      ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q2 @ R ) )
     => ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q5 @ R4 ) )
       => ( Q2 = Q5 ) ) ) ).

% unique_quotient
thf(fact_3285_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_3286_div__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R ) )
     => ( ( divide_divide_int @ K @ L )
        = Q2 ) ) ).

% div_int_unique
thf(fact_3287_mod__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R ) )
     => ( ( modulo_modulo_int @ K @ L )
        = R ) ) ).

% mod_int_unique
thf(fact_3288_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q2: int] :
      ( ( L != zero_zero_int )
     => ( ( K
          = ( times_times_int @ Q2 @ L ) )
       => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ zero_zero_int ) ) ) ) ).

% eucl_rel_int_dividesI
thf(fact_3289_eucl__rel__int,axiom,
    ! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L ) @ ( modulo_modulo_int @ K @ L ) ) ) ).

% eucl_rel_int
thf(fact_3290_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R ) )
      = ( ( K
          = ( plus_plus_int @ ( times_times_int @ L @ Q2 ) @ R ) )
        & ( ( ord_less_int @ zero_zero_int @ L )
         => ( ( ord_less_eq_int @ zero_zero_int @ R )
            & ( ord_less_int @ R @ L ) ) )
        & ( ~ ( ord_less_int @ zero_zero_int @ L )
         => ( ( ( ord_less_int @ L @ zero_zero_int )
             => ( ( ord_less_int @ L @ R )
                & ( ord_less_eq_int @ R @ zero_zero_int ) ) )
            & ( ~ ( ord_less_int @ L @ zero_zero_int )
             => ( Q2 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_3291_mult__less__iff1,axiom,
    ! [Z: real,X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_real @ ( times_times_real @ X3 @ Z ) @ ( times_times_real @ Y3 @ Z ) )
        = ( ord_less_real @ X3 @ Y3 ) ) ) ).

% mult_less_iff1
thf(fact_3292_mult__less__iff1,axiom,
    ! [Z: rat,X3: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_rat @ ( times_times_rat @ X3 @ Z ) @ ( times_times_rat @ Y3 @ Z ) )
        = ( ord_less_rat @ X3 @ Y3 ) ) ) ).

% mult_less_iff1
thf(fact_3293_mult__less__iff1,axiom,
    ! [Z: int,X3: int,Y3: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_int @ ( times_times_int @ X3 @ Z ) @ ( times_times_int @ Y3 @ Z ) )
        = ( ord_less_int @ X3 @ Y3 ) ) ) ).

% mult_less_iff1
thf(fact_3294_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A: int,Q2: int,R: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B2 )
     => ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q2 @ R ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ ( product_Pair_int_int @ Q2 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_3295_obtain__set__pred,axiom,
    ! [Z: nat,X3: nat,A4: set_nat] :
      ( ( ord_less_nat @ Z @ X3 )
     => ( ( vEBT_VEBT_min_in_set @ A4 @ Z )
       => ( ( finite_finite_nat @ A4 )
         => ? [X_1: nat] : ( vEBT_is_pred_in_set @ A4 @ X3 @ X_1 ) ) ) ) ).

% obtain_set_pred
thf(fact_3296_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_finite
thf(fact_3297_pred__none__empty,axiom,
    ! [Xs2: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : ( vEBT_is_pred_in_set @ Xs2 @ A @ X_1 )
     => ( ( finite_finite_nat @ Xs2 )
       => ~ ? [X6: nat] :
              ( ( member_nat @ X6 @ Xs2 )
              & ( ord_less_nat @ X6 @ A ) ) ) ) ).

% pred_none_empty
thf(fact_3298_prod_Oinject,axiom,
    ! [X1: int,X2: int,Y1: int,Y2: int] :
      ( ( ( product_Pair_int_int @ X1 @ X2 )
        = ( product_Pair_int_int @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_3299_prod_Oinject,axiom,
    ! [X1: code_integer > option6357759511663192854e_term,X2: produc8923325533196201883nteger,Y1: code_integer > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
      ( ( ( produc6137756002093451184nteger @ X1 @ X2 )
        = ( produc6137756002093451184nteger @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_3300_prod_Oinject,axiom,
    ! [X1: produc6241069584506657477e_term > option6357759511663192854e_term,X2: produc8923325533196201883nteger,Y1: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ X1 @ X2 )
        = ( produc8603105652947943368nteger @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_3301_prod_Oinject,axiom,
    ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X2: product_prod_int_int,Y1: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ X1 @ X2 )
        = ( produc5700946648718959541nt_int @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_3302_prod_Oinject,axiom,
    ! [X1: int > option6357759511663192854e_term,X2: product_prod_int_int,Y1: int > option6357759511663192854e_term,Y2: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ X1 @ X2 )
        = ( produc4305682042979456191nt_int @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_3303_old_Oprod_Oinject,axiom,
    ! [A: int,B2: int,A2: int,B: int] :
      ( ( ( product_Pair_int_int @ A @ B2 )
        = ( product_Pair_int_int @ A2 @ B ) )
      = ( ( A = A2 )
        & ( B2 = B ) ) ) ).

% old.prod.inject
thf(fact_3304_old_Oprod_Oinject,axiom,
    ! [A: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger,A2: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( ( produc6137756002093451184nteger @ A @ B2 )
        = ( produc6137756002093451184nteger @ A2 @ B ) )
      = ( ( A = A2 )
        & ( B2 = B ) ) ) ).

% old.prod.inject
thf(fact_3305_old_Oprod_Oinject,axiom,
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger,A2: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ A @ B2 )
        = ( produc8603105652947943368nteger @ A2 @ B ) )
      = ( ( A = A2 )
        & ( B2 = B ) ) ) ).

% old.prod.inject
thf(fact_3306_old_Oprod_Oinject,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int,A2: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ A @ B2 )
        = ( produc5700946648718959541nt_int @ A2 @ B ) )
      = ( ( A = A2 )
        & ( B2 = B ) ) ) ).

% old.prod.inject
thf(fact_3307_old_Oprod_Oinject,axiom,
    ! [A: int > option6357759511663192854e_term,B2: product_prod_int_int,A2: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ A @ B2 )
        = ( produc4305682042979456191nt_int @ A2 @ B ) )
      = ( ( A = A2 )
        & ( B2 = B ) ) ) ).

% old.prod.inject
thf(fact_3308_List_Ofinite__set,axiom,
    ! [Xs2: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_3309_List_Ofinite__set,axiom,
    ! [Xs2: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_3310_List_Ofinite__set,axiom,
    ! [Xs2: list_int] : ( finite_finite_int @ ( set_int2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_3311_List_Ofinite__set,axiom,
    ! [Xs2: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_3312_infinite__Icc__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B2 ) ) )
      = ( ord_less_rat @ A @ B2 ) ) ).

% infinite_Icc_iff
thf(fact_3313_infinite__Icc__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B2 ) ) )
      = ( ord_less_real @ A @ B2 ) ) ).

% infinite_Icc_iff
thf(fact_3314_bounded__nat__set__is__finite,axiom,
    ! [N4: set_nat,N: nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ N4 )
         => ( ord_less_nat @ X5 @ N ) )
     => ( finite_finite_nat @ N4 ) ) ).

% bounded_nat_set_is_finite
thf(fact_3315_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N6: set_nat] :
        ? [M3: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N6 )
         => ( ord_less_nat @ X @ M3 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_3316_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N6: set_nat] :
        ? [M3: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N6 )
         => ( ord_less_eq_nat @ X @ M3 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_3317_finite__list,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ? [Xs3: list_VEBT_VEBT] :
          ( ( set_VEBT_VEBT2 @ Xs3 )
          = A4 ) ) ).

% finite_list
thf(fact_3318_finite__list,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ? [Xs3: list_nat] :
          ( ( set_nat2 @ Xs3 )
          = A4 ) ) ).

% finite_list
thf(fact_3319_finite__list,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ? [Xs3: list_int] :
          ( ( set_int2 @ Xs3 )
          = A4 ) ) ).

% finite_list
thf(fact_3320_finite__list,axiom,
    ! [A4: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ? [Xs3: list_complex] :
          ( ( set_complex2 @ Xs3 )
          = A4 ) ) ).

% finite_list
thf(fact_3321_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K2: nat] :
            ( ( P @ K2 )
            & ( ord_less_nat @ K2 @ I ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_3322_finite__less__ub,axiom,
    ! [F: nat > nat,U: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_3323_finite__lists__length__eq,axiom,
    ! [A4: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A4 )
              & ( ( size_s3451745648224563538omplex @ Xs )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_3324_finite__lists__length__eq,axiom,
    ! [A4: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A4 )
              & ( ( size_s6755466524823107622T_VEBT @ Xs )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_3325_finite__lists__length__eq,axiom,
    ! [A4: set_o,N: nat] :
      ( ( finite_finite_o @ A4 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A4 )
              & ( ( size_size_list_o @ Xs )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_3326_finite__lists__length__eq,axiom,
    ! [A4: set_nat,N: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A4 )
              & ( ( size_size_list_nat @ Xs )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_3327_finite__lists__length__eq,axiom,
    ! [A4: set_int,N: nat] :
      ( ( finite_finite_int @ A4 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A4 )
              & ( ( size_size_list_int @ Xs )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_3328_infinite__Icc,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B2 ) ) ) ).

% infinite_Icc
thf(fact_3329_infinite__Icc,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B2 ) ) ) ).

% infinite_Icc
thf(fact_3330_finite__lists__length__le,axiom,
    ! [A4: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A4 )
              & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_3331_finite__lists__length__le,axiom,
    ! [A4: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A4 )
              & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_3332_finite__lists__length__le,axiom,
    ! [A4: set_o,N: nat] :
      ( ( finite_finite_o @ A4 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A4 )
              & ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_3333_finite__lists__length__le,axiom,
    ! [A4: set_nat,N: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A4 )
              & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_3334_finite__lists__length__le,axiom,
    ! [A4: set_int,N: nat] :
      ( ( finite_finite_int @ A4 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A4 )
              & ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_3335_old_Oprod_Oexhaust,axiom,
    ! [Y3: product_prod_int_int] :
      ~ ! [A3: int,B3: int] :
          ( Y3
         != ( product_Pair_int_int @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_3336_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc8763457246119570046nteger] :
      ~ ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
          ( Y3
         != ( produc6137756002093451184nteger @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_3337_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc1908205239877642774nteger] :
      ~ ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
          ( Y3
         != ( produc8603105652947943368nteger @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_3338_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc2285326912895808259nt_int] :
      ~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
          ( Y3
         != ( produc5700946648718959541nt_int @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_3339_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc7773217078559923341nt_int] :
      ~ ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
          ( Y3
         != ( produc4305682042979456191nt_int @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_3340_surj__pair,axiom,
    ! [P4: product_prod_int_int] :
    ? [X5: int,Y4: int] :
      ( P4
      = ( product_Pair_int_int @ X5 @ Y4 ) ) ).

% surj_pair
thf(fact_3341_surj__pair,axiom,
    ! [P4: produc8763457246119570046nteger] :
    ? [X5: code_integer > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
      ( P4
      = ( produc6137756002093451184nteger @ X5 @ Y4 ) ) ).

% surj_pair
thf(fact_3342_surj__pair,axiom,
    ! [P4: produc1908205239877642774nteger] :
    ? [X5: produc6241069584506657477e_term > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
      ( P4
      = ( produc8603105652947943368nteger @ X5 @ Y4 ) ) ).

% surj_pair
thf(fact_3343_surj__pair,axiom,
    ! [P4: produc2285326912895808259nt_int] :
    ? [X5: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int] :
      ( P4
      = ( produc5700946648718959541nt_int @ X5 @ Y4 ) ) ).

% surj_pair
thf(fact_3344_surj__pair,axiom,
    ! [P4: produc7773217078559923341nt_int] :
    ? [X5: int > option6357759511663192854e_term,Y4: product_prod_int_int] :
      ( P4
      = ( produc4305682042979456191nt_int @ X5 @ Y4 ) ) ).

% surj_pair
thf(fact_3345_prod__cases,axiom,
    ! [P: product_prod_int_int > $o,P4: product_prod_int_int] :
      ( ! [A3: int,B3: int] : ( P @ ( product_Pair_int_int @ A3 @ B3 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_3346_prod__cases,axiom,
    ! [P: produc8763457246119570046nteger > $o,P4: produc8763457246119570046nteger] :
      ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] : ( P @ ( produc6137756002093451184nteger @ A3 @ B3 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_3347_prod__cases,axiom,
    ! [P: produc1908205239877642774nteger > $o,P4: produc1908205239877642774nteger] :
      ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] : ( P @ ( produc8603105652947943368nteger @ A3 @ B3 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_3348_prod__cases,axiom,
    ! [P: produc2285326912895808259nt_int > $o,P4: produc2285326912895808259nt_int] :
      ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] : ( P @ ( produc5700946648718959541nt_int @ A3 @ B3 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_3349_prod__cases,axiom,
    ! [P: produc7773217078559923341nt_int > $o,P4: produc7773217078559923341nt_int] :
      ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] : ( P @ ( produc4305682042979456191nt_int @ A3 @ B3 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_3350_Pair__inject,axiom,
    ! [A: int,B2: int,A2: int,B: int] :
      ( ( ( product_Pair_int_int @ A @ B2 )
        = ( product_Pair_int_int @ A2 @ B ) )
     => ~ ( ( A = A2 )
         => ( B2 != B ) ) ) ).

% Pair_inject
thf(fact_3351_Pair__inject,axiom,
    ! [A: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger,A2: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( ( produc6137756002093451184nteger @ A @ B2 )
        = ( produc6137756002093451184nteger @ A2 @ B ) )
     => ~ ( ( A = A2 )
         => ( B2 != B ) ) ) ).

% Pair_inject
thf(fact_3352_Pair__inject,axiom,
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger,A2: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ A @ B2 )
        = ( produc8603105652947943368nteger @ A2 @ B ) )
     => ~ ( ( A = A2 )
         => ( B2 != B ) ) ) ).

% Pair_inject
thf(fact_3353_Pair__inject,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int,A2: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ A @ B2 )
        = ( produc5700946648718959541nt_int @ A2 @ B ) )
     => ~ ( ( A = A2 )
         => ( B2 != B ) ) ) ).

% Pair_inject
thf(fact_3354_Pair__inject,axiom,
    ! [A: int > option6357759511663192854e_term,B2: product_prod_int_int,A2: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ A @ B2 )
        = ( produc4305682042979456191nt_int @ A2 @ B ) )
     => ~ ( ( A = A2 )
         => ( B2 != B ) ) ) ).

% Pair_inject
thf(fact_3355_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N4: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
     => ( finite_finite_nat @ N4 ) ) ).

% subset_eq_atLeast0_atMost_finite
thf(fact_3356_prod__cases3,axiom,
    ! [Y3: produc8763457246119570046nteger] :
      ~ ! [A3: code_integer > option6357759511663192854e_term,B3: code_integer,C2: code_integer] :
          ( Y3
         != ( produc6137756002093451184nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C2 ) ) ) ).

% prod_cases3
thf(fact_3357_prod__cases3,axiom,
    ! [Y3: produc1908205239877642774nteger] :
      ~ ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: code_integer,C2: code_integer] :
          ( Y3
         != ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C2 ) ) ) ).

% prod_cases3
thf(fact_3358_prod__cases3,axiom,
    ! [Y3: produc2285326912895808259nt_int] :
      ~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: int,C2: int] :
          ( Y3
         != ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C2 ) ) ) ).

% prod_cases3
thf(fact_3359_prod__cases3,axiom,
    ! [Y3: produc7773217078559923341nt_int] :
      ~ ! [A3: int > option6357759511663192854e_term,B3: int,C2: int] :
          ( Y3
         != ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C2 ) ) ) ).

% prod_cases3
thf(fact_3360_prod__induct3,axiom,
    ! [P: produc8763457246119570046nteger > $o,X3: produc8763457246119570046nteger] :
      ( ! [A3: code_integer > option6357759511663192854e_term,B3: code_integer,C2: code_integer] : ( P @ ( produc6137756002093451184nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C2 ) ) )
     => ( P @ X3 ) ) ).

% prod_induct3
thf(fact_3361_prod__induct3,axiom,
    ! [P: produc1908205239877642774nteger > $o,X3: produc1908205239877642774nteger] :
      ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: code_integer,C2: code_integer] : ( P @ ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C2 ) ) )
     => ( P @ X3 ) ) ).

% prod_induct3
thf(fact_3362_prod__induct3,axiom,
    ! [P: produc2285326912895808259nt_int > $o,X3: produc2285326912895808259nt_int] :
      ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: int,C2: int] : ( P @ ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C2 ) ) )
     => ( P @ X3 ) ) ).

% prod_induct3
thf(fact_3363_prod__induct3,axiom,
    ! [P: produc7773217078559923341nt_int > $o,X3: produc7773217078559923341nt_int] :
      ( ! [A3: int > option6357759511663192854e_term,B3: int,C2: int] : ( P @ ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C2 ) ) )
     => ( P @ X3 ) ) ).

% prod_induct3
thf(fact_3364_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_3365_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_3366_finite__Collect__subsets,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_3367_finite__Collect__subsets,axiom,
    ! [A4: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( finite6551019134538273531omplex
        @ ( collect_set_complex
          @ ^ [B6: set_complex] : ( ord_le211207098394363844omplex @ B6 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_3368_finite__Collect__subsets,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( finite6197958912794628473et_int
        @ ( collect_set_int
          @ ^ [B6: set_int] : ( ord_less_eq_set_int @ B6 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_3369_finite__Collect__bounded__ex,axiom,
    ! [P: real > $o,Q: real > real > $o] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [X: real] :
              ? [Y: real] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: real] :
              ( ( P @ Y )
             => ( finite_finite_real
                @ ( collect_real
                  @ ^ [X: real] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_3370_finite__Collect__bounded__ex,axiom,
    ! [P: real > $o,Q: nat > real > $o] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
              ? [Y: real] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: real] :
              ( ( P @ Y )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_3371_finite__Collect__bounded__ex,axiom,
    ! [P: real > $o,Q: int > real > $o] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [X: int] :
              ? [Y: real] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: real] :
              ( ( P @ Y )
             => ( finite_finite_int
                @ ( collect_int
                  @ ^ [X: int] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_3372_finite__Collect__bounded__ex,axiom,
    ! [P: real > $o,Q: complex > real > $o] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [X: complex] :
              ? [Y: real] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: real] :
              ( ( P @ Y )
             => ( finite3207457112153483333omplex
                @ ( collect_complex
                  @ ^ [X: complex] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_3373_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q: real > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [X: real] :
              ? [Y: nat] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: nat] :
              ( ( P @ Y )
             => ( finite_finite_real
                @ ( collect_real
                  @ ^ [X: real] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_3374_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q: nat > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
              ? [Y: nat] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: nat] :
              ( ( P @ Y )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_3375_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q: int > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [X: int] :
              ? [Y: nat] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: nat] :
              ( ( P @ Y )
             => ( finite_finite_int
                @ ( collect_int
                  @ ^ [X: int] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_3376_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q: complex > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [X: complex] :
              ? [Y: nat] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: nat] :
              ( ( P @ Y )
             => ( finite3207457112153483333omplex
                @ ( collect_complex
                  @ ^ [X: complex] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_3377_finite__Collect__bounded__ex,axiom,
    ! [P: int > $o,Q: real > int > $o] :
      ( ( finite_finite_int @ ( collect_int @ P ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [X: real] :
              ? [Y: int] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: int] :
              ( ( P @ Y )
             => ( finite_finite_real
                @ ( collect_real
                  @ ^ [X: real] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_3378_finite__Collect__bounded__ex,axiom,
    ! [P: int > $o,Q: nat > int > $o] :
      ( ( finite_finite_int @ ( collect_int @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
              ? [Y: int] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: int] :
              ( ( P @ Y )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_3379_finite__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Z2: real] :
              ( ( power_power_real @ Z2 @ N )
              = one_one_real ) ) ) ) ).

% finite_roots_unity
thf(fact_3380_finite__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z2: complex] :
              ( ( power_power_complex @ Z2 @ N )
              = one_one_complex ) ) ) ) ).

% finite_roots_unity
thf(fact_3381_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X3: real > complex,Y3: real > complex] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I4: real] :
              ( ( member_real @ I4 @ I5 )
              & ( ( X3 @ I4 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( times_times_complex @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3382_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X3: nat > complex,Y3: nat > complex] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I4: nat] :
              ( ( member_nat @ I4 @ I5 )
              & ( ( X3 @ I4 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( times_times_complex @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3383_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X3: int > complex,Y3: int > complex] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I4: int] :
              ( ( member_int @ I4 @ I5 )
              & ( ( X3 @ I4 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( times_times_complex @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3384_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X3: complex > complex,Y3: complex > complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I4: complex] :
              ( ( member_complex @ I4 @ I5 )
              & ( ( X3 @ I4 )
               != one_one_complex ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != one_one_complex ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( times_times_complex @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3385_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X3: real > real,Y3: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I4: real] :
              ( ( member_real @ I4 @ I5 )
              & ( ( X3 @ I4 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != one_one_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( times_times_real @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3386_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X3: nat > real,Y3: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I4: nat] :
              ( ( member_nat @ I4 @ I5 )
              & ( ( X3 @ I4 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != one_one_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( times_times_real @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3387_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X3: int > real,Y3: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I4: int] :
              ( ( member_int @ I4 @ I5 )
              & ( ( X3 @ I4 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != one_one_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( times_times_real @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3388_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X3: complex > real,Y3: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I4: complex] :
              ( ( member_complex @ I4 @ I5 )
              & ( ( X3 @ I4 )
               != one_one_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != one_one_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( times_times_real @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3389_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X3: real > rat,Y3: real > rat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I4: real] :
              ( ( member_real @ I4 @ I5 )
              & ( ( X3 @ I4 )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != one_one_rat ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( times_times_rat @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3390_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X3: nat > rat,Y3: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I4: nat] :
              ( ( member_nat @ I4 @ I5 )
              & ( ( X3 @ I4 )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != one_one_rat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( times_times_rat @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3391_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X3: real > complex,Y3: real > complex] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I4: real] :
              ( ( member_real @ I4 @ I5 )
              & ( ( X3 @ I4 )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( plus_plus_complex @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3392_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X3: nat > complex,Y3: nat > complex] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I4: nat] :
              ( ( member_nat @ I4 @ I5 )
              & ( ( X3 @ I4 )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( plus_plus_complex @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3393_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X3: int > complex,Y3: int > complex] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I4: int] :
              ( ( member_int @ I4 @ I5 )
              & ( ( X3 @ I4 )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( plus_plus_complex @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3394_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X3: complex > complex,Y3: complex > complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I4: complex] :
              ( ( member_complex @ I4 @ I5 )
              & ( ( X3 @ I4 )
               != zero_zero_complex ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != zero_zero_complex ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( plus_plus_complex @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3395_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X3: real > real,Y3: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I4: real] :
              ( ( member_real @ I4 @ I5 )
              & ( ( X3 @ I4 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( plus_plus_real @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3396_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X3: nat > real,Y3: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I4: nat] :
              ( ( member_nat @ I4 @ I5 )
              & ( ( X3 @ I4 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( plus_plus_real @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3397_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X3: int > real,Y3: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I4: int] :
              ( ( member_int @ I4 @ I5 )
              & ( ( X3 @ I4 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( plus_plus_real @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3398_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X3: complex > real,Y3: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I4: complex] :
              ( ( member_complex @ I4 @ I5 )
              & ( ( X3 @ I4 )
               != zero_zero_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != zero_zero_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( plus_plus_real @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3399_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X3: real > rat,Y3: real > rat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I4: real] :
              ( ( member_real @ I4 @ I5 )
              & ( ( X3 @ I4 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( plus_plus_rat @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3400_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X3: nat > rat,Y3: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I4: nat] :
              ( ( member_nat @ I4 @ I5 )
              & ( ( X3 @ I4 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( Y3 @ I4 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( plus_plus_rat @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3401_finite__interval__int1,axiom,
    ! [A: int,B2: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I4: int] :
            ( ( ord_less_eq_int @ A @ I4 )
            & ( ord_less_eq_int @ I4 @ B2 ) ) ) ) ).

% finite_interval_int1
thf(fact_3402_finite__interval__int4,axiom,
    ! [A: int,B2: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I4: int] :
            ( ( ord_less_int @ A @ I4 )
            & ( ord_less_int @ I4 @ B2 ) ) ) ) ).

% finite_interval_int4
thf(fact_3403_finite__interval__int3,axiom,
    ! [A: int,B2: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I4: int] :
            ( ( ord_less_int @ A @ I4 )
            & ( ord_less_eq_int @ I4 @ B2 ) ) ) ) ).

% finite_interval_int3
thf(fact_3404_finite__interval__int2,axiom,
    ! [A: int,B2: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I4: int] :
            ( ( ord_less_eq_int @ A @ I4 )
            & ( ord_less_int @ I4 @ B2 ) ) ) ) ).

% finite_interval_int2
thf(fact_3405_finite__maxlen,axiom,
    ! [M7: set_list_VEBT_VEBT] :
      ( ( finite3004134309566078307T_VEBT @ M7 )
     => ? [N3: nat] :
        ! [X6: list_VEBT_VEBT] :
          ( ( member2936631157270082147T_VEBT @ X6 @ M7 )
         => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X6 ) @ N3 ) ) ) ).

% finite_maxlen
thf(fact_3406_finite__maxlen,axiom,
    ! [M7: set_list_o] :
      ( ( finite_finite_list_o @ M7 )
     => ? [N3: nat] :
        ! [X6: list_o] :
          ( ( member_list_o @ X6 @ M7 )
         => ( ord_less_nat @ ( size_size_list_o @ X6 ) @ N3 ) ) ) ).

% finite_maxlen
thf(fact_3407_finite__maxlen,axiom,
    ! [M7: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ M7 )
     => ? [N3: nat] :
        ! [X6: list_nat] :
          ( ( member_list_nat @ X6 @ M7 )
         => ( ord_less_nat @ ( size_size_list_nat @ X6 ) @ N3 ) ) ) ).

% finite_maxlen
thf(fact_3408_finite__has__minimal2,axiom,
    ! [A4: set_real,A: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ A @ A4 )
       => ? [X5: real] :
            ( ( member_real @ X5 @ A4 )
            & ( ord_less_eq_real @ X5 @ A )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A4 )
               => ( ( ord_less_eq_real @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_3409_finite__has__minimal2,axiom,
    ! [A4: set_set_int,A: set_int] :
      ( ( finite6197958912794628473et_int @ A4 )
     => ( ( member_set_int @ A @ A4 )
       => ? [X5: set_int] :
            ( ( member_set_int @ X5 @ A4 )
            & ( ord_less_eq_set_int @ X5 @ A )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A4 )
               => ( ( ord_less_eq_set_int @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_3410_finite__has__minimal2,axiom,
    ! [A4: set_rat,A: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( member_rat @ A @ A4 )
       => ? [X5: rat] :
            ( ( member_rat @ X5 @ A4 )
            & ( ord_less_eq_rat @ X5 @ A )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A4 )
               => ( ( ord_less_eq_rat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_3411_finite__has__minimal2,axiom,
    ! [A4: set_num,A: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( member_num @ A @ A4 )
       => ? [X5: num] :
            ( ( member_num @ X5 @ A4 )
            & ( ord_less_eq_num @ X5 @ A )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A4 )
               => ( ( ord_less_eq_num @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_3412_finite__has__minimal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ A @ A4 )
       => ? [X5: nat] :
            ( ( member_nat @ X5 @ A4 )
            & ( ord_less_eq_nat @ X5 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A4 )
               => ( ( ord_less_eq_nat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_3413_finite__has__minimal2,axiom,
    ! [A4: set_int,A: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ A @ A4 )
       => ? [X5: int] :
            ( ( member_int @ X5 @ A4 )
            & ( ord_less_eq_int @ X5 @ A )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A4 )
               => ( ( ord_less_eq_int @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_3414_finite__has__maximal2,axiom,
    ! [A4: set_real,A: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ A @ A4 )
       => ? [X5: real] :
            ( ( member_real @ X5 @ A4 )
            & ( ord_less_eq_real @ A @ X5 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A4 )
               => ( ( ord_less_eq_real @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_3415_finite__has__maximal2,axiom,
    ! [A4: set_set_int,A: set_int] :
      ( ( finite6197958912794628473et_int @ A4 )
     => ( ( member_set_int @ A @ A4 )
       => ? [X5: set_int] :
            ( ( member_set_int @ X5 @ A4 )
            & ( ord_less_eq_set_int @ A @ X5 )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A4 )
               => ( ( ord_less_eq_set_int @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_3416_finite__has__maximal2,axiom,
    ! [A4: set_rat,A: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( member_rat @ A @ A4 )
       => ? [X5: rat] :
            ( ( member_rat @ X5 @ A4 )
            & ( ord_less_eq_rat @ A @ X5 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A4 )
               => ( ( ord_less_eq_rat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_3417_finite__has__maximal2,axiom,
    ! [A4: set_num,A: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( member_num @ A @ A4 )
       => ? [X5: num] :
            ( ( member_num @ X5 @ A4 )
            & ( ord_less_eq_num @ A @ X5 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A4 )
               => ( ( ord_less_eq_num @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_3418_finite__has__maximal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ A @ A4 )
       => ? [X5: nat] :
            ( ( member_nat @ X5 @ A4 )
            & ( ord_less_eq_nat @ A @ X5 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A4 )
               => ( ( ord_less_eq_nat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_3419_finite__has__maximal2,axiom,
    ! [A4: set_int,A: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ A @ A4 )
       => ? [X5: int] :
            ( ( member_int @ X5 @ A4 )
            & ( ord_less_eq_int @ A @ X5 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A4 )
               => ( ( ord_less_eq_int @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_3420_rev__finite__subset,axiom,
    ! [B5: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( finite_finite_nat @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_3421_rev__finite__subset,axiom,
    ! [B5: set_complex,A4: set_complex] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( finite3207457112153483333omplex @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_3422_rev__finite__subset,axiom,
    ! [B5: set_int,A4: set_int] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( finite_finite_int @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_3423_infinite__super,axiom,
    ! [S2: set_nat,T3: set_nat] :
      ( ( ord_less_eq_set_nat @ S2 @ T3 )
     => ( ~ ( finite_finite_nat @ S2 )
       => ~ ( finite_finite_nat @ T3 ) ) ) ).

% infinite_super
thf(fact_3424_infinite__super,axiom,
    ! [S2: set_complex,T3: set_complex] :
      ( ( ord_le211207098394363844omplex @ S2 @ T3 )
     => ( ~ ( finite3207457112153483333omplex @ S2 )
       => ~ ( finite3207457112153483333omplex @ T3 ) ) ) ).

% infinite_super
thf(fact_3425_infinite__super,axiom,
    ! [S2: set_int,T3: set_int] :
      ( ( ord_less_eq_set_int @ S2 @ T3 )
     => ( ~ ( finite_finite_int @ S2 )
       => ~ ( finite_finite_int @ T3 ) ) ) ).

% infinite_super
thf(fact_3426_finite__subset,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B5 )
     => ( ( finite_finite_nat @ B5 )
       => ( finite_finite_nat @ A4 ) ) ) ).

% finite_subset
thf(fact_3427_finite__subset,axiom,
    ! [A4: set_complex,B5: set_complex] :
      ( ( ord_le211207098394363844omplex @ A4 @ B5 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( finite3207457112153483333omplex @ A4 ) ) ) ).

% finite_subset
thf(fact_3428_finite__subset,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B5 )
     => ( ( finite_finite_int @ B5 )
       => ( finite_finite_int @ A4 ) ) ) ).

% finite_subset
thf(fact_3429_finite__psubset__induct,axiom,
    ! [A4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [A7: set_nat] :
            ( ( finite_finite_nat @ A7 )
           => ( ! [B7: set_nat] :
                  ( ( ord_less_set_nat @ B7 @ A7 )
                 => ( P @ B7 ) )
             => ( P @ A7 ) ) )
       => ( P @ A4 ) ) ) ).

% finite_psubset_induct
thf(fact_3430_finite__psubset__induct,axiom,
    ! [A4: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ! [A7: set_int] :
            ( ( finite_finite_int @ A7 )
           => ( ! [B7: set_int] :
                  ( ( ord_less_set_int @ B7 @ A7 )
                 => ( P @ B7 ) )
             => ( P @ A7 ) ) )
       => ( P @ A4 ) ) ) ).

% finite_psubset_induct
thf(fact_3431_finite__psubset__induct,axiom,
    ! [A4: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [A7: set_complex] :
            ( ( finite3207457112153483333omplex @ A7 )
           => ( ! [B7: set_complex] :
                  ( ( ord_less_set_complex @ B7 @ A7 )
                 => ( P @ B7 ) )
             => ( P @ A7 ) ) )
       => ( P @ A4 ) ) ) ).

% finite_psubset_induct
thf(fact_3432_finite__image__set,axiom,
    ! [P: real > $o,F: real > real] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Uu3: real] :
            ? [X: real] :
              ( ( Uu3
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_3433_finite__image__set,axiom,
    ! [P: real > $o,F: real > nat] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Uu3: nat] :
            ? [X: real] :
              ( ( Uu3
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_3434_finite__image__set,axiom,
    ! [P: real > $o,F: real > int] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [Uu3: int] :
            ? [X: real] :
              ( ( Uu3
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_3435_finite__image__set,axiom,
    ! [P: real > $o,F: real > complex] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Uu3: complex] :
            ? [X: real] :
              ( ( Uu3
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_3436_finite__image__set,axiom,
    ! [P: nat > $o,F: nat > real] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Uu3: real] :
            ? [X: nat] :
              ( ( Uu3
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_3437_finite__image__set,axiom,
    ! [P: nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Uu3: nat] :
            ? [X: nat] :
              ( ( Uu3
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_3438_finite__image__set,axiom,
    ! [P: nat > $o,F: nat > int] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [Uu3: int] :
            ? [X: nat] :
              ( ( Uu3
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_3439_finite__image__set,axiom,
    ! [P: nat > $o,F: nat > complex] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Uu3: complex] :
            ? [X: nat] :
              ( ( Uu3
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_3440_finite__image__set,axiom,
    ! [P: int > $o,F: int > real] :
      ( ( finite_finite_int @ ( collect_int @ P ) )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Uu3: real] :
            ? [X: int] :
              ( ( Uu3
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_3441_finite__image__set,axiom,
    ! [P: int > $o,F: int > nat] :
      ( ( finite_finite_int @ ( collect_int @ P ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Uu3: nat] :
            ? [X: int] :
              ( ( Uu3
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_3442_finite__image__set2,axiom,
    ! [P: real > $o,Q: real > $o,F: real > real > real] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_real @ ( collect_real @ Q ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [Uu3: real] :
              ? [X: real,Y: real] :
                ( ( Uu3
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_3443_finite__image__set2,axiom,
    ! [P: real > $o,Q: real > $o,F: real > real > nat] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_real @ ( collect_real @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu3: nat] :
              ? [X: real,Y: real] :
                ( ( Uu3
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_3444_finite__image__set2,axiom,
    ! [P: real > $o,Q: real > $o,F: real > real > int] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_real @ ( collect_real @ Q ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [Uu3: int] :
              ? [X: real,Y: real] :
                ( ( Uu3
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_3445_finite__image__set2,axiom,
    ! [P: real > $o,Q: real > $o,F: real > real > complex] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_real @ ( collect_real @ Q ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [Uu3: complex] :
              ? [X: real,Y: real] :
                ( ( Uu3
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_3446_finite__image__set2,axiom,
    ! [P: real > $o,Q: nat > $o,F: real > nat > real] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [Uu3: real] :
              ? [X: real,Y: nat] :
                ( ( Uu3
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_3447_finite__image__set2,axiom,
    ! [P: real > $o,Q: nat > $o,F: real > nat > nat] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu3: nat] :
              ? [X: real,Y: nat] :
                ( ( Uu3
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_3448_finite__image__set2,axiom,
    ! [P: real > $o,Q: nat > $o,F: real > nat > int] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [Uu3: int] :
              ? [X: real,Y: nat] :
                ( ( Uu3
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_3449_finite__image__set2,axiom,
    ! [P: real > $o,Q: nat > $o,F: real > nat > complex] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [Uu3: complex] :
              ? [X: real,Y: nat] :
                ( ( Uu3
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_3450_finite__image__set2,axiom,
    ! [P: real > $o,Q: int > $o,F: real > int > real] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_int @ ( collect_int @ Q ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [Uu3: real] :
              ? [X: real,Y: int] :
                ( ( Uu3
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_3451_finite__image__set2,axiom,
    ! [P: real > $o,Q: int > $o,F: real > int > nat] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_int @ ( collect_int @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu3: nat] :
              ? [X: real,Y: int] :
                ( ( Uu3
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_3452_finite__has__minimal,axiom,
    ! [A4: set_real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ? [X5: real] :
            ( ( member_real @ X5 @ A4 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A4 )
               => ( ( ord_less_eq_real @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_3453_finite__has__minimal,axiom,
    ! [A4: set_set_int] :
      ( ( finite6197958912794628473et_int @ A4 )
     => ( ( A4 != bot_bot_set_set_int )
       => ? [X5: set_int] :
            ( ( member_set_int @ X5 @ A4 )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A4 )
               => ( ( ord_less_eq_set_int @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_3454_finite__has__minimal,axiom,
    ! [A4: set_rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ? [X5: rat] :
            ( ( member_rat @ X5 @ A4 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A4 )
               => ( ( ord_less_eq_rat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_3455_finite__has__minimal,axiom,
    ! [A4: set_num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ? [X5: num] :
            ( ( member_num @ X5 @ A4 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A4 )
               => ( ( ord_less_eq_num @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_3456_finite__has__minimal,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ? [X5: nat] :
            ( ( member_nat @ X5 @ A4 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A4 )
               => ( ( ord_less_eq_nat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_3457_finite__has__minimal,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ? [X5: int] :
            ( ( member_int @ X5 @ A4 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A4 )
               => ( ( ord_less_eq_int @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_3458_finite__has__maximal,axiom,
    ! [A4: set_real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ? [X5: real] :
            ( ( member_real @ X5 @ A4 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A4 )
               => ( ( ord_less_eq_real @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_3459_finite__has__maximal,axiom,
    ! [A4: set_set_int] :
      ( ( finite6197958912794628473et_int @ A4 )
     => ( ( A4 != bot_bot_set_set_int )
       => ? [X5: set_int] :
            ( ( member_set_int @ X5 @ A4 )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A4 )
               => ( ( ord_less_eq_set_int @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_3460_finite__has__maximal,axiom,
    ! [A4: set_rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ? [X5: rat] :
            ( ( member_rat @ X5 @ A4 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A4 )
               => ( ( ord_less_eq_rat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_3461_finite__has__maximal,axiom,
    ! [A4: set_num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ? [X5: num] :
            ( ( member_num @ X5 @ A4 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A4 )
               => ( ( ord_less_eq_num @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_3462_finite__has__maximal,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ? [X5: nat] :
            ( ( member_nat @ X5 @ A4 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A4 )
               => ( ( ord_less_eq_nat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_3463_finite__has__maximal,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ? [X5: int] :
            ( ( member_int @ X5 @ A4 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A4 )
               => ( ( ord_less_eq_int @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_3464_finite__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z2: complex] :
              ( ( power_power_complex @ Z2 @ N )
              = C ) ) ) ) ).

% finite_nth_roots
thf(fact_3465_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList2: list_VEBT_VEBT,X3: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( ( ord_less_nat @ X3 @ Mi )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X3 != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X3 @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_excp
thf(fact_3466_insert__simp__norm,axiom,
    ! [X3: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( ( ord_less_nat @ Mi @ X3 )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X3 != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X3 @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_norm
thf(fact_3467_gcd__nat__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
     => ( ! [M4: nat,N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( ( P @ N3 @ ( modulo_modulo_nat @ M4 @ N3 ) )
             => ( P @ M4 @ N3 ) ) )
       => ( P @ M @ N ) ) ) ).

% gcd_nat_induct
thf(fact_3468_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_3469_order__refl,axiom,
    ! [X3: set_int] : ( ord_less_eq_set_int @ X3 @ X3 ) ).

% order_refl
thf(fact_3470_order__refl,axiom,
    ! [X3: rat] : ( ord_less_eq_rat @ X3 @ X3 ) ).

% order_refl
thf(fact_3471_order__refl,axiom,
    ! [X3: num] : ( ord_less_eq_num @ X3 @ X3 ) ).

% order_refl
thf(fact_3472_order__refl,axiom,
    ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).

% order_refl
thf(fact_3473_order__refl,axiom,
    ! [X3: int] : ( ord_less_eq_int @ X3 @ X3 ) ).

% order_refl
thf(fact_3474_dual__order_Orefl,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).

% dual_order.refl
thf(fact_3475_dual__order_Orefl,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% dual_order.refl
thf(fact_3476_dual__order_Orefl,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% dual_order.refl
thf(fact_3477_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_3478_dual__order_Orefl,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% dual_order.refl
thf(fact_3479_length__list__update,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X3: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X3 ) )
      = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).

% length_list_update
thf(fact_3480_length__list__update,axiom,
    ! [Xs2: list_o,I: nat,X3: $o] :
      ( ( size_size_list_o @ ( list_update_o @ Xs2 @ I @ X3 ) )
      = ( size_size_list_o @ Xs2 ) ) ).

% length_list_update
thf(fact_3481_length__list__update,axiom,
    ! [Xs2: list_nat,I: nat,X3: nat] :
      ( ( size_size_list_nat @ ( list_update_nat @ Xs2 @ I @ X3 ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_list_update
thf(fact_3482_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_3483_max__0R,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ N @ zero_zero_nat )
      = N ) ).

% max_0R
thf(fact_3484_max__0L,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N )
      = N ) ).

% max_0L
thf(fact_3485_max__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ zero_zero_nat )
      = A ) ).

% max_nat.right_neutral
thf(fact_3486_max__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( zero_zero_nat
        = ( ord_max_nat @ A @ B2 ) )
      = ( ( A = zero_zero_nat )
        & ( B2 = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
thf(fact_3487_max__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A )
      = A ) ).

% max_nat.left_neutral
thf(fact_3488_max__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( ord_max_nat @ A @ B2 )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( B2 = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
thf(fact_3489_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs2: list_int,X3: int] :
      ( ( I != J )
     => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X3 ) @ J )
        = ( nth_int @ Xs2 @ J ) ) ) ).

% nth_list_update_neq
thf(fact_3490_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs2: list_nat,X3: nat] :
      ( ( I != J )
     => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X3 ) @ J )
        = ( nth_nat @ Xs2 @ J ) ) ) ).

% nth_list_update_neq
thf(fact_3491_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs2: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( ( I != J )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X3 ) @ J )
        = ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ).

% nth_list_update_neq
thf(fact_3492_list__update__id,axiom,
    ! [Xs2: list_int,I: nat] :
      ( ( list_update_int @ Xs2 @ I @ ( nth_int @ Xs2 @ I ) )
      = Xs2 ) ).

% list_update_id
thf(fact_3493_list__update__id,axiom,
    ! [Xs2: list_nat,I: nat] :
      ( ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ I ) )
      = Xs2 ) ).

% list_update_id
thf(fact_3494_list__update__id,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat] :
      ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( nth_VEBT_VEBT @ Xs2 @ I ) )
      = Xs2 ) ).

% list_update_id
thf(fact_3495_concat__bit__0,axiom,
    ! [K: int,L: int] :
      ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
      = L ) ).

% concat_bit_0
thf(fact_3496_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
       => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
          = ( numera1916890842035813515d_enat @ V ) ) )
      & ( ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
       => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
          = ( numera1916890842035813515d_enat @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_3497_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
          = ( numeral_numeral_real @ V ) ) )
      & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
          = ( numeral_numeral_real @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_3498_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ V ) ) )
      & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_3499_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
       => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
          = ( numeral_numeral_nat @ V ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
       => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
          = ( numeral_numeral_nat @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_3500_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
          = ( numeral_numeral_int @ V ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
          = ( numeral_numeral_int @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_3501_max__0__1_I4_J,axiom,
    ! [X3: num] :
      ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X3 ) @ zero_z5237406670263579293d_enat )
      = ( numera1916890842035813515d_enat @ X3 ) ) ).

% max_0_1(4)
thf(fact_3502_max__0__1_I4_J,axiom,
    ! [X3: num] :
      ( ( ord_max_real @ ( numeral_numeral_real @ X3 ) @ zero_zero_real )
      = ( numeral_numeral_real @ X3 ) ) ).

% max_0_1(4)
thf(fact_3503_max__0__1_I4_J,axiom,
    ! [X3: num] :
      ( ( ord_max_rat @ ( numeral_numeral_rat @ X3 ) @ zero_zero_rat )
      = ( numeral_numeral_rat @ X3 ) ) ).

% max_0_1(4)
thf(fact_3504_max__0__1_I4_J,axiom,
    ! [X3: num] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ X3 ) @ zero_zero_nat )
      = ( numeral_numeral_nat @ X3 ) ) ).

% max_0_1(4)
thf(fact_3505_max__0__1_I4_J,axiom,
    ! [X3: num] :
      ( ( ord_max_int @ ( numeral_numeral_int @ X3 ) @ zero_zero_int )
      = ( numeral_numeral_int @ X3 ) ) ).

% max_0_1(4)
thf(fact_3506_max__0__1_I3_J,axiom,
    ! [X3: num] :
      ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ X3 ) )
      = ( numera1916890842035813515d_enat @ X3 ) ) ).

% max_0_1(3)
thf(fact_3507_max__0__1_I3_J,axiom,
    ! [X3: num] :
      ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X3 ) )
      = ( numeral_numeral_real @ X3 ) ) ).

% max_0_1(3)
thf(fact_3508_max__0__1_I3_J,axiom,
    ! [X3: num] :
      ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X3 ) )
      = ( numeral_numeral_rat @ X3 ) ) ).

% max_0_1(3)
thf(fact_3509_max__0__1_I3_J,axiom,
    ! [X3: num] :
      ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X3 ) )
      = ( numeral_numeral_nat @ X3 ) ) ).

% max_0_1(3)
thf(fact_3510_max__0__1_I3_J,axiom,
    ! [X3: num] :
      ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X3 ) )
      = ( numeral_numeral_int @ X3 ) ) ).

% max_0_1(3)
thf(fact_3511_max__0__1_I1_J,axiom,
    ( ( ord_max_real @ zero_zero_real @ one_one_real )
    = one_one_real ) ).

% max_0_1(1)
thf(fact_3512_max__0__1_I1_J,axiom,
    ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
    = one_one_rat ) ).

% max_0_1(1)
thf(fact_3513_max__0__1_I1_J,axiom,
    ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
    = one_one_nat ) ).

% max_0_1(1)
thf(fact_3514_max__0__1_I1_J,axiom,
    ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat )
    = one_on7984719198319812577d_enat ) ).

% max_0_1(1)
thf(fact_3515_max__0__1_I1_J,axiom,
    ( ( ord_max_int @ zero_zero_int @ one_one_int )
    = one_one_int ) ).

% max_0_1(1)
thf(fact_3516_max__0__1_I2_J,axiom,
    ( ( ord_max_real @ one_one_real @ zero_zero_real )
    = one_one_real ) ).

% max_0_1(2)
thf(fact_3517_max__0__1_I2_J,axiom,
    ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
    = one_one_rat ) ).

% max_0_1(2)
thf(fact_3518_max__0__1_I2_J,axiom,
    ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
    = one_one_nat ) ).

% max_0_1(2)
thf(fact_3519_max__0__1_I2_J,axiom,
    ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat )
    = one_on7984719198319812577d_enat ) ).

% max_0_1(2)
thf(fact_3520_max__0__1_I2_J,axiom,
    ( ( ord_max_int @ one_one_int @ zero_zero_int )
    = one_one_int ) ).

% max_0_1(2)
thf(fact_3521_max__0__1_I6_J,axiom,
    ! [X3: num] :
      ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X3 ) @ one_on7984719198319812577d_enat )
      = ( numera1916890842035813515d_enat @ X3 ) ) ).

% max_0_1(6)
thf(fact_3522_max__0__1_I6_J,axiom,
    ! [X3: num] :
      ( ( ord_max_real @ ( numeral_numeral_real @ X3 ) @ one_one_real )
      = ( numeral_numeral_real @ X3 ) ) ).

% max_0_1(6)
thf(fact_3523_max__0__1_I6_J,axiom,
    ! [X3: num] :
      ( ( ord_max_rat @ ( numeral_numeral_rat @ X3 ) @ one_one_rat )
      = ( numeral_numeral_rat @ X3 ) ) ).

% max_0_1(6)
thf(fact_3524_max__0__1_I6_J,axiom,
    ! [X3: num] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ X3 ) @ one_one_nat )
      = ( numeral_numeral_nat @ X3 ) ) ).

% max_0_1(6)
thf(fact_3525_max__0__1_I6_J,axiom,
    ! [X3: num] :
      ( ( ord_max_int @ ( numeral_numeral_int @ X3 ) @ one_one_int )
      = ( numeral_numeral_int @ X3 ) ) ).

% max_0_1(6)
thf(fact_3526_max__0__1_I5_J,axiom,
    ! [X3: num] :
      ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X3 ) )
      = ( numera1916890842035813515d_enat @ X3 ) ) ).

% max_0_1(5)
thf(fact_3527_max__0__1_I5_J,axiom,
    ! [X3: num] :
      ( ( ord_max_real @ one_one_real @ ( numeral_numeral_real @ X3 ) )
      = ( numeral_numeral_real @ X3 ) ) ).

% max_0_1(5)
thf(fact_3528_max__0__1_I5_J,axiom,
    ! [X3: num] :
      ( ( ord_max_rat @ one_one_rat @ ( numeral_numeral_rat @ X3 ) )
      = ( numeral_numeral_rat @ X3 ) ) ).

% max_0_1(5)
thf(fact_3529_max__0__1_I5_J,axiom,
    ! [X3: num] :
      ( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X3 ) )
      = ( numeral_numeral_nat @ X3 ) ) ).

% max_0_1(5)
thf(fact_3530_max__0__1_I5_J,axiom,
    ! [X3: num] :
      ( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X3 ) )
      = ( numeral_numeral_int @ X3 ) ) ).

% max_0_1(5)
thf(fact_3531_list__update__beyond,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X3: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ I )
     => ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X3 )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_3532_list__update__beyond,axiom,
    ! [Xs2: list_o,I: nat,X3: $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ I )
     => ( ( list_update_o @ Xs2 @ I @ X3 )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_3533_list__update__beyond,axiom,
    ! [Xs2: list_nat,I: nat,X3: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ I )
     => ( ( list_update_nat @ Xs2 @ I @ X3 )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_3534_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_3535_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_3536_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_int,X3: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X3 ) @ I )
        = X3 ) ) ).

% nth_list_update_eq
thf(fact_3537_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X3 ) @ I )
        = X3 ) ) ).

% nth_list_update_eq
thf(fact_3538_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_o,X3: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_o @ ( list_update_o @ Xs2 @ I @ X3 ) @ I )
        = X3 ) ) ).

% nth_list_update_eq
thf(fact_3539_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_nat,X3: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X3 ) @ I )
        = X3 ) ) ).

% nth_list_update_eq
thf(fact_3540_set__swap,axiom,
    ! [I: nat,Xs2: list_int,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs2 ) )
       => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs2 @ I @ ( nth_int @ Xs2 @ J ) ) @ J @ ( nth_int @ Xs2 @ I ) ) )
          = ( set_int2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_3541_set__swap,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( nth_VEBT_VEBT @ Xs2 @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs2 @ I ) ) )
          = ( set_VEBT_VEBT2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_3542_set__swap,axiom,
    ! [I: nat,Xs2: list_o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs2 ) )
       => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs2 @ I @ ( nth_o @ Xs2 @ J ) ) @ J @ ( nth_o @ Xs2 @ I ) ) )
          = ( set_o2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_3543_set__swap,axiom,
    ! [I: nat,Xs2: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
       => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ J ) ) @ J @ ( nth_nat @ Xs2 @ I ) ) )
          = ( set_nat2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_3544_max__def,axiom,
    ( ord_ma741700101516333627d_enat
    = ( ^ [A5: extended_enat,B4: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def
thf(fact_3545_max__def,axiom,
    ( ord_max_set_int
    = ( ^ [A5: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def
thf(fact_3546_max__def,axiom,
    ( ord_max_rat
    = ( ^ [A5: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def
thf(fact_3547_max__def,axiom,
    ( ord_max_num
    = ( ^ [A5: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def
thf(fact_3548_max__def,axiom,
    ( ord_max_nat
    = ( ^ [A5: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def
thf(fact_3549_max__def,axiom,
    ( ord_max_int
    = ( ^ [A5: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def
thf(fact_3550_max__absorb1,axiom,
    ! [Y3: extended_enat,X3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Y3 @ X3 )
     => ( ( ord_ma741700101516333627d_enat @ X3 @ Y3 )
        = X3 ) ) ).

% max_absorb1
thf(fact_3551_max__absorb1,axiom,
    ! [Y3: set_int,X3: set_int] :
      ( ( ord_less_eq_set_int @ Y3 @ X3 )
     => ( ( ord_max_set_int @ X3 @ Y3 )
        = X3 ) ) ).

% max_absorb1
thf(fact_3552_max__absorb1,axiom,
    ! [Y3: rat,X3: rat] :
      ( ( ord_less_eq_rat @ Y3 @ X3 )
     => ( ( ord_max_rat @ X3 @ Y3 )
        = X3 ) ) ).

% max_absorb1
thf(fact_3553_max__absorb1,axiom,
    ! [Y3: num,X3: num] :
      ( ( ord_less_eq_num @ Y3 @ X3 )
     => ( ( ord_max_num @ X3 @ Y3 )
        = X3 ) ) ).

% max_absorb1
thf(fact_3554_max__absorb1,axiom,
    ! [Y3: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X3 )
     => ( ( ord_max_nat @ X3 @ Y3 )
        = X3 ) ) ).

% max_absorb1
thf(fact_3555_max__absorb1,axiom,
    ! [Y3: int,X3: int] :
      ( ( ord_less_eq_int @ Y3 @ X3 )
     => ( ( ord_max_int @ X3 @ Y3 )
        = X3 ) ) ).

% max_absorb1
thf(fact_3556_max__absorb2,axiom,
    ! [X3: extended_enat,Y3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
     => ( ( ord_ma741700101516333627d_enat @ X3 @ Y3 )
        = Y3 ) ) ).

% max_absorb2
thf(fact_3557_max__absorb2,axiom,
    ! [X3: set_int,Y3: set_int] :
      ( ( ord_less_eq_set_int @ X3 @ Y3 )
     => ( ( ord_max_set_int @ X3 @ Y3 )
        = Y3 ) ) ).

% max_absorb2
thf(fact_3558_max__absorb2,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y3 )
     => ( ( ord_max_rat @ X3 @ Y3 )
        = Y3 ) ) ).

% max_absorb2
thf(fact_3559_max__absorb2,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_eq_num @ X3 @ Y3 )
     => ( ( ord_max_num @ X3 @ Y3 )
        = Y3 ) ) ).

% max_absorb2
thf(fact_3560_max__absorb2,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_max_nat @ X3 @ Y3 )
        = Y3 ) ) ).

% max_absorb2
thf(fact_3561_max__absorb2,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ X3 @ Y3 )
     => ( ( ord_max_int @ X3 @ Y3 )
        = Y3 ) ) ).

% max_absorb2
thf(fact_3562_max__add__distrib__left,axiom,
    ! [X3: real,Y3: real,Z: real] :
      ( ( plus_plus_real @ ( ord_max_real @ X3 @ Y3 ) @ Z )
      = ( ord_max_real @ ( plus_plus_real @ X3 @ Z ) @ ( plus_plus_real @ Y3 @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_3563_max__add__distrib__left,axiom,
    ! [X3: rat,Y3: rat,Z: rat] :
      ( ( plus_plus_rat @ ( ord_max_rat @ X3 @ Y3 ) @ Z )
      = ( ord_max_rat @ ( plus_plus_rat @ X3 @ Z ) @ ( plus_plus_rat @ Y3 @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_3564_max__add__distrib__left,axiom,
    ! [X3: nat,Y3: nat,Z: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ X3 @ Y3 ) @ Z )
      = ( ord_max_nat @ ( plus_plus_nat @ X3 @ Z ) @ ( plus_plus_nat @ Y3 @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_3565_max__add__distrib__left,axiom,
    ! [X3: int,Y3: int,Z: int] :
      ( ( plus_plus_int @ ( ord_max_int @ X3 @ Y3 ) @ Z )
      = ( ord_max_int @ ( plus_plus_int @ X3 @ Z ) @ ( plus_plus_int @ Y3 @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_3566_max__add__distrib__right,axiom,
    ! [X3: real,Y3: real,Z: real] :
      ( ( plus_plus_real @ X3 @ ( ord_max_real @ Y3 @ Z ) )
      = ( ord_max_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( plus_plus_real @ X3 @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_3567_max__add__distrib__right,axiom,
    ! [X3: rat,Y3: rat,Z: rat] :
      ( ( plus_plus_rat @ X3 @ ( ord_max_rat @ Y3 @ Z ) )
      = ( ord_max_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( plus_plus_rat @ X3 @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_3568_max__add__distrib__right,axiom,
    ! [X3: nat,Y3: nat,Z: nat] :
      ( ( plus_plus_nat @ X3 @ ( ord_max_nat @ Y3 @ Z ) )
      = ( ord_max_nat @ ( plus_plus_nat @ X3 @ Y3 ) @ ( plus_plus_nat @ X3 @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_3569_max__add__distrib__right,axiom,
    ! [X3: int,Y3: int,Z: int] :
      ( ( plus_plus_int @ X3 @ ( ord_max_int @ Y3 @ Z ) )
      = ( ord_max_int @ ( plus_plus_int @ X3 @ Y3 ) @ ( plus_plus_int @ X3 @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_3570_max__diff__distrib__left,axiom,
    ! [X3: real,Y3: real,Z: real] :
      ( ( minus_minus_real @ ( ord_max_real @ X3 @ Y3 ) @ Z )
      = ( ord_max_real @ ( minus_minus_real @ X3 @ Z ) @ ( minus_minus_real @ Y3 @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_3571_max__diff__distrib__left,axiom,
    ! [X3: rat,Y3: rat,Z: rat] :
      ( ( minus_minus_rat @ ( ord_max_rat @ X3 @ Y3 ) @ Z )
      = ( ord_max_rat @ ( minus_minus_rat @ X3 @ Z ) @ ( minus_minus_rat @ Y3 @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_3572_max__diff__distrib__left,axiom,
    ! [X3: int,Y3: int,Z: int] :
      ( ( minus_minus_int @ ( ord_max_int @ X3 @ Y3 ) @ Z )
      = ( ord_max_int @ ( minus_minus_int @ X3 @ Z ) @ ( minus_minus_int @ Y3 @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_3573_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).

% nat_add_max_left
thf(fact_3574_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).

% nat_add_max_right
thf(fact_3575_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
      = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).

% nat_mult_max_right
thf(fact_3576_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
      = ( ord_max_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).

% nat_mult_max_left
thf(fact_3577_max__def__raw,axiom,
    ( ord_ma741700101516333627d_enat
    = ( ^ [A5: extended_enat,B4: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def_raw
thf(fact_3578_max__def__raw,axiom,
    ( ord_max_set_int
    = ( ^ [A5: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def_raw
thf(fact_3579_max__def__raw,axiom,
    ( ord_max_rat
    = ( ^ [A5: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def_raw
thf(fact_3580_max__def__raw,axiom,
    ( ord_max_num
    = ( ^ [A5: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def_raw
thf(fact_3581_max__def__raw,axiom,
    ( ord_max_nat
    = ( ^ [A5: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def_raw
thf(fact_3582_max__def__raw,axiom,
    ( ord_max_int
    = ( ^ [A5: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def_raw
thf(fact_3583_concat__bit__assoc,axiom,
    ! [N: nat,K: int,M: nat,L: int,R: int] :
      ( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L @ R ) )
      = ( bit_concat_bit @ ( plus_plus_nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L ) @ R ) ) ).

% concat_bit_assoc
thf(fact_3584_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_3585_set__update__subsetI,axiom,
    ! [Xs2: list_real,A4: set_real,X3: real,I: nat] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ A4 )
     => ( ( member_real @ X3 @ A4 )
       => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs2 @ I @ X3 ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_3586_set__update__subsetI,axiom,
    ! [Xs2: list_complex,A4: set_complex,X3: complex,I: nat] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A4 )
     => ( ( member_complex @ X3 @ A4 )
       => ( ord_le211207098394363844omplex @ ( set_complex2 @ ( list_update_complex @ Xs2 @ I @ X3 ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_3587_set__update__subsetI,axiom,
    ! [Xs2: list_nat,A4: set_nat,X3: nat,I: nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A4 )
     => ( ( member_nat @ X3 @ A4 )
       => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I @ X3 ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_3588_set__update__subsetI,axiom,
    ! [Xs2: list_VEBT_VEBT,A4: set_VEBT_VEBT,X3: vEBT_VEBT,I: nat] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A4 )
     => ( ( member_VEBT_VEBT @ X3 @ A4 )
       => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X3 ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_3589_set__update__subsetI,axiom,
    ! [Xs2: list_int,A4: set_int,X3: int,I: nat] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A4 )
     => ( ( member_int @ X3 @ A4 )
       => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs2 @ I @ X3 ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_3590_set__update__memI,axiom,
    ! [N: nat,Xs2: list_real,X3: real] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
     => ( member_real @ X3 @ ( set_real2 @ ( list_update_real @ Xs2 @ N @ X3 ) ) ) ) ).

% set_update_memI
thf(fact_3591_set__update__memI,axiom,
    ! [N: nat,Xs2: list_complex,X3: complex] :
      ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
     => ( member_complex @ X3 @ ( set_complex2 @ ( list_update_complex @ Xs2 @ N @ X3 ) ) ) ) ).

% set_update_memI
thf(fact_3592_set__update__memI,axiom,
    ! [N: nat,Xs2: list_int,X3: int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
     => ( member_int @ X3 @ ( set_int2 @ ( list_update_int @ Xs2 @ N @ X3 ) ) ) ) ).

% set_update_memI
thf(fact_3593_set__update__memI,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ N @ X3 ) ) ) ) ).

% set_update_memI
thf(fact_3594_set__update__memI,axiom,
    ! [N: nat,Xs2: list_o,X3: $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
     => ( member_o @ X3 @ ( set_o2 @ ( list_update_o @ Xs2 @ N @ X3 ) ) ) ) ).

% set_update_memI
thf(fact_3595_set__update__memI,axiom,
    ! [N: nat,Xs2: list_nat,X3: nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( member_nat @ X3 @ ( set_nat2 @ ( list_update_nat @ Xs2 @ N @ X3 ) ) ) ) ).

% set_update_memI
thf(fact_3596_nle__le,axiom,
    ! [A: rat,B2: rat] :
      ( ( ~ ( ord_less_eq_rat @ A @ B2 ) )
      = ( ( ord_less_eq_rat @ B2 @ A )
        & ( B2 != A ) ) ) ).

% nle_le
thf(fact_3597_nle__le,axiom,
    ! [A: num,B2: num] :
      ( ( ~ ( ord_less_eq_num @ A @ B2 ) )
      = ( ( ord_less_eq_num @ B2 @ A )
        & ( B2 != A ) ) ) ).

% nle_le
thf(fact_3598_nle__le,axiom,
    ! [A: nat,B2: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
      = ( ( ord_less_eq_nat @ B2 @ A )
        & ( B2 != A ) ) ) ).

% nle_le
thf(fact_3599_nle__le,axiom,
    ! [A: int,B2: int] :
      ( ( ~ ( ord_less_eq_int @ A @ B2 ) )
      = ( ( ord_less_eq_int @ B2 @ A )
        & ( B2 != A ) ) ) ).

% nle_le
thf(fact_3600_le__cases3,axiom,
    ! [X3: rat,Y3: rat,Z: rat] :
      ( ( ( ord_less_eq_rat @ X3 @ Y3 )
       => ~ ( ord_less_eq_rat @ Y3 @ Z ) )
     => ( ( ( ord_less_eq_rat @ Y3 @ X3 )
         => ~ ( ord_less_eq_rat @ X3 @ Z ) )
       => ( ( ( ord_less_eq_rat @ X3 @ Z )
           => ~ ( ord_less_eq_rat @ Z @ Y3 ) )
         => ( ( ( ord_less_eq_rat @ Z @ Y3 )
             => ~ ( ord_less_eq_rat @ Y3 @ X3 ) )
           => ( ( ( ord_less_eq_rat @ Y3 @ Z )
               => ~ ( ord_less_eq_rat @ Z @ X3 ) )
             => ~ ( ( ord_less_eq_rat @ Z @ X3 )
                 => ~ ( ord_less_eq_rat @ X3 @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_3601_le__cases3,axiom,
    ! [X3: num,Y3: num,Z: num] :
      ( ( ( ord_less_eq_num @ X3 @ Y3 )
       => ~ ( ord_less_eq_num @ Y3 @ Z ) )
     => ( ( ( ord_less_eq_num @ Y3 @ X3 )
         => ~ ( ord_less_eq_num @ X3 @ Z ) )
       => ( ( ( ord_less_eq_num @ X3 @ Z )
           => ~ ( ord_less_eq_num @ Z @ Y3 ) )
         => ( ( ( ord_less_eq_num @ Z @ Y3 )
             => ~ ( ord_less_eq_num @ Y3 @ X3 ) )
           => ( ( ( ord_less_eq_num @ Y3 @ Z )
               => ~ ( ord_less_eq_num @ Z @ X3 ) )
             => ~ ( ( ord_less_eq_num @ Z @ X3 )
                 => ~ ( ord_less_eq_num @ X3 @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_3602_le__cases3,axiom,
    ! [X3: nat,Y3: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X3 @ Y3 )
       => ~ ( ord_less_eq_nat @ Y3 @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y3 @ X3 )
         => ~ ( ord_less_eq_nat @ X3 @ Z ) )
       => ( ( ( ord_less_eq_nat @ X3 @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y3 ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y3 )
             => ~ ( ord_less_eq_nat @ Y3 @ X3 ) )
           => ( ( ( ord_less_eq_nat @ Y3 @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X3 ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X3 )
                 => ~ ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_3603_le__cases3,axiom,
    ! [X3: int,Y3: int,Z: int] :
      ( ( ( ord_less_eq_int @ X3 @ Y3 )
       => ~ ( ord_less_eq_int @ Y3 @ Z ) )
     => ( ( ( ord_less_eq_int @ Y3 @ X3 )
         => ~ ( ord_less_eq_int @ X3 @ Z ) )
       => ( ( ( ord_less_eq_int @ X3 @ Z )
           => ~ ( ord_less_eq_int @ Z @ Y3 ) )
         => ( ( ( ord_less_eq_int @ Z @ Y3 )
             => ~ ( ord_less_eq_int @ Y3 @ X3 ) )
           => ( ( ( ord_less_eq_int @ Y3 @ Z )
               => ~ ( ord_less_eq_int @ Z @ X3 ) )
             => ~ ( ( ord_less_eq_int @ Z @ X3 )
                 => ~ ( ord_less_eq_int @ X3 @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_3604_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: set_int,Z5: set_int] : Y6 = Z5 )
    = ( ^ [X: set_int,Y: set_int] :
          ( ( ord_less_eq_set_int @ X @ Y )
          & ( ord_less_eq_set_int @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_3605_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: rat,Z5: rat] : Y6 = Z5 )
    = ( ^ [X: rat,Y: rat] :
          ( ( ord_less_eq_rat @ X @ Y )
          & ( ord_less_eq_rat @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_3606_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: num,Z5: num] : Y6 = Z5 )
    = ( ^ [X: num,Y: num] :
          ( ( ord_less_eq_num @ X @ Y )
          & ( ord_less_eq_num @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_3607_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: nat,Z5: nat] : Y6 = Z5 )
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_eq_nat @ X @ Y )
          & ( ord_less_eq_nat @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_3608_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: int,Z5: int] : Y6 = Z5 )
    = ( ^ [X: int,Y: int] :
          ( ( ord_less_eq_int @ X @ Y )
          & ( ord_less_eq_int @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_3609_ord__eq__le__trans,axiom,
    ! [A: set_int,B2: set_int,C: set_int] :
      ( ( A = B2 )
     => ( ( ord_less_eq_set_int @ B2 @ C )
       => ( ord_less_eq_set_int @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_3610_ord__eq__le__trans,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( A = B2 )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_3611_ord__eq__le__trans,axiom,
    ! [A: num,B2: num,C: num] :
      ( ( A = B2 )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_3612_ord__eq__le__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( A = B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_3613_ord__eq__le__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( A = B2 )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_3614_ord__le__eq__trans,axiom,
    ! [A: set_int,B2: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_set_int @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_3615_ord__le__eq__trans,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_3616_ord__le__eq__trans,axiom,
    ! [A: num,B2: num,C: num] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_3617_ord__le__eq__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_3618_ord__le__eq__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_3619_order__antisym,axiom,
    ! [X3: set_int,Y3: set_int] :
      ( ( ord_less_eq_set_int @ X3 @ Y3 )
     => ( ( ord_less_eq_set_int @ Y3 @ X3 )
       => ( X3 = Y3 ) ) ) ).

% order_antisym
thf(fact_3620_order__antisym,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y3 )
     => ( ( ord_less_eq_rat @ Y3 @ X3 )
       => ( X3 = Y3 ) ) ) ).

% order_antisym
thf(fact_3621_order__antisym,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_eq_num @ X3 @ Y3 )
     => ( ( ord_less_eq_num @ Y3 @ X3 )
       => ( X3 = Y3 ) ) ) ).

% order_antisym
thf(fact_3622_order__antisym,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ X3 )
       => ( X3 = Y3 ) ) ) ).

% order_antisym
thf(fact_3623_order__antisym,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ X3 @ Y3 )
     => ( ( ord_less_eq_int @ Y3 @ X3 )
       => ( X3 = Y3 ) ) ) ).

% order_antisym
thf(fact_3624_order_Otrans,axiom,
    ! [A: set_int,B2: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ B2 )
     => ( ( ord_less_eq_set_int @ B2 @ C )
       => ( ord_less_eq_set_int @ A @ C ) ) ) ).

% order.trans
thf(fact_3625_order_Otrans,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% order.trans
thf(fact_3626_order_Otrans,axiom,
    ! [A: num,B2: num,C: num] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% order.trans
thf(fact_3627_order_Otrans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_3628_order_Otrans,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% order.trans
thf(fact_3629_order__trans,axiom,
    ! [X3: set_int,Y3: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ X3 @ Y3 )
     => ( ( ord_less_eq_set_int @ Y3 @ Z )
       => ( ord_less_eq_set_int @ X3 @ Z ) ) ) ).

% order_trans
thf(fact_3630_order__trans,axiom,
    ! [X3: rat,Y3: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y3 )
     => ( ( ord_less_eq_rat @ Y3 @ Z )
       => ( ord_less_eq_rat @ X3 @ Z ) ) ) ).

% order_trans
thf(fact_3631_order__trans,axiom,
    ! [X3: num,Y3: num,Z: num] :
      ( ( ord_less_eq_num @ X3 @ Y3 )
     => ( ( ord_less_eq_num @ Y3 @ Z )
       => ( ord_less_eq_num @ X3 @ Z ) ) ) ).

% order_trans
thf(fact_3632_order__trans,axiom,
    ! [X3: nat,Y3: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z )
       => ( ord_less_eq_nat @ X3 @ Z ) ) ) ).

% order_trans
thf(fact_3633_order__trans,axiom,
    ! [X3: int,Y3: int,Z: int] :
      ( ( ord_less_eq_int @ X3 @ Y3 )
     => ( ( ord_less_eq_int @ Y3 @ Z )
       => ( ord_less_eq_int @ X3 @ Z ) ) ) ).

% order_trans
thf(fact_3634_linorder__wlog,axiom,
    ! [P: rat > rat > $o,A: rat,B2: rat] :
      ( ! [A3: rat,B3: rat] :
          ( ( ord_less_eq_rat @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: rat,B3: rat] :
            ( ( P @ B3 @ A3 )
           => ( P @ A3 @ B3 ) )
       => ( P @ A @ B2 ) ) ) ).

% linorder_wlog
thf(fact_3635_linorder__wlog,axiom,
    ! [P: num > num > $o,A: num,B2: num] :
      ( ! [A3: num,B3: num] :
          ( ( ord_less_eq_num @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: num,B3: num] :
            ( ( P @ B3 @ A3 )
           => ( P @ A3 @ B3 ) )
       => ( P @ A @ B2 ) ) ) ).

% linorder_wlog
thf(fact_3636_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B2: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: nat,B3: nat] :
            ( ( P @ B3 @ A3 )
           => ( P @ A3 @ B3 ) )
       => ( P @ A @ B2 ) ) ) ).

% linorder_wlog
thf(fact_3637_linorder__wlog,axiom,
    ! [P: int > int > $o,A: int,B2: int] :
      ( ! [A3: int,B3: int] :
          ( ( ord_less_eq_int @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: int,B3: int] :
            ( ( P @ B3 @ A3 )
           => ( P @ A3 @ B3 ) )
       => ( P @ A @ B2 ) ) ) ).

% linorder_wlog
thf(fact_3638_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: set_int,Z5: set_int] : Y6 = Z5 )
    = ( ^ [A5: set_int,B4: set_int] :
          ( ( ord_less_eq_set_int @ B4 @ A5 )
          & ( ord_less_eq_set_int @ A5 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_3639_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: rat,Z5: rat] : Y6 = Z5 )
    = ( ^ [A5: rat,B4: rat] :
          ( ( ord_less_eq_rat @ B4 @ A5 )
          & ( ord_less_eq_rat @ A5 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_3640_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: num,Z5: num] : Y6 = Z5 )
    = ( ^ [A5: num,B4: num] :
          ( ( ord_less_eq_num @ B4 @ A5 )
          & ( ord_less_eq_num @ A5 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_3641_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: nat,Z5: nat] : Y6 = Z5 )
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A5 )
          & ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_3642_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: int,Z5: int] : Y6 = Z5 )
    = ( ^ [A5: int,B4: int] :
          ( ( ord_less_eq_int @ B4 @ A5 )
          & ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_3643_dual__order_Oantisym,axiom,
    ! [B2: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ B2 @ A )
     => ( ( ord_less_eq_set_int @ A @ B2 )
       => ( A = B2 ) ) ) ).

% dual_order.antisym
thf(fact_3644_dual__order_Oantisym,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_eq_rat @ B2 @ A )
     => ( ( ord_less_eq_rat @ A @ B2 )
       => ( A = B2 ) ) ) ).

% dual_order.antisym
thf(fact_3645_dual__order_Oantisym,axiom,
    ! [B2: num,A: num] :
      ( ( ord_less_eq_num @ B2 @ A )
     => ( ( ord_less_eq_num @ A @ B2 )
       => ( A = B2 ) ) ) ).

% dual_order.antisym
thf(fact_3646_dual__order_Oantisym,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( ord_less_eq_nat @ A @ B2 )
       => ( A = B2 ) ) ) ).

% dual_order.antisym
thf(fact_3647_dual__order_Oantisym,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ( ord_less_eq_int @ A @ B2 )
       => ( A = B2 ) ) ) ).

% dual_order.antisym
thf(fact_3648_dual__order_Otrans,axiom,
    ! [B2: set_int,A: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ B2 @ A )
     => ( ( ord_less_eq_set_int @ C @ B2 )
       => ( ord_less_eq_set_int @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_3649_dual__order_Otrans,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B2 @ A )
     => ( ( ord_less_eq_rat @ C @ B2 )
       => ( ord_less_eq_rat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_3650_dual__order_Otrans,axiom,
    ! [B2: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B2 @ A )
     => ( ( ord_less_eq_num @ C @ B2 )
       => ( ord_less_eq_num @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_3651_dual__order_Otrans,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_3652_dual__order_Otrans,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ( ord_less_eq_int @ C @ B2 )
       => ( ord_less_eq_int @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_3653_antisym,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A @ B2 )
     => ( ( ord_less_eq_set_int @ B2 @ A )
       => ( A = B2 ) ) ) ).

% antisym
thf(fact_3654_antisym,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ B2 @ A )
       => ( A = B2 ) ) ) ).

% antisym
thf(fact_3655_antisym,axiom,
    ! [A: num,B2: num] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ord_less_eq_num @ B2 @ A )
       => ( A = B2 ) ) ) ).

% antisym
thf(fact_3656_antisym,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ A )
       => ( A = B2 ) ) ) ).

% antisym
thf(fact_3657_antisym,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ B2 @ A )
       => ( A = B2 ) ) ) ).

% antisym
thf(fact_3658_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: set_int,Z5: set_int] : Y6 = Z5 )
    = ( ^ [A5: set_int,B4: set_int] :
          ( ( ord_less_eq_set_int @ A5 @ B4 )
          & ( ord_less_eq_set_int @ B4 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_3659_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: rat,Z5: rat] : Y6 = Z5 )
    = ( ^ [A5: rat,B4: rat] :
          ( ( ord_less_eq_rat @ A5 @ B4 )
          & ( ord_less_eq_rat @ B4 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_3660_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: num,Z5: num] : Y6 = Z5 )
    = ( ^ [A5: num,B4: num] :
          ( ( ord_less_eq_num @ A5 @ B4 )
          & ( ord_less_eq_num @ B4 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_3661_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: nat,Z5: nat] : Y6 = Z5 )
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A5 @ B4 )
          & ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_3662_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: int,Z5: int] : Y6 = Z5 )
    = ( ^ [A5: int,B4: int] :
          ( ( ord_less_eq_int @ A5 @ B4 )
          & ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_3663_order__subst1,axiom,
    ! [A: rat,F: rat > rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3664_order__subst1,axiom,
    ! [A: rat,F: num > rat,B2: num,C: num] :
      ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3665_order__subst1,axiom,
    ! [A: rat,F: nat > rat,B2: nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3666_order__subst1,axiom,
    ! [A: rat,F: int > rat,B2: int,C: int] :
      ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ! [X5: int,Y4: int] :
              ( ( ord_less_eq_int @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3667_order__subst1,axiom,
    ! [A: num,F: rat > num,B2: rat,C: rat] :
      ( ( ord_less_eq_num @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3668_order__subst1,axiom,
    ! [A: num,F: num > num,B2: num,C: num] :
      ( ( ord_less_eq_num @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3669_order__subst1,axiom,
    ! [A: num,F: nat > num,B2: nat,C: nat] :
      ( ( ord_less_eq_num @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3670_order__subst1,axiom,
    ! [A: num,F: int > num,B2: int,C: int] :
      ( ( ord_less_eq_num @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ! [X5: int,Y4: int] :
              ( ( ord_less_eq_int @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3671_order__subst1,axiom,
    ! [A: nat,F: rat > nat,B2: rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3672_order__subst1,axiom,
    ! [A: nat,F: num > nat,B2: num,C: num] :
      ( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3673_order__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3674_order__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3675_order__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3676_order__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3677_order__subst2,axiom,
    ! [A: num,B2: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3678_order__subst2,axiom,
    ! [A: num,B2: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3679_order__subst2,axiom,
    ! [A: num,B2: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3680_order__subst2,axiom,
    ! [A: num,B2: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3681_order__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3682_order__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3683_order__eq__refl,axiom,
    ! [X3: set_int,Y3: set_int] :
      ( ( X3 = Y3 )
     => ( ord_less_eq_set_int @ X3 @ Y3 ) ) ).

% order_eq_refl
thf(fact_3684_order__eq__refl,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( X3 = Y3 )
     => ( ord_less_eq_rat @ X3 @ Y3 ) ) ).

% order_eq_refl
thf(fact_3685_order__eq__refl,axiom,
    ! [X3: num,Y3: num] :
      ( ( X3 = Y3 )
     => ( ord_less_eq_num @ X3 @ Y3 ) ) ).

% order_eq_refl
thf(fact_3686_order__eq__refl,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( X3 = Y3 )
     => ( ord_less_eq_nat @ X3 @ Y3 ) ) ).

% order_eq_refl
thf(fact_3687_order__eq__refl,axiom,
    ! [X3: int,Y3: int] :
      ( ( X3 = Y3 )
     => ( ord_less_eq_int @ X3 @ Y3 ) ) ).

% order_eq_refl
thf(fact_3688_linorder__linear,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y3 )
      | ( ord_less_eq_rat @ Y3 @ X3 ) ) ).

% linorder_linear
thf(fact_3689_linorder__linear,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_eq_num @ X3 @ Y3 )
      | ( ord_less_eq_num @ Y3 @ X3 ) ) ).

% linorder_linear
thf(fact_3690_linorder__linear,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
      | ( ord_less_eq_nat @ Y3 @ X3 ) ) ).

% linorder_linear
thf(fact_3691_linorder__linear,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ X3 @ Y3 )
      | ( ord_less_eq_int @ Y3 @ X3 ) ) ).

% linorder_linear
thf(fact_3692_ord__eq__le__subst,axiom,
    ! [A: rat,F: rat > rat,B2: rat,C: rat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3693_ord__eq__le__subst,axiom,
    ! [A: num,F: rat > num,B2: rat,C: rat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3694_ord__eq__le__subst,axiom,
    ! [A: nat,F: rat > nat,B2: rat,C: rat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3695_ord__eq__le__subst,axiom,
    ! [A: int,F: rat > int,B2: rat,C: rat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3696_ord__eq__le__subst,axiom,
    ! [A: rat,F: num > rat,B2: num,C: num] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3697_ord__eq__le__subst,axiom,
    ! [A: num,F: num > num,B2: num,C: num] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3698_ord__eq__le__subst,axiom,
    ! [A: nat,F: num > nat,B2: num,C: num] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3699_ord__eq__le__subst,axiom,
    ! [A: int,F: num > int,B2: num,C: num] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3700_ord__eq__le__subst,axiom,
    ! [A: rat,F: nat > rat,B2: nat,C: nat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3701_ord__eq__le__subst,axiom,
    ! [A: num,F: nat > num,B2: nat,C: nat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3702_ord__le__eq__subst,axiom,
    ! [A: rat,B2: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3703_ord__le__eq__subst,axiom,
    ! [A: rat,B2: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3704_ord__le__eq__subst,axiom,
    ! [A: rat,B2: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3705_ord__le__eq__subst,axiom,
    ! [A: rat,B2: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3706_ord__le__eq__subst,axiom,
    ! [A: num,B2: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3707_ord__le__eq__subst,axiom,
    ! [A: num,B2: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3708_ord__le__eq__subst,axiom,
    ! [A: num,B2: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3709_ord__le__eq__subst,axiom,
    ! [A: num,B2: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3710_ord__le__eq__subst,axiom,
    ! [A: nat,B2: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3711_ord__le__eq__subst,axiom,
    ! [A: nat,B2: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3712_linorder__le__cases,axiom,
    ! [X3: rat,Y3: rat] :
      ( ~ ( ord_less_eq_rat @ X3 @ Y3 )
     => ( ord_less_eq_rat @ Y3 @ X3 ) ) ).

% linorder_le_cases
thf(fact_3713_linorder__le__cases,axiom,
    ! [X3: num,Y3: num] :
      ( ~ ( ord_less_eq_num @ X3 @ Y3 )
     => ( ord_less_eq_num @ Y3 @ X3 ) ) ).

% linorder_le_cases
thf(fact_3714_linorder__le__cases,axiom,
    ! [X3: nat,Y3: nat] :
      ( ~ ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ord_less_eq_nat @ Y3 @ X3 ) ) ).

% linorder_le_cases
thf(fact_3715_linorder__le__cases,axiom,
    ! [X3: int,Y3: int] :
      ( ~ ( ord_less_eq_int @ X3 @ Y3 )
     => ( ord_less_eq_int @ Y3 @ X3 ) ) ).

% linorder_le_cases
thf(fact_3716_order__antisym__conv,axiom,
    ! [Y3: set_int,X3: set_int] :
      ( ( ord_less_eq_set_int @ Y3 @ X3 )
     => ( ( ord_less_eq_set_int @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_3717_order__antisym__conv,axiom,
    ! [Y3: rat,X3: rat] :
      ( ( ord_less_eq_rat @ Y3 @ X3 )
     => ( ( ord_less_eq_rat @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_3718_order__antisym__conv,axiom,
    ! [Y3: num,X3: num] :
      ( ( ord_less_eq_num @ Y3 @ X3 )
     => ( ( ord_less_eq_num @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_3719_order__antisym__conv,axiom,
    ! [Y3: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X3 )
     => ( ( ord_less_eq_nat @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_3720_order__antisym__conv,axiom,
    ! [Y3: int,X3: int] :
      ( ( ord_less_eq_int @ Y3 @ X3 )
     => ( ( ord_less_eq_int @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_3721_nth__list__update,axiom,
    ! [I: nat,Xs2: list_int,J: nat,X3: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X3 ) @ J )
            = X3 ) )
        & ( ( I != J )
         => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X3 ) @ J )
            = ( nth_int @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_3722_nth__list__update,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,J: nat,X3: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X3 ) @ J )
            = X3 ) )
        & ( ( I != J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X3 ) @ J )
            = ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_3723_nth__list__update,axiom,
    ! [I: nat,Xs2: list_o,X3: $o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_o @ ( list_update_o @ Xs2 @ I @ X3 ) @ J )
        = ( ( ( I = J )
           => X3 )
          & ( ( I != J )
           => ( nth_o @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_3724_nth__list__update,axiom,
    ! [I: nat,Xs2: list_nat,J: nat,X3: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X3 ) @ J )
            = X3 ) )
        & ( ( I != J )
         => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X3 ) @ J )
            = ( nth_nat @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_3725_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_int,X3: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( ( list_update_int @ Xs2 @ I @ X3 )
          = Xs2 )
        = ( ( nth_int @ Xs2 @ I )
          = X3 ) ) ) ).

% list_update_same_conv
thf(fact_3726_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X3 )
          = Xs2 )
        = ( ( nth_VEBT_VEBT @ Xs2 @ I )
          = X3 ) ) ) ).

% list_update_same_conv
thf(fact_3727_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_o,X3: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( ( list_update_o @ Xs2 @ I @ X3 )
          = Xs2 )
        = ( ( nth_o @ Xs2 @ I )
          = X3 ) ) ) ).

% list_update_same_conv
thf(fact_3728_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_nat,X3: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ( list_update_nat @ Xs2 @ I @ X3 )
          = Xs2 )
        = ( ( nth_nat @ Xs2 @ I )
          = X3 ) ) ) ).

% list_update_same_conv
thf(fact_3729_lt__ex,axiom,
    ! [X3: real] :
    ? [Y4: real] : ( ord_less_real @ Y4 @ X3 ) ).

% lt_ex
thf(fact_3730_lt__ex,axiom,
    ! [X3: rat] :
    ? [Y4: rat] : ( ord_less_rat @ Y4 @ X3 ) ).

% lt_ex
thf(fact_3731_lt__ex,axiom,
    ! [X3: int] :
    ? [Y4: int] : ( ord_less_int @ Y4 @ X3 ) ).

% lt_ex
thf(fact_3732_gt__ex,axiom,
    ! [X3: real] :
    ? [X_1: real] : ( ord_less_real @ X3 @ X_1 ) ).

% gt_ex
thf(fact_3733_gt__ex,axiom,
    ! [X3: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X3 @ X_1 ) ).

% gt_ex
thf(fact_3734_gt__ex,axiom,
    ! [X3: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).

% gt_ex
thf(fact_3735_gt__ex,axiom,
    ! [X3: int] :
    ? [X_1: int] : ( ord_less_int @ X3 @ X_1 ) ).

% gt_ex
thf(fact_3736_dense,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ? [Z4: real] :
          ( ( ord_less_real @ X3 @ Z4 )
          & ( ord_less_real @ Z4 @ Y3 ) ) ) ).

% dense
thf(fact_3737_dense,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ X3 @ Y3 )
     => ? [Z4: rat] :
          ( ( ord_less_rat @ X3 @ Z4 )
          & ( ord_less_rat @ Z4 @ Y3 ) ) ) ).

% dense
thf(fact_3738_less__imp__neq,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% less_imp_neq
thf(fact_3739_less__imp__neq,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% less_imp_neq
thf(fact_3740_less__imp__neq,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_num @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% less_imp_neq
thf(fact_3741_less__imp__neq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% less_imp_neq
thf(fact_3742_less__imp__neq,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_int @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% less_imp_neq
thf(fact_3743_order_Oasym,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ~ ( ord_less_real @ B2 @ A ) ) ).

% order.asym
thf(fact_3744_order_Oasym,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ~ ( ord_less_rat @ B2 @ A ) ) ).

% order.asym
thf(fact_3745_order_Oasym,axiom,
    ! [A: num,B2: num] :
      ( ( ord_less_num @ A @ B2 )
     => ~ ( ord_less_num @ B2 @ A ) ) ).

% order.asym
thf(fact_3746_order_Oasym,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ~ ( ord_less_nat @ B2 @ A ) ) ).

% order.asym
thf(fact_3747_order_Oasym,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ B2 )
     => ~ ( ord_less_int @ B2 @ A ) ) ).

% order.asym
thf(fact_3748_ord__eq__less__trans,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( A = B2 )
     => ( ( ord_less_real @ B2 @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_3749_ord__eq__less__trans,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( A = B2 )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_3750_ord__eq__less__trans,axiom,
    ! [A: num,B2: num,C: num] :
      ( ( A = B2 )
     => ( ( ord_less_num @ B2 @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_3751_ord__eq__less__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( A = B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_3752_ord__eq__less__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( A = B2 )
     => ( ( ord_less_int @ B2 @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_3753_ord__less__eq__trans,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_real @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_3754_ord__less__eq__trans,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_3755_ord__less__eq__trans,axiom,
    ! [A: num,B2: num,C: num] :
      ( ( ord_less_num @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_num @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_3756_ord__less__eq__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_3757_ord__less__eq__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( B2 = C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_3758_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X5: nat] :
          ( ! [Y5: nat] :
              ( ( ord_less_nat @ Y5 @ X5 )
             => ( P @ Y5 ) )
         => ( P @ X5 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_3759_antisym__conv3,axiom,
    ! [Y3: real,X3: real] :
      ( ~ ( ord_less_real @ Y3 @ X3 )
     => ( ( ~ ( ord_less_real @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_3760_antisym__conv3,axiom,
    ! [Y3: rat,X3: rat] :
      ( ~ ( ord_less_rat @ Y3 @ X3 )
     => ( ( ~ ( ord_less_rat @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_3761_antisym__conv3,axiom,
    ! [Y3: num,X3: num] :
      ( ~ ( ord_less_num @ Y3 @ X3 )
     => ( ( ~ ( ord_less_num @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_3762_antisym__conv3,axiom,
    ! [Y3: nat,X3: nat] :
      ( ~ ( ord_less_nat @ Y3 @ X3 )
     => ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_3763_antisym__conv3,axiom,
    ! [Y3: int,X3: int] :
      ( ~ ( ord_less_int @ Y3 @ X3 )
     => ( ( ~ ( ord_less_int @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_3764_linorder__cases,axiom,
    ! [X3: real,Y3: real] :
      ( ~ ( ord_less_real @ X3 @ Y3 )
     => ( ( X3 != Y3 )
       => ( ord_less_real @ Y3 @ X3 ) ) ) ).

% linorder_cases
thf(fact_3765_linorder__cases,axiom,
    ! [X3: rat,Y3: rat] :
      ( ~ ( ord_less_rat @ X3 @ Y3 )
     => ( ( X3 != Y3 )
       => ( ord_less_rat @ Y3 @ X3 ) ) ) ).

% linorder_cases
thf(fact_3766_linorder__cases,axiom,
    ! [X3: num,Y3: num] :
      ( ~ ( ord_less_num @ X3 @ Y3 )
     => ( ( X3 != Y3 )
       => ( ord_less_num @ Y3 @ X3 ) ) ) ).

% linorder_cases
thf(fact_3767_linorder__cases,axiom,
    ! [X3: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y3 )
     => ( ( X3 != Y3 )
       => ( ord_less_nat @ Y3 @ X3 ) ) ) ).

% linorder_cases
thf(fact_3768_linorder__cases,axiom,
    ! [X3: int,Y3: int] :
      ( ~ ( ord_less_int @ X3 @ Y3 )
     => ( ( X3 != Y3 )
       => ( ord_less_int @ Y3 @ X3 ) ) ) ).

% linorder_cases
thf(fact_3769_dual__order_Oasym,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ B2 @ A )
     => ~ ( ord_less_real @ A @ B2 ) ) ).

% dual_order.asym
thf(fact_3770_dual__order_Oasym,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_rat @ B2 @ A )
     => ~ ( ord_less_rat @ A @ B2 ) ) ).

% dual_order.asym
thf(fact_3771_dual__order_Oasym,axiom,
    ! [B2: num,A: num] :
      ( ( ord_less_num @ B2 @ A )
     => ~ ( ord_less_num @ A @ B2 ) ) ).

% dual_order.asym
thf(fact_3772_dual__order_Oasym,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ~ ( ord_less_nat @ A @ B2 ) ) ).

% dual_order.asym
thf(fact_3773_dual__order_Oasym,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ B2 @ A )
     => ~ ( ord_less_int @ A @ B2 ) ) ).

% dual_order.asym
thf(fact_3774_dual__order_Oirrefl,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% dual_order.irrefl
thf(fact_3775_dual__order_Oirrefl,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% dual_order.irrefl
thf(fact_3776_dual__order_Oirrefl,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% dual_order.irrefl
thf(fact_3777_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_3778_dual__order_Oirrefl,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% dual_order.irrefl
thf(fact_3779_exists__least__iff,axiom,
    ( ( ^ [P2: nat > $o] :
        ? [X4: nat] : ( P2 @ X4 ) )
    = ( ^ [P3: nat > $o] :
        ? [N2: nat] :
          ( ( P3 @ N2 )
          & ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N2 )
             => ~ ( P3 @ M3 ) ) ) ) ) ).

% exists_least_iff
thf(fact_3780_linorder__less__wlog,axiom,
    ! [P: real > real > $o,A: real,B2: real] :
      ( ! [A3: real,B3: real] :
          ( ( ord_less_real @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: real] : ( P @ A3 @ A3 )
       => ( ! [A3: real,B3: real] :
              ( ( P @ B3 @ A3 )
             => ( P @ A3 @ B3 ) )
         => ( P @ A @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_3781_linorder__less__wlog,axiom,
    ! [P: rat > rat > $o,A: rat,B2: rat] :
      ( ! [A3: rat,B3: rat] :
          ( ( ord_less_rat @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: rat] : ( P @ A3 @ A3 )
       => ( ! [A3: rat,B3: rat] :
              ( ( P @ B3 @ A3 )
             => ( P @ A3 @ B3 ) )
         => ( P @ A @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_3782_linorder__less__wlog,axiom,
    ! [P: num > num > $o,A: num,B2: num] :
      ( ! [A3: num,B3: num] :
          ( ( ord_less_num @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: num] : ( P @ A3 @ A3 )
       => ( ! [A3: num,B3: num] :
              ( ( P @ B3 @ A3 )
             => ( P @ A3 @ B3 ) )
         => ( P @ A @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_3783_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B2: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_nat @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: nat] : ( P @ A3 @ A3 )
       => ( ! [A3: nat,B3: nat] :
              ( ( P @ B3 @ A3 )
             => ( P @ A3 @ B3 ) )
         => ( P @ A @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_3784_linorder__less__wlog,axiom,
    ! [P: int > int > $o,A: int,B2: int] :
      ( ! [A3: int,B3: int] :
          ( ( ord_less_int @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: int] : ( P @ A3 @ A3 )
       => ( ! [A3: int,B3: int] :
              ( ( P @ B3 @ A3 )
             => ( P @ A3 @ B3 ) )
         => ( P @ A @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_3785_order_Ostrict__trans,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ B2 @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_3786_order_Ostrict__trans,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_3787_order_Ostrict__trans,axiom,
    ! [A: num,B2: num,C: num] :
      ( ( ord_less_num @ A @ B2 )
     => ( ( ord_less_num @ B2 @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_3788_order_Ostrict__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_3789_order_Ostrict__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_int @ B2 @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_3790_not__less__iff__gr__or__eq,axiom,
    ! [X3: real,Y3: real] :
      ( ( ~ ( ord_less_real @ X3 @ Y3 ) )
      = ( ( ord_less_real @ Y3 @ X3 )
        | ( X3 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_3791_not__less__iff__gr__or__eq,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ~ ( ord_less_rat @ X3 @ Y3 ) )
      = ( ( ord_less_rat @ Y3 @ X3 )
        | ( X3 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_3792_not__less__iff__gr__or__eq,axiom,
    ! [X3: num,Y3: num] :
      ( ( ~ ( ord_less_num @ X3 @ Y3 ) )
      = ( ( ord_less_num @ Y3 @ X3 )
        | ( X3 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_3793_not__less__iff__gr__or__eq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
      = ( ( ord_less_nat @ Y3 @ X3 )
        | ( X3 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_3794_not__less__iff__gr__or__eq,axiom,
    ! [X3: int,Y3: int] :
      ( ( ~ ( ord_less_int @ X3 @ Y3 ) )
      = ( ( ord_less_int @ Y3 @ X3 )
        | ( X3 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_3795_dual__order_Ostrict__trans,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ( ord_less_real @ C @ B2 )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_3796_dual__order_Ostrict__trans,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B2 @ A )
     => ( ( ord_less_rat @ C @ B2 )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_3797_dual__order_Ostrict__trans,axiom,
    ! [B2: num,A: num,C: num] :
      ( ( ord_less_num @ B2 @ A )
     => ( ( ord_less_num @ C @ B2 )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_3798_dual__order_Ostrict__trans,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ( ord_less_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_3799_dual__order_Ostrict__trans,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_int @ B2 @ A )
     => ( ( ord_less_int @ C @ B2 )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_3800_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_3801_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_3802_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B2: num] :
      ( ( ord_less_num @ A @ B2 )
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_3803_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_3804_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_3805_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_3806_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_rat @ B2 @ A )
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_3807_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: num,A: num] :
      ( ( ord_less_num @ B2 @ A )
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_3808_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_3809_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ B2 @ A )
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_3810_linorder__neqE,axiom,
    ! [X3: real,Y3: real] :
      ( ( X3 != Y3 )
     => ( ~ ( ord_less_real @ X3 @ Y3 )
       => ( ord_less_real @ Y3 @ X3 ) ) ) ).

% linorder_neqE
thf(fact_3811_linorder__neqE,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( X3 != Y3 )
     => ( ~ ( ord_less_rat @ X3 @ Y3 )
       => ( ord_less_rat @ Y3 @ X3 ) ) ) ).

% linorder_neqE
thf(fact_3812_linorder__neqE,axiom,
    ! [X3: num,Y3: num] :
      ( ( X3 != Y3 )
     => ( ~ ( ord_less_num @ X3 @ Y3 )
       => ( ord_less_num @ Y3 @ X3 ) ) ) ).

% linorder_neqE
thf(fact_3813_linorder__neqE,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( X3 != Y3 )
     => ( ~ ( ord_less_nat @ X3 @ Y3 )
       => ( ord_less_nat @ Y3 @ X3 ) ) ) ).

% linorder_neqE
thf(fact_3814_linorder__neqE,axiom,
    ! [X3: int,Y3: int] :
      ( ( X3 != Y3 )
     => ( ~ ( ord_less_int @ X3 @ Y3 )
       => ( ord_less_int @ Y3 @ X3 ) ) ) ).

% linorder_neqE
thf(fact_3815_order__less__asym,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ~ ( ord_less_real @ Y3 @ X3 ) ) ).

% order_less_asym
thf(fact_3816_order__less__asym,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ X3 @ Y3 )
     => ~ ( ord_less_rat @ Y3 @ X3 ) ) ).

% order_less_asym
thf(fact_3817_order__less__asym,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_num @ X3 @ Y3 )
     => ~ ( ord_less_num @ Y3 @ X3 ) ) ).

% order_less_asym
thf(fact_3818_order__less__asym,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).

% order_less_asym
thf(fact_3819_order__less__asym,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_int @ X3 @ Y3 )
     => ~ ( ord_less_int @ Y3 @ X3 ) ) ).

% order_less_asym
thf(fact_3820_linorder__neq__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( X3 != Y3 )
      = ( ( ord_less_real @ X3 @ Y3 )
        | ( ord_less_real @ Y3 @ X3 ) ) ) ).

% linorder_neq_iff
thf(fact_3821_linorder__neq__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( X3 != Y3 )
      = ( ( ord_less_rat @ X3 @ Y3 )
        | ( ord_less_rat @ Y3 @ X3 ) ) ) ).

% linorder_neq_iff
thf(fact_3822_linorder__neq__iff,axiom,
    ! [X3: num,Y3: num] :
      ( ( X3 != Y3 )
      = ( ( ord_less_num @ X3 @ Y3 )
        | ( ord_less_num @ Y3 @ X3 ) ) ) ).

% linorder_neq_iff
thf(fact_3823_linorder__neq__iff,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( X3 != Y3 )
      = ( ( ord_less_nat @ X3 @ Y3 )
        | ( ord_less_nat @ Y3 @ X3 ) ) ) ).

% linorder_neq_iff
thf(fact_3824_linorder__neq__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( X3 != Y3 )
      = ( ( ord_less_int @ X3 @ Y3 )
        | ( ord_less_int @ Y3 @ X3 ) ) ) ).

% linorder_neq_iff
thf(fact_3825_order__less__asym_H,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ~ ( ord_less_real @ B2 @ A ) ) ).

% order_less_asym'
thf(fact_3826_order__less__asym_H,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ~ ( ord_less_rat @ B2 @ A ) ) ).

% order_less_asym'
thf(fact_3827_order__less__asym_H,axiom,
    ! [A: num,B2: num] :
      ( ( ord_less_num @ A @ B2 )
     => ~ ( ord_less_num @ B2 @ A ) ) ).

% order_less_asym'
thf(fact_3828_order__less__asym_H,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ~ ( ord_less_nat @ B2 @ A ) ) ).

% order_less_asym'
thf(fact_3829_order__less__asym_H,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ B2 )
     => ~ ( ord_less_int @ B2 @ A ) ) ).

% order_less_asym'
thf(fact_3830_order__less__trans,axiom,
    ! [X3: real,Y3: real,Z: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ( ( ord_less_real @ Y3 @ Z )
       => ( ord_less_real @ X3 @ Z ) ) ) ).

% order_less_trans
thf(fact_3831_order__less__trans,axiom,
    ! [X3: rat,Y3: rat,Z: rat] :
      ( ( ord_less_rat @ X3 @ Y3 )
     => ( ( ord_less_rat @ Y3 @ Z )
       => ( ord_less_rat @ X3 @ Z ) ) ) ).

% order_less_trans
thf(fact_3832_order__less__trans,axiom,
    ! [X3: num,Y3: num,Z: num] :
      ( ( ord_less_num @ X3 @ Y3 )
     => ( ( ord_less_num @ Y3 @ Z )
       => ( ord_less_num @ X3 @ Z ) ) ) ).

% order_less_trans
thf(fact_3833_order__less__trans,axiom,
    ! [X3: nat,Y3: nat,Z: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ Z )
       => ( ord_less_nat @ X3 @ Z ) ) ) ).

% order_less_trans
thf(fact_3834_order__less__trans,axiom,
    ! [X3: int,Y3: int,Z: int] :
      ( ( ord_less_int @ X3 @ Y3 )
     => ( ( ord_less_int @ Y3 @ Z )
       => ( ord_less_int @ X3 @ Z ) ) ) ).

% order_less_trans
thf(fact_3835_ord__eq__less__subst,axiom,
    ! [A: real,F: real > real,B2: real,C: real] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_3836_ord__eq__less__subst,axiom,
    ! [A: rat,F: real > rat,B2: real,C: real] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_3837_ord__eq__less__subst,axiom,
    ! [A: num,F: real > num,B2: real,C: real] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_3838_ord__eq__less__subst,axiom,
    ! [A: nat,F: real > nat,B2: real,C: real] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_3839_ord__eq__less__subst,axiom,
    ! [A: int,F: real > int,B2: real,C: real] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_3840_ord__eq__less__subst,axiom,
    ! [A: real,F: rat > real,B2: rat,C: rat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_3841_ord__eq__less__subst,axiom,
    ! [A: rat,F: rat > rat,B2: rat,C: rat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_3842_ord__eq__less__subst,axiom,
    ! [A: num,F: rat > num,B2: rat,C: rat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_3843_ord__eq__less__subst,axiom,
    ! [A: nat,F: rat > nat,B2: rat,C: rat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_3844_ord__eq__less__subst,axiom,
    ! [A: int,F: rat > int,B2: rat,C: rat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_3845_ord__less__eq__subst,axiom,
    ! [A: real,B2: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_3846_ord__less__eq__subst,axiom,
    ! [A: real,B2: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_3847_ord__less__eq__subst,axiom,
    ! [A: real,B2: real,F: real > num,C: num] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_3848_ord__less__eq__subst,axiom,
    ! [A: real,B2: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_3849_ord__less__eq__subst,axiom,
    ! [A: real,B2: real,F: real > int,C: int] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_3850_ord__less__eq__subst,axiom,
    ! [A: rat,B2: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_3851_ord__less__eq__subst,axiom,
    ! [A: rat,B2: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_3852_ord__less__eq__subst,axiom,
    ! [A: rat,B2: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_3853_ord__less__eq__subst,axiom,
    ! [A: rat,B2: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_3854_ord__less__eq__subst,axiom,
    ! [A: rat,B2: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_3855_order__less__irrefl,axiom,
    ! [X3: real] :
      ~ ( ord_less_real @ X3 @ X3 ) ).

% order_less_irrefl
thf(fact_3856_order__less__irrefl,axiom,
    ! [X3: rat] :
      ~ ( ord_less_rat @ X3 @ X3 ) ).

% order_less_irrefl
thf(fact_3857_order__less__irrefl,axiom,
    ! [X3: num] :
      ~ ( ord_less_num @ X3 @ X3 ) ).

% order_less_irrefl
thf(fact_3858_order__less__irrefl,axiom,
    ! [X3: nat] :
      ~ ( ord_less_nat @ X3 @ X3 ) ).

% order_less_irrefl
thf(fact_3859_order__less__irrefl,axiom,
    ! [X3: int] :
      ~ ( ord_less_int @ X3 @ X3 ) ).

% order_less_irrefl
thf(fact_3860_order__less__subst1,axiom,
    ! [A: real,F: real > real,B2: real,C: real] :
      ( ( ord_less_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_3861_order__less__subst1,axiom,
    ! [A: real,F: rat > real,B2: rat,C: rat] :
      ( ( ord_less_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_3862_order__less__subst1,axiom,
    ! [A: real,F: num > real,B2: num,C: num] :
      ( ( ord_less_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_num @ B2 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_num @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_3863_order__less__subst1,axiom,
    ! [A: real,F: nat > real,B2: nat,C: nat] :
      ( ( ord_less_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_nat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_3864_order__less__subst1,axiom,
    ! [A: real,F: int > real,B2: int,C: int] :
      ( ( ord_less_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_int @ B2 @ C )
       => ( ! [X5: int,Y4: int] :
              ( ( ord_less_int @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_3865_order__less__subst1,axiom,
    ! [A: rat,F: real > rat,B2: real,C: real] :
      ( ( ord_less_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_3866_order__less__subst1,axiom,
    ! [A: rat,F: rat > rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_3867_order__less__subst1,axiom,
    ! [A: rat,F: num > rat,B2: num,C: num] :
      ( ( ord_less_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_num @ B2 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_num @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_3868_order__less__subst1,axiom,
    ! [A: rat,F: nat > rat,B2: nat,C: nat] :
      ( ( ord_less_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_nat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_3869_order__less__subst1,axiom,
    ! [A: rat,F: int > rat,B2: int,C: int] :
      ( ( ord_less_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_int @ B2 @ C )
       => ( ! [X5: int,Y4: int] :
              ( ( ord_less_int @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_3870_order__less__subst2,axiom,
    ! [A: real,B2: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ ( F @ B2 ) @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_3871_order__less__subst2,axiom,
    ! [A: real,B2: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_rat @ ( F @ B2 ) @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_3872_order__less__subst2,axiom,
    ! [A: real,B2: real,F: real > num,C: num] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_num @ ( F @ B2 ) @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_3873_order__less__subst2,axiom,
    ! [A: real,B2: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_3874_order__less__subst2,axiom,
    ! [A: real,B2: real,F: real > int,C: int] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_int @ ( F @ B2 ) @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_3875_order__less__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_real @ ( F @ B2 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_3876_order__less__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_rat @ ( F @ B2 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_3877_order__less__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_num @ ( F @ B2 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_3878_order__less__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_3879_order__less__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_int @ ( F @ B2 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_3880_order__less__not__sym,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ~ ( ord_less_real @ Y3 @ X3 ) ) ).

% order_less_not_sym
thf(fact_3881_order__less__not__sym,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ X3 @ Y3 )
     => ~ ( ord_less_rat @ Y3 @ X3 ) ) ).

% order_less_not_sym
thf(fact_3882_order__less__not__sym,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_num @ X3 @ Y3 )
     => ~ ( ord_less_num @ Y3 @ X3 ) ) ).

% order_less_not_sym
thf(fact_3883_order__less__not__sym,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).

% order_less_not_sym
thf(fact_3884_order__less__not__sym,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_int @ X3 @ Y3 )
     => ~ ( ord_less_int @ Y3 @ X3 ) ) ).

% order_less_not_sym
thf(fact_3885_order__less__imp__triv,axiom,
    ! [X3: real,Y3: real,P: $o] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ( ( ord_less_real @ Y3 @ X3 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_3886_order__less__imp__triv,axiom,
    ! [X3: rat,Y3: rat,P: $o] :
      ( ( ord_less_rat @ X3 @ Y3 )
     => ( ( ord_less_rat @ Y3 @ X3 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_3887_order__less__imp__triv,axiom,
    ! [X3: num,Y3: num,P: $o] :
      ( ( ord_less_num @ X3 @ Y3 )
     => ( ( ord_less_num @ Y3 @ X3 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_3888_order__less__imp__triv,axiom,
    ! [X3: nat,Y3: nat,P: $o] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ X3 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_3889_order__less__imp__triv,axiom,
    ! [X3: int,Y3: int,P: $o] :
      ( ( ord_less_int @ X3 @ Y3 )
     => ( ( ord_less_int @ Y3 @ X3 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_3890_linorder__less__linear,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
      | ( X3 = Y3 )
      | ( ord_less_real @ Y3 @ X3 ) ) ).

% linorder_less_linear
thf(fact_3891_linorder__less__linear,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ X3 @ Y3 )
      | ( X3 = Y3 )
      | ( ord_less_rat @ Y3 @ X3 ) ) ).

% linorder_less_linear
thf(fact_3892_linorder__less__linear,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_num @ X3 @ Y3 )
      | ( X3 = Y3 )
      | ( ord_less_num @ Y3 @ X3 ) ) ).

% linorder_less_linear
thf(fact_3893_linorder__less__linear,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
      | ( X3 = Y3 )
      | ( ord_less_nat @ Y3 @ X3 ) ) ).

% linorder_less_linear
thf(fact_3894_linorder__less__linear,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_int @ X3 @ Y3 )
      | ( X3 = Y3 )
      | ( ord_less_int @ Y3 @ X3 ) ) ).

% linorder_less_linear
thf(fact_3895_order__less__imp__not__eq,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_3896_order__less__imp__not__eq,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_3897_order__less__imp__not__eq,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_num @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_3898_order__less__imp__not__eq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_3899_order__less__imp__not__eq,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_int @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_3900_order__less__imp__not__eq2,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ( Y3 != X3 ) ) ).

% order_less_imp_not_eq2
thf(fact_3901_order__less__imp__not__eq2,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ X3 @ Y3 )
     => ( Y3 != X3 ) ) ).

% order_less_imp_not_eq2
thf(fact_3902_order__less__imp__not__eq2,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_num @ X3 @ Y3 )
     => ( Y3 != X3 ) ) ).

% order_less_imp_not_eq2
thf(fact_3903_order__less__imp__not__eq2,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( Y3 != X3 ) ) ).

% order_less_imp_not_eq2
thf(fact_3904_order__less__imp__not__eq2,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_int @ X3 @ Y3 )
     => ( Y3 != X3 ) ) ).

% order_less_imp_not_eq2
thf(fact_3905_order__less__imp__not__less,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ~ ( ord_less_real @ Y3 @ X3 ) ) ).

% order_less_imp_not_less
thf(fact_3906_order__less__imp__not__less,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ X3 @ Y3 )
     => ~ ( ord_less_rat @ Y3 @ X3 ) ) ).

% order_less_imp_not_less
thf(fact_3907_order__less__imp__not__less,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_num @ X3 @ Y3 )
     => ~ ( ord_less_num @ Y3 @ X3 ) ) ).

% order_less_imp_not_less
thf(fact_3908_order__less__imp__not__less,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).

% order_less_imp_not_less
thf(fact_3909_order__less__imp__not__less,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_int @ X3 @ Y3 )
     => ~ ( ord_less_int @ Y3 @ X3 ) ) ).

% order_less_imp_not_less
thf(fact_3910_small__lazy_H_Ocases,axiom,
    ! [X3: product_prod_int_int] :
      ~ ! [D3: int,I3: int] :
          ( X3
         != ( product_Pair_int_int @ D3 @ I3 ) ) ).

% small_lazy'.cases
thf(fact_3911_exhaustive__int_H_Ocases,axiom,
    ! [X3: produc7773217078559923341nt_int] :
      ~ ! [F2: int > option6357759511663192854e_term,D3: int,I3: int] :
          ( X3
         != ( produc4305682042979456191nt_int @ F2 @ ( product_Pair_int_int @ D3 @ I3 ) ) ) ).

% exhaustive_int'.cases
thf(fact_3912_full__exhaustive__int_H_Ocases,axiom,
    ! [X3: produc2285326912895808259nt_int] :
      ~ ! [F2: produc8551481072490612790e_term > option6357759511663192854e_term,D3: int,I3: int] :
          ( X3
         != ( produc5700946648718959541nt_int @ F2 @ ( product_Pair_int_int @ D3 @ I3 ) ) ) ).

% full_exhaustive_int'.cases
thf(fact_3913_leD,axiom,
    ! [Y3: real,X3: real] :
      ( ( ord_less_eq_real @ Y3 @ X3 )
     => ~ ( ord_less_real @ X3 @ Y3 ) ) ).

% leD
thf(fact_3914_leD,axiom,
    ! [Y3: set_int,X3: set_int] :
      ( ( ord_less_eq_set_int @ Y3 @ X3 )
     => ~ ( ord_less_set_int @ X3 @ Y3 ) ) ).

% leD
thf(fact_3915_leD,axiom,
    ! [Y3: rat,X3: rat] :
      ( ( ord_less_eq_rat @ Y3 @ X3 )
     => ~ ( ord_less_rat @ X3 @ Y3 ) ) ).

% leD
thf(fact_3916_leD,axiom,
    ! [Y3: num,X3: num] :
      ( ( ord_less_eq_num @ Y3 @ X3 )
     => ~ ( ord_less_num @ X3 @ Y3 ) ) ).

% leD
thf(fact_3917_leD,axiom,
    ! [Y3: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X3 )
     => ~ ( ord_less_nat @ X3 @ Y3 ) ) ).

% leD
thf(fact_3918_leD,axiom,
    ! [Y3: int,X3: int] :
      ( ( ord_less_eq_int @ Y3 @ X3 )
     => ~ ( ord_less_int @ X3 @ Y3 ) ) ).

% leD
thf(fact_3919_leI,axiom,
    ! [X3: real,Y3: real] :
      ( ~ ( ord_less_real @ X3 @ Y3 )
     => ( ord_less_eq_real @ Y3 @ X3 ) ) ).

% leI
thf(fact_3920_leI,axiom,
    ! [X3: rat,Y3: rat] :
      ( ~ ( ord_less_rat @ X3 @ Y3 )
     => ( ord_less_eq_rat @ Y3 @ X3 ) ) ).

% leI
thf(fact_3921_leI,axiom,
    ! [X3: num,Y3: num] :
      ( ~ ( ord_less_num @ X3 @ Y3 )
     => ( ord_less_eq_num @ Y3 @ X3 ) ) ).

% leI
thf(fact_3922_leI,axiom,
    ! [X3: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y3 )
     => ( ord_less_eq_nat @ Y3 @ X3 ) ) ).

% leI
thf(fact_3923_leI,axiom,
    ! [X3: int,Y3: int] :
      ( ~ ( ord_less_int @ X3 @ Y3 )
     => ( ord_less_eq_int @ Y3 @ X3 ) ) ).

% leI
thf(fact_3924_nless__le,axiom,
    ! [A: real,B2: real] :
      ( ( ~ ( ord_less_real @ A @ B2 ) )
      = ( ~ ( ord_less_eq_real @ A @ B2 )
        | ( A = B2 ) ) ) ).

% nless_le
thf(fact_3925_nless__le,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( ~ ( ord_less_set_int @ A @ B2 ) )
      = ( ~ ( ord_less_eq_set_int @ A @ B2 )
        | ( A = B2 ) ) ) ).

% nless_le
thf(fact_3926_nless__le,axiom,
    ! [A: rat,B2: rat] :
      ( ( ~ ( ord_less_rat @ A @ B2 ) )
      = ( ~ ( ord_less_eq_rat @ A @ B2 )
        | ( A = B2 ) ) ) ).

% nless_le
thf(fact_3927_nless__le,axiom,
    ! [A: num,B2: num] :
      ( ( ~ ( ord_less_num @ A @ B2 ) )
      = ( ~ ( ord_less_eq_num @ A @ B2 )
        | ( A = B2 ) ) ) ).

% nless_le
thf(fact_3928_nless__le,axiom,
    ! [A: nat,B2: nat] :
      ( ( ~ ( ord_less_nat @ A @ B2 ) )
      = ( ~ ( ord_less_eq_nat @ A @ B2 )
        | ( A = B2 ) ) ) ).

% nless_le
thf(fact_3929_nless__le,axiom,
    ! [A: int,B2: int] :
      ( ( ~ ( ord_less_int @ A @ B2 ) )
      = ( ~ ( ord_less_eq_int @ A @ B2 )
        | ( A = B2 ) ) ) ).

% nless_le
thf(fact_3930_antisym__conv1,axiom,
    ! [X3: real,Y3: real] :
      ( ~ ( ord_less_real @ X3 @ Y3 )
     => ( ( ord_less_eq_real @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_3931_antisym__conv1,axiom,
    ! [X3: set_int,Y3: set_int] :
      ( ~ ( ord_less_set_int @ X3 @ Y3 )
     => ( ( ord_less_eq_set_int @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_3932_antisym__conv1,axiom,
    ! [X3: rat,Y3: rat] :
      ( ~ ( ord_less_rat @ X3 @ Y3 )
     => ( ( ord_less_eq_rat @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_3933_antisym__conv1,axiom,
    ! [X3: num,Y3: num] :
      ( ~ ( ord_less_num @ X3 @ Y3 )
     => ( ( ord_less_eq_num @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_3934_antisym__conv1,axiom,
    ! [X3: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_3935_antisym__conv1,axiom,
    ! [X3: int,Y3: int] :
      ( ~ ( ord_less_int @ X3 @ Y3 )
     => ( ( ord_less_eq_int @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_3936_antisym__conv2,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ X3 @ Y3 )
     => ( ( ~ ( ord_less_real @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_3937_antisym__conv2,axiom,
    ! [X3: set_int,Y3: set_int] :
      ( ( ord_less_eq_set_int @ X3 @ Y3 )
     => ( ( ~ ( ord_less_set_int @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_3938_antisym__conv2,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y3 )
     => ( ( ~ ( ord_less_rat @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_3939_antisym__conv2,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_eq_num @ X3 @ Y3 )
     => ( ( ~ ( ord_less_num @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_3940_antisym__conv2,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_3941_antisym__conv2,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ X3 @ Y3 )
     => ( ( ~ ( ord_less_int @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_3942_dense__ge,axiom,
    ! [Z: real,Y3: real] :
      ( ! [X5: real] :
          ( ( ord_less_real @ Z @ X5 )
         => ( ord_less_eq_real @ Y3 @ X5 ) )
     => ( ord_less_eq_real @ Y3 @ Z ) ) ).

% dense_ge
thf(fact_3943_dense__ge,axiom,
    ! [Z: rat,Y3: rat] :
      ( ! [X5: rat] :
          ( ( ord_less_rat @ Z @ X5 )
         => ( ord_less_eq_rat @ Y3 @ X5 ) )
     => ( ord_less_eq_rat @ Y3 @ Z ) ) ).

% dense_ge
thf(fact_3944_dense__le,axiom,
    ! [Y3: real,Z: real] :
      ( ! [X5: real] :
          ( ( ord_less_real @ X5 @ Y3 )
         => ( ord_less_eq_real @ X5 @ Z ) )
     => ( ord_less_eq_real @ Y3 @ Z ) ) ).

% dense_le
thf(fact_3945_dense__le,axiom,
    ! [Y3: rat,Z: rat] :
      ( ! [X5: rat] :
          ( ( ord_less_rat @ X5 @ Y3 )
         => ( ord_less_eq_rat @ X5 @ Z ) )
     => ( ord_less_eq_rat @ Y3 @ Z ) ) ).

% dense_le
thf(fact_3946_less__le__not__le,axiom,
    ( ord_less_real
    = ( ^ [X: real,Y: real] :
          ( ( ord_less_eq_real @ X @ Y )
          & ~ ( ord_less_eq_real @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_3947_less__le__not__le,axiom,
    ( ord_less_set_int
    = ( ^ [X: set_int,Y: set_int] :
          ( ( ord_less_eq_set_int @ X @ Y )
          & ~ ( ord_less_eq_set_int @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_3948_less__le__not__le,axiom,
    ( ord_less_rat
    = ( ^ [X: rat,Y: rat] :
          ( ( ord_less_eq_rat @ X @ Y )
          & ~ ( ord_less_eq_rat @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_3949_less__le__not__le,axiom,
    ( ord_less_num
    = ( ^ [X: num,Y: num] :
          ( ( ord_less_eq_num @ X @ Y )
          & ~ ( ord_less_eq_num @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_3950_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_eq_nat @ X @ Y )
          & ~ ( ord_less_eq_nat @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_3951_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X: int,Y: int] :
          ( ( ord_less_eq_int @ X @ Y )
          & ~ ( ord_less_eq_int @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_3952_not__le__imp__less,axiom,
    ! [Y3: real,X3: real] :
      ( ~ ( ord_less_eq_real @ Y3 @ X3 )
     => ( ord_less_real @ X3 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_3953_not__le__imp__less,axiom,
    ! [Y3: rat,X3: rat] :
      ( ~ ( ord_less_eq_rat @ Y3 @ X3 )
     => ( ord_less_rat @ X3 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_3954_not__le__imp__less,axiom,
    ! [Y3: num,X3: num] :
      ( ~ ( ord_less_eq_num @ Y3 @ X3 )
     => ( ord_less_num @ X3 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_3955_not__le__imp__less,axiom,
    ! [Y3: nat,X3: nat] :
      ( ~ ( ord_less_eq_nat @ Y3 @ X3 )
     => ( ord_less_nat @ X3 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_3956_not__le__imp__less,axiom,
    ! [Y3: int,X3: int] :
      ( ~ ( ord_less_eq_int @ Y3 @ X3 )
     => ( ord_less_int @ X3 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_3957_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [A5: real,B4: real] :
          ( ( ord_less_real @ A5 @ B4 )
          | ( A5 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_3958_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B4: set_int] :
          ( ( ord_less_set_int @ A5 @ B4 )
          | ( A5 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_3959_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [A5: rat,B4: rat] :
          ( ( ord_less_rat @ A5 @ B4 )
          | ( A5 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_3960_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [A5: num,B4: num] :
          ( ( ord_less_num @ A5 @ B4 )
          | ( A5 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_3961_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_nat @ A5 @ B4 )
          | ( A5 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_3962_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B4: int] :
          ( ( ord_less_int @ A5 @ B4 )
          | ( A5 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_3963_order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [A5: real,B4: real] :
          ( ( ord_less_eq_real @ A5 @ B4 )
          & ( A5 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_3964_order_Ostrict__iff__order,axiom,
    ( ord_less_set_int
    = ( ^ [A5: set_int,B4: set_int] :
          ( ( ord_less_eq_set_int @ A5 @ B4 )
          & ( A5 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_3965_order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [A5: rat,B4: rat] :
          ( ( ord_less_eq_rat @ A5 @ B4 )
          & ( A5 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_3966_order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [A5: num,B4: num] :
          ( ( ord_less_eq_num @ A5 @ B4 )
          & ( A5 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_3967_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A5 @ B4 )
          & ( A5 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_3968_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A5: int,B4: int] :
          ( ( ord_less_eq_int @ A5 @ B4 )
          & ( A5 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_3969_order_Ostrict__trans1,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_real @ B2 @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_3970_order_Ostrict__trans1,axiom,
    ! [A: set_int,B2: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ B2 )
     => ( ( ord_less_set_int @ B2 @ C )
       => ( ord_less_set_int @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_3971_order_Ostrict__trans1,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_3972_order_Ostrict__trans1,axiom,
    ! [A: num,B2: num,C: num] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ord_less_num @ B2 @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_3973_order_Ostrict__trans1,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_3974_order_Ostrict__trans1,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_int @ B2 @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_3975_order_Ostrict__trans2,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_eq_real @ B2 @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_3976_order_Ostrict__trans2,axiom,
    ! [A: set_int,B2: set_int,C: set_int] :
      ( ( ord_less_set_int @ A @ B2 )
     => ( ( ord_less_eq_set_int @ B2 @ C )
       => ( ord_less_set_int @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_3977_order_Ostrict__trans2,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_3978_order_Ostrict__trans2,axiom,
    ! [A: num,B2: num,C: num] :
      ( ( ord_less_num @ A @ B2 )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_3979_order_Ostrict__trans2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_3980_order_Ostrict__trans2,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_eq_int @ B2 @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_3981_order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [A5: real,B4: real] :
          ( ( ord_less_eq_real @ A5 @ B4 )
          & ~ ( ord_less_eq_real @ B4 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_3982_order_Ostrict__iff__not,axiom,
    ( ord_less_set_int
    = ( ^ [A5: set_int,B4: set_int] :
          ( ( ord_less_eq_set_int @ A5 @ B4 )
          & ~ ( ord_less_eq_set_int @ B4 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_3983_order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [A5: rat,B4: rat] :
          ( ( ord_less_eq_rat @ A5 @ B4 )
          & ~ ( ord_less_eq_rat @ B4 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_3984_order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [A5: num,B4: num] :
          ( ( ord_less_eq_num @ A5 @ B4 )
          & ~ ( ord_less_eq_num @ B4 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_3985_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A5 @ B4 )
          & ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_3986_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A5: int,B4: int] :
          ( ( ord_less_eq_int @ A5 @ B4 )
          & ~ ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_3987_dense__ge__bounded,axiom,
    ! [Z: real,X3: real,Y3: real] :
      ( ( ord_less_real @ Z @ X3 )
     => ( ! [W2: real] :
            ( ( ord_less_real @ Z @ W2 )
           => ( ( ord_less_real @ W2 @ X3 )
             => ( ord_less_eq_real @ Y3 @ W2 ) ) )
       => ( ord_less_eq_real @ Y3 @ Z ) ) ) ).

% dense_ge_bounded
thf(fact_3988_dense__ge__bounded,axiom,
    ! [Z: rat,X3: rat,Y3: rat] :
      ( ( ord_less_rat @ Z @ X3 )
     => ( ! [W2: rat] :
            ( ( ord_less_rat @ Z @ W2 )
           => ( ( ord_less_rat @ W2 @ X3 )
             => ( ord_less_eq_rat @ Y3 @ W2 ) ) )
       => ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).

% dense_ge_bounded
thf(fact_3989_dense__le__bounded,axiom,
    ! [X3: real,Y3: real,Z: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ( ! [W2: real] :
            ( ( ord_less_real @ X3 @ W2 )
           => ( ( ord_less_real @ W2 @ Y3 )
             => ( ord_less_eq_real @ W2 @ Z ) ) )
       => ( ord_less_eq_real @ Y3 @ Z ) ) ) ).

% dense_le_bounded
thf(fact_3990_dense__le__bounded,axiom,
    ! [X3: rat,Y3: rat,Z: rat] :
      ( ( ord_less_rat @ X3 @ Y3 )
     => ( ! [W2: rat] :
            ( ( ord_less_rat @ X3 @ W2 )
           => ( ( ord_less_rat @ W2 @ Y3 )
             => ( ord_less_eq_rat @ W2 @ Z ) ) )
       => ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).

% dense_le_bounded
thf(fact_3991_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [B4: real,A5: real] :
          ( ( ord_less_real @ B4 @ A5 )
          | ( A5 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_3992_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_int
    = ( ^ [B4: set_int,A5: set_int] :
          ( ( ord_less_set_int @ B4 @ A5 )
          | ( A5 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_3993_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A5: rat] :
          ( ( ord_less_rat @ B4 @ A5 )
          | ( A5 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_3994_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [B4: num,A5: num] :
          ( ( ord_less_num @ B4 @ A5 )
          | ( A5 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_3995_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( ord_less_nat @ B4 @ A5 )
          | ( A5 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_3996_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A5: int] :
          ( ( ord_less_int @ B4 @ A5 )
          | ( A5 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_3997_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [B4: real,A5: real] :
          ( ( ord_less_eq_real @ B4 @ A5 )
          & ( A5 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_3998_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_int
    = ( ^ [B4: set_int,A5: set_int] :
          ( ( ord_less_eq_set_int @ B4 @ A5 )
          & ( A5 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_3999_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [B4: rat,A5: rat] :
          ( ( ord_less_eq_rat @ B4 @ A5 )
          & ( A5 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_4000_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [B4: num,A5: num] :
          ( ( ord_less_eq_num @ B4 @ A5 )
          & ( A5 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_4001_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( ord_less_eq_nat @ B4 @ A5 )
          & ( A5 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_4002_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B4: int,A5: int] :
          ( ( ord_less_eq_int @ B4 @ A5 )
          & ( A5 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_4003_dual__order_Ostrict__trans1,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B2 @ A )
     => ( ( ord_less_real @ C @ B2 )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_4004_dual__order_Ostrict__trans1,axiom,
    ! [B2: set_int,A: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ B2 @ A )
     => ( ( ord_less_set_int @ C @ B2 )
       => ( ord_less_set_int @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_4005_dual__order_Ostrict__trans1,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B2 @ A )
     => ( ( ord_less_rat @ C @ B2 )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_4006_dual__order_Ostrict__trans1,axiom,
    ! [B2: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B2 @ A )
     => ( ( ord_less_num @ C @ B2 )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_4007_dual__order_Ostrict__trans1,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( ord_less_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_4008_dual__order_Ostrict__trans1,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ( ord_less_int @ C @ B2 )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_4009_dual__order_Ostrict__trans2,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ( ord_less_eq_real @ C @ B2 )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_4010_dual__order_Ostrict__trans2,axiom,
    ! [B2: set_int,A: set_int,C: set_int] :
      ( ( ord_less_set_int @ B2 @ A )
     => ( ( ord_less_eq_set_int @ C @ B2 )
       => ( ord_less_set_int @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_4011_dual__order_Ostrict__trans2,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B2 @ A )
     => ( ( ord_less_eq_rat @ C @ B2 )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_4012_dual__order_Ostrict__trans2,axiom,
    ! [B2: num,A: num,C: num] :
      ( ( ord_less_num @ B2 @ A )
     => ( ( ord_less_eq_num @ C @ B2 )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_4013_dual__order_Ostrict__trans2,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_4014_dual__order_Ostrict__trans2,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_int @ B2 @ A )
     => ( ( ord_less_eq_int @ C @ B2 )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_4015_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [B4: real,A5: real] :
          ( ( ord_less_eq_real @ B4 @ A5 )
          & ~ ( ord_less_eq_real @ A5 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_4016_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_int
    = ( ^ [B4: set_int,A5: set_int] :
          ( ( ord_less_eq_set_int @ B4 @ A5 )
          & ~ ( ord_less_eq_set_int @ A5 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_4017_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [B4: rat,A5: rat] :
          ( ( ord_less_eq_rat @ B4 @ A5 )
          & ~ ( ord_less_eq_rat @ A5 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_4018_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [B4: num,A5: num] :
          ( ( ord_less_eq_num @ B4 @ A5 )
          & ~ ( ord_less_eq_num @ A5 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_4019_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( ord_less_eq_nat @ B4 @ A5 )
          & ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_4020_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B4: int,A5: int] :
          ( ( ord_less_eq_int @ B4 @ A5 )
          & ~ ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_4021_order_Ostrict__implies__order,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ord_less_eq_real @ A @ B2 ) ) ).

% order.strict_implies_order
thf(fact_4022_order_Ostrict__implies__order,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( ord_less_set_int @ A @ B2 )
     => ( ord_less_eq_set_int @ A @ B2 ) ) ).

% order.strict_implies_order
thf(fact_4023_order_Ostrict__implies__order,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ord_less_eq_rat @ A @ B2 ) ) ).

% order.strict_implies_order
thf(fact_4024_order_Ostrict__implies__order,axiom,
    ! [A: num,B2: num] :
      ( ( ord_less_num @ A @ B2 )
     => ( ord_less_eq_num @ A @ B2 ) ) ).

% order.strict_implies_order
thf(fact_4025_order_Ostrict__implies__order,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ord_less_eq_nat @ A @ B2 ) ) ).

% order.strict_implies_order
thf(fact_4026_order_Ostrict__implies__order,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ord_less_eq_int @ A @ B2 ) ) ).

% order.strict_implies_order
thf(fact_4027_dual__order_Ostrict__implies__order,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ord_less_eq_real @ B2 @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_4028_dual__order_Ostrict__implies__order,axiom,
    ! [B2: set_int,A: set_int] :
      ( ( ord_less_set_int @ B2 @ A )
     => ( ord_less_eq_set_int @ B2 @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_4029_dual__order_Ostrict__implies__order,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_rat @ B2 @ A )
     => ( ord_less_eq_rat @ B2 @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_4030_dual__order_Ostrict__implies__order,axiom,
    ! [B2: num,A: num] :
      ( ( ord_less_num @ B2 @ A )
     => ( ord_less_eq_num @ B2 @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_4031_dual__order_Ostrict__implies__order,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ord_less_eq_nat @ B2 @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_4032_dual__order_Ostrict__implies__order,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ B2 @ A )
     => ( ord_less_eq_int @ B2 @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_4033_order__le__less,axiom,
    ( ord_less_eq_real
    = ( ^ [X: real,Y: real] :
          ( ( ord_less_real @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_4034_order__le__less,axiom,
    ( ord_less_eq_set_int
    = ( ^ [X: set_int,Y: set_int] :
          ( ( ord_less_set_int @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_4035_order__le__less,axiom,
    ( ord_less_eq_rat
    = ( ^ [X: rat,Y: rat] :
          ( ( ord_less_rat @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_4036_order__le__less,axiom,
    ( ord_less_eq_num
    = ( ^ [X: num,Y: num] :
          ( ( ord_less_num @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_4037_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_nat @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_4038_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X: int,Y: int] :
          ( ( ord_less_int @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_4039_order__less__le,axiom,
    ( ord_less_real
    = ( ^ [X: real,Y: real] :
          ( ( ord_less_eq_real @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_4040_order__less__le,axiom,
    ( ord_less_set_int
    = ( ^ [X: set_int,Y: set_int] :
          ( ( ord_less_eq_set_int @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_4041_order__less__le,axiom,
    ( ord_less_rat
    = ( ^ [X: rat,Y: rat] :
          ( ( ord_less_eq_rat @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_4042_order__less__le,axiom,
    ( ord_less_num
    = ( ^ [X: num,Y: num] :
          ( ( ord_less_eq_num @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_4043_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_eq_nat @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_4044_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X: int,Y: int] :
          ( ( ord_less_eq_int @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_4045_linorder__not__le,axiom,
    ! [X3: real,Y3: real] :
      ( ( ~ ( ord_less_eq_real @ X3 @ Y3 ) )
      = ( ord_less_real @ Y3 @ X3 ) ) ).

% linorder_not_le
thf(fact_4046_linorder__not__le,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ~ ( ord_less_eq_rat @ X3 @ Y3 ) )
      = ( ord_less_rat @ Y3 @ X3 ) ) ).

% linorder_not_le
thf(fact_4047_linorder__not__le,axiom,
    ! [X3: num,Y3: num] :
      ( ( ~ ( ord_less_eq_num @ X3 @ Y3 ) )
      = ( ord_less_num @ Y3 @ X3 ) ) ).

% linorder_not_le
thf(fact_4048_linorder__not__le,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ~ ( ord_less_eq_nat @ X3 @ Y3 ) )
      = ( ord_less_nat @ Y3 @ X3 ) ) ).

% linorder_not_le
thf(fact_4049_linorder__not__le,axiom,
    ! [X3: int,Y3: int] :
      ( ( ~ ( ord_less_eq_int @ X3 @ Y3 ) )
      = ( ord_less_int @ Y3 @ X3 ) ) ).

% linorder_not_le
thf(fact_4050_linorder__not__less,axiom,
    ! [X3: real,Y3: real] :
      ( ( ~ ( ord_less_real @ X3 @ Y3 ) )
      = ( ord_less_eq_real @ Y3 @ X3 ) ) ).

% linorder_not_less
thf(fact_4051_linorder__not__less,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ~ ( ord_less_rat @ X3 @ Y3 ) )
      = ( ord_less_eq_rat @ Y3 @ X3 ) ) ).

% linorder_not_less
thf(fact_4052_linorder__not__less,axiom,
    ! [X3: num,Y3: num] :
      ( ( ~ ( ord_less_num @ X3 @ Y3 ) )
      = ( ord_less_eq_num @ Y3 @ X3 ) ) ).

% linorder_not_less
thf(fact_4053_linorder__not__less,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
      = ( ord_less_eq_nat @ Y3 @ X3 ) ) ).

% linorder_not_less
thf(fact_4054_linorder__not__less,axiom,
    ! [X3: int,Y3: int] :
      ( ( ~ ( ord_less_int @ X3 @ Y3 ) )
      = ( ord_less_eq_int @ Y3 @ X3 ) ) ).

% linorder_not_less
thf(fact_4055_order__less__imp__le,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ( ord_less_eq_real @ X3 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_4056_order__less__imp__le,axiom,
    ! [X3: set_int,Y3: set_int] :
      ( ( ord_less_set_int @ X3 @ Y3 )
     => ( ord_less_eq_set_int @ X3 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_4057_order__less__imp__le,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ X3 @ Y3 )
     => ( ord_less_eq_rat @ X3 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_4058_order__less__imp__le,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_num @ X3 @ Y3 )
     => ( ord_less_eq_num @ X3 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_4059_order__less__imp__le,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( ord_less_eq_nat @ X3 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_4060_order__less__imp__le,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_int @ X3 @ Y3 )
     => ( ord_less_eq_int @ X3 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_4061_order__le__neq__trans,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( A != B2 )
       => ( ord_less_real @ A @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_4062_order__le__neq__trans,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A @ B2 )
     => ( ( A != B2 )
       => ( ord_less_set_int @ A @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_4063_order__le__neq__trans,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( A != B2 )
       => ( ord_less_rat @ A @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_4064_order__le__neq__trans,axiom,
    ! [A: num,B2: num] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( A != B2 )
       => ( ord_less_num @ A @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_4065_order__le__neq__trans,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( A != B2 )
       => ( ord_less_nat @ A @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_4066_order__le__neq__trans,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( A != B2 )
       => ( ord_less_int @ A @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_4067_order__neq__le__trans,axiom,
    ! [A: real,B2: real] :
      ( ( A != B2 )
     => ( ( ord_less_eq_real @ A @ B2 )
       => ( ord_less_real @ A @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_4068_order__neq__le__trans,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( A != B2 )
     => ( ( ord_less_eq_set_int @ A @ B2 )
       => ( ord_less_set_int @ A @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_4069_order__neq__le__trans,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != B2 )
     => ( ( ord_less_eq_rat @ A @ B2 )
       => ( ord_less_rat @ A @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_4070_order__neq__le__trans,axiom,
    ! [A: num,B2: num] :
      ( ( A != B2 )
     => ( ( ord_less_eq_num @ A @ B2 )
       => ( ord_less_num @ A @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_4071_order__neq__le__trans,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != B2 )
     => ( ( ord_less_eq_nat @ A @ B2 )
       => ( ord_less_nat @ A @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_4072_order__neq__le__trans,axiom,
    ! [A: int,B2: int] :
      ( ( A != B2 )
     => ( ( ord_less_eq_int @ A @ B2 )
       => ( ord_less_int @ A @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_4073_order__le__less__trans,axiom,
    ! [X3: real,Y3: real,Z: real] :
      ( ( ord_less_eq_real @ X3 @ Y3 )
     => ( ( ord_less_real @ Y3 @ Z )
       => ( ord_less_real @ X3 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_4074_order__le__less__trans,axiom,
    ! [X3: set_int,Y3: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ X3 @ Y3 )
     => ( ( ord_less_set_int @ Y3 @ Z )
       => ( ord_less_set_int @ X3 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_4075_order__le__less__trans,axiom,
    ! [X3: rat,Y3: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y3 )
     => ( ( ord_less_rat @ Y3 @ Z )
       => ( ord_less_rat @ X3 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_4076_order__le__less__trans,axiom,
    ! [X3: num,Y3: num,Z: num] :
      ( ( ord_less_eq_num @ X3 @ Y3 )
     => ( ( ord_less_num @ Y3 @ Z )
       => ( ord_less_num @ X3 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_4077_order__le__less__trans,axiom,
    ! [X3: nat,Y3: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ Z )
       => ( ord_less_nat @ X3 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_4078_order__le__less__trans,axiom,
    ! [X3: int,Y3: int,Z: int] :
      ( ( ord_less_eq_int @ X3 @ Y3 )
     => ( ( ord_less_int @ Y3 @ Z )
       => ( ord_less_int @ X3 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_4079_order__less__le__trans,axiom,
    ! [X3: real,Y3: real,Z: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ Z )
       => ( ord_less_real @ X3 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_4080_order__less__le__trans,axiom,
    ! [X3: set_int,Y3: set_int,Z: set_int] :
      ( ( ord_less_set_int @ X3 @ Y3 )
     => ( ( ord_less_eq_set_int @ Y3 @ Z )
       => ( ord_less_set_int @ X3 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_4081_order__less__le__trans,axiom,
    ! [X3: rat,Y3: rat,Z: rat] :
      ( ( ord_less_rat @ X3 @ Y3 )
     => ( ( ord_less_eq_rat @ Y3 @ Z )
       => ( ord_less_rat @ X3 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_4082_order__less__le__trans,axiom,
    ! [X3: num,Y3: num,Z: num] :
      ( ( ord_less_num @ X3 @ Y3 )
     => ( ( ord_less_eq_num @ Y3 @ Z )
       => ( ord_less_num @ X3 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_4083_order__less__le__trans,axiom,
    ! [X3: nat,Y3: nat,Z: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z )
       => ( ord_less_nat @ X3 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_4084_order__less__le__trans,axiom,
    ! [X3: int,Y3: int,Z: int] :
      ( ( ord_less_int @ X3 @ Y3 )
     => ( ( ord_less_eq_int @ Y3 @ Z )
       => ( ord_less_int @ X3 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_4085_order__le__less__subst1,axiom,
    ! [A: real,F: real > real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4086_order__le__less__subst1,axiom,
    ! [A: real,F: rat > real,B2: rat,C: rat] :
      ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4087_order__le__less__subst1,axiom,
    ! [A: real,F: num > real,B2: num,C: num] :
      ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_num @ B2 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_num @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4088_order__le__less__subst1,axiom,
    ! [A: real,F: nat > real,B2: nat,C: nat] :
      ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_nat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4089_order__le__less__subst1,axiom,
    ! [A: real,F: int > real,B2: int,C: int] :
      ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_int @ B2 @ C )
       => ( ! [X5: int,Y4: int] :
              ( ( ord_less_int @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4090_order__le__less__subst1,axiom,
    ! [A: rat,F: real > rat,B2: real,C: real] :
      ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_real @ B2 @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4091_order__le__less__subst1,axiom,
    ! [A: rat,F: rat > rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4092_order__le__less__subst1,axiom,
    ! [A: rat,F: num > rat,B2: num,C: num] :
      ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_num @ B2 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_num @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4093_order__le__less__subst1,axiom,
    ! [A: rat,F: nat > rat,B2: nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_nat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4094_order__le__less__subst1,axiom,
    ! [A: rat,F: int > rat,B2: int,C: int] :
      ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_int @ B2 @ C )
       => ( ! [X5: int,Y4: int] :
              ( ( ord_less_int @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4095_order__le__less__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > real,C: real] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_real @ ( F @ B2 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_4096_order__le__less__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_rat @ ( F @ B2 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_4097_order__le__less__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_num @ ( F @ B2 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_4098_order__le__less__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_4099_order__le__less__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_int @ ( F @ B2 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_4100_order__le__less__subst2,axiom,
    ! [A: num,B2: num,F: num > real,C: real] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ord_less_real @ ( F @ B2 ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_4101_order__le__less__subst2,axiom,
    ! [A: num,B2: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ord_less_rat @ ( F @ B2 ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_4102_order__le__less__subst2,axiom,
    ! [A: num,B2: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ord_less_num @ ( F @ B2 ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_4103_order__le__less__subst2,axiom,
    ! [A: num,B2: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_4104_order__le__less__subst2,axiom,
    ! [A: num,B2: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ord_less_int @ ( F @ B2 ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_4105_order__less__le__subst1,axiom,
    ! [A: real,F: rat > real,B2: rat,C: rat] :
      ( ( ord_less_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4106_order__less__le__subst1,axiom,
    ! [A: rat,F: rat > rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4107_order__less__le__subst1,axiom,
    ! [A: num,F: rat > num,B2: rat,C: rat] :
      ( ( ord_less_num @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4108_order__less__le__subst1,axiom,
    ! [A: nat,F: rat > nat,B2: rat,C: rat] :
      ( ( ord_less_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4109_order__less__le__subst1,axiom,
    ! [A: int,F: rat > int,B2: rat,C: rat] :
      ( ( ord_less_int @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4110_order__less__le__subst1,axiom,
    ! [A: real,F: num > real,B2: num,C: num] :
      ( ( ord_less_real @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4111_order__less__le__subst1,axiom,
    ! [A: rat,F: num > rat,B2: num,C: num] :
      ( ( ord_less_rat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4112_order__less__le__subst1,axiom,
    ! [A: num,F: num > num,B2: num,C: num] :
      ( ( ord_less_num @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4113_order__less__le__subst1,axiom,
    ! [A: nat,F: num > nat,B2: num,C: num] :
      ( ( ord_less_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4114_order__less__le__subst1,axiom,
    ! [A: int,F: num > int,B2: num,C: num] :
      ( ( ord_less_int @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4115_order__less__le__subst2,axiom,
    ! [A: real,B2: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_4116_order__less__le__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_4117_order__less__le__subst2,axiom,
    ! [A: num,B2: num,F: num > real,C: real] :
      ( ( ord_less_num @ A @ B2 )
     => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_num @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_4118_order__less__le__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > real,C: real] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_nat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_4119_order__less__le__subst2,axiom,
    ! [A: int,B2: int,F: int > real,C: real] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
       => ( ! [X5: int,Y4: int] :
              ( ( ord_less_int @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_4120_order__less__le__subst2,axiom,
    ! [A: real,B2: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_4121_order__less__le__subst2,axiom,
    ! [A: rat,B2: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_4122_order__less__le__subst2,axiom,
    ! [A: num,B2: num,F: num > rat,C: rat] :
      ( ( ord_less_num @ A @ B2 )
     => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_num @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_4123_order__less__le__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > rat,C: rat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_nat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_4124_order__less__le__subst2,axiom,
    ! [A: int,B2: int,F: int > rat,C: rat] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
       => ( ! [X5: int,Y4: int] :
              ( ( ord_less_int @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_4125_linorder__le__less__linear,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ X3 @ Y3 )
      | ( ord_less_real @ Y3 @ X3 ) ) ).

% linorder_le_less_linear
thf(fact_4126_linorder__le__less__linear,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y3 )
      | ( ord_less_rat @ Y3 @ X3 ) ) ).

% linorder_le_less_linear
thf(fact_4127_linorder__le__less__linear,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_eq_num @ X3 @ Y3 )
      | ( ord_less_num @ Y3 @ X3 ) ) ).

% linorder_le_less_linear
thf(fact_4128_linorder__le__less__linear,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
      | ( ord_less_nat @ Y3 @ X3 ) ) ).

% linorder_le_less_linear
thf(fact_4129_linorder__le__less__linear,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ X3 @ Y3 )
      | ( ord_less_int @ Y3 @ X3 ) ) ).

% linorder_le_less_linear
thf(fact_4130_order__le__imp__less__or__eq,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ X3 @ Y3 )
     => ( ( ord_less_real @ X3 @ Y3 )
        | ( X3 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_4131_order__le__imp__less__or__eq,axiom,
    ! [X3: set_int,Y3: set_int] :
      ( ( ord_less_eq_set_int @ X3 @ Y3 )
     => ( ( ord_less_set_int @ X3 @ Y3 )
        | ( X3 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_4132_order__le__imp__less__or__eq,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y3 )
     => ( ( ord_less_rat @ X3 @ Y3 )
        | ( X3 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_4133_order__le__imp__less__or__eq,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_eq_num @ X3 @ Y3 )
     => ( ( ord_less_num @ X3 @ Y3 )
        | ( X3 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_4134_order__le__imp__less__or__eq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_nat @ X3 @ Y3 )
        | ( X3 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_4135_order__le__imp__less__or__eq,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ X3 @ Y3 )
     => ( ( ord_less_int @ X3 @ Y3 )
        | ( X3 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_4136_bot_Oextremum,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% bot.extremum
thf(fact_4137_bot_Oextremum,axiom,
    ! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ bot_bo4199563552545308370d_enat @ A ) ).

% bot.extremum
thf(fact_4138_bot_Oextremum,axiom,
    ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).

% bot.extremum
thf(fact_4139_bot_Oextremum,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).

% bot.extremum
thf(fact_4140_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_4141_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
      = ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
thf(fact_4142_bot_Oextremum__unique,axiom,
    ! [A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ bot_bo4199563552545308370d_enat )
      = ( A = bot_bo4199563552545308370d_enat ) ) ).

% bot.extremum_unique
thf(fact_4143_bot_Oextremum__unique,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
      = ( A = bot_bot_set_real ) ) ).

% bot.extremum_unique
thf(fact_4144_bot_Oextremum__unique,axiom,
    ! [A: set_int] :
      ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
      = ( A = bot_bot_set_int ) ) ).

% bot.extremum_unique
thf(fact_4145_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_4146_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_4147_bot_Oextremum__uniqueI,axiom,
    ! [A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ bot_bo4199563552545308370d_enat )
     => ( A = bot_bo4199563552545308370d_enat ) ) ).

% bot.extremum_uniqueI
thf(fact_4148_bot_Oextremum__uniqueI,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
     => ( A = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
thf(fact_4149_bot_Oextremum__uniqueI,axiom,
    ! [A: set_int] :
      ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
     => ( A = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
thf(fact_4150_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_4151_bot_Oextremum__strict,axiom,
    ! [A: set_nat] :
      ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).

% bot.extremum_strict
thf(fact_4152_bot_Oextremum__strict,axiom,
    ! [A: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ A @ bot_bo4199563552545308370d_enat ) ).

% bot.extremum_strict
thf(fact_4153_bot_Oextremum__strict,axiom,
    ! [A: set_int] :
      ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).

% bot.extremum_strict
thf(fact_4154_bot_Oextremum__strict,axiom,
    ! [A: set_real] :
      ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).

% bot.extremum_strict
thf(fact_4155_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_4156_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
      = ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_4157_bot_Onot__eq__extremum,axiom,
    ! [A: extended_enat] :
      ( ( A != bot_bo4199563552545308370d_enat )
      = ( ord_le72135733267957522d_enat @ bot_bo4199563552545308370d_enat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_4158_bot_Onot__eq__extremum,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
      = ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).

% bot.not_eq_extremum
thf(fact_4159_bot_Onot__eq__extremum,axiom,
    ! [A: set_real] :
      ( ( A != bot_bot_set_real )
      = ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).

% bot.not_eq_extremum
thf(fact_4160_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_4161_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A: nat,B2: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( P @ A3 @ B3 )
          = ( P @ B3 @ A3 ) )
     => ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
       => ( ! [A3: nat,B3: nat] :
              ( ( P @ A3 @ B3 )
             => ( P @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
         => ( P @ A @ B2 ) ) ) ) ).

% Euclid_induct
thf(fact_4162_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X3 )
      = ( if_VEBT_VEBT
        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
          & ~ ( ( X3 = Mi )
              | ( X3 = Ma ) ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ X3 @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ Mi @ X3 ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) ) ).

% vebt_insert.simps(5)
thf(fact_4163_vebt__insert_Oelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X3 @ Xa2 )
        = Y3 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X3
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => ( Y3
                    = ( vEBT_Leaf @ $true @ B3 ) ) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => ( Y3
                        = ( vEBT_Leaf @ A3 @ $true ) ) )
                    & ( ( Xa2 != one_one_nat )
                     => ( Y3
                        = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
              ( ( X3
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
             => ( Y3
               != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
               => ( Y3
                 != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) ) )
           => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
                 => ( Y3
                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X3
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                   => ( Y3
                     != ( if_VEBT_VEBT
                        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                          & ~ ( ( Xa2 = Mi2 )
                              | ( Xa2 = Ma2 ) ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
thf(fact_4164_vebt__insert_Opelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X3 @ Xa2 )
        = Y3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X3
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( ( ( Xa2 = zero_zero_nat )
                   => ( Y3
                      = ( vEBT_Leaf @ $true @ B3 ) ) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => ( Y3
                          = ( vEBT_Leaf @ A3 @ $true ) ) )
                      & ( ( Xa2 != one_one_nat )
                       => ( Y3
                          = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
               => ( ( Y3
                    = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) @ Xa2 ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
                 => ( ( Y3
                      = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ Xa2 ) ) ) )
             => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X3
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
                   => ( ( Y3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( Y3
                          = ( if_VEBT_VEBT
                            @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                              & ~ ( ( Xa2 = Mi2 )
                                  | ( Xa2 = Ma2 ) ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
thf(fact_4165_max__less__iff__conj,axiom,
    ! [X3: extended_enat,Y3: extended_enat,Z: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ X3 @ Y3 ) @ Z )
      = ( ( ord_le72135733267957522d_enat @ X3 @ Z )
        & ( ord_le72135733267957522d_enat @ Y3 @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_4166_max__less__iff__conj,axiom,
    ! [X3: real,Y3: real,Z: real] :
      ( ( ord_less_real @ ( ord_max_real @ X3 @ Y3 ) @ Z )
      = ( ( ord_less_real @ X3 @ Z )
        & ( ord_less_real @ Y3 @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_4167_max__less__iff__conj,axiom,
    ! [X3: rat,Y3: rat,Z: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ X3 @ Y3 ) @ Z )
      = ( ( ord_less_rat @ X3 @ Z )
        & ( ord_less_rat @ Y3 @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_4168_max__less__iff__conj,axiom,
    ! [X3: num,Y3: num,Z: num] :
      ( ( ord_less_num @ ( ord_max_num @ X3 @ Y3 ) @ Z )
      = ( ( ord_less_num @ X3 @ Z )
        & ( ord_less_num @ Y3 @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_4169_max__less__iff__conj,axiom,
    ! [X3: nat,Y3: nat,Z: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ X3 @ Y3 ) @ Z )
      = ( ( ord_less_nat @ X3 @ Z )
        & ( ord_less_nat @ Y3 @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_4170_max__less__iff__conj,axiom,
    ! [X3: int,Y3: int,Z: int] :
      ( ( ord_less_int @ ( ord_max_int @ X3 @ Y3 ) @ Z )
      = ( ( ord_less_int @ X3 @ Z )
        & ( ord_less_int @ Y3 @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_4171_max_Oabsorb4,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B2 )
     => ( ( ord_ma741700101516333627d_enat @ A @ B2 )
        = B2 ) ) ).

% max.absorb4
thf(fact_4172_max_Oabsorb4,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_max_real @ A @ B2 )
        = B2 ) ) ).

% max.absorb4
thf(fact_4173_max_Oabsorb4,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ( ord_max_rat @ A @ B2 )
        = B2 ) ) ).

% max.absorb4
thf(fact_4174_max_Oabsorb4,axiom,
    ! [A: num,B2: num] :
      ( ( ord_less_num @ A @ B2 )
     => ( ( ord_max_num @ A @ B2 )
        = B2 ) ) ).

% max.absorb4
thf(fact_4175_max_Oabsorb4,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_max_nat @ A @ B2 )
        = B2 ) ) ).

% max.absorb4
thf(fact_4176_max_Oabsorb4,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ( ord_max_int @ A @ B2 )
        = B2 ) ) ).

% max.absorb4
thf(fact_4177_max_Oabsorb3,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B2 @ A )
     => ( ( ord_ma741700101516333627d_enat @ A @ B2 )
        = A ) ) ).

% max.absorb3
thf(fact_4178_max_Oabsorb3,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( ( ord_max_real @ A @ B2 )
        = A ) ) ).

% max.absorb3
thf(fact_4179_max_Oabsorb3,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_rat @ B2 @ A )
     => ( ( ord_max_rat @ A @ B2 )
        = A ) ) ).

% max.absorb3
thf(fact_4180_max_Oabsorb3,axiom,
    ! [B2: num,A: num] :
      ( ( ord_less_num @ B2 @ A )
     => ( ( ord_max_num @ A @ B2 )
        = A ) ) ).

% max.absorb3
thf(fact_4181_max_Oabsorb3,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ( ord_max_nat @ A @ B2 )
        = A ) ) ).

% max.absorb3
thf(fact_4182_max_Oabsorb3,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ B2 @ A )
     => ( ( ord_max_int @ A @ B2 )
        = A ) ) ).

% max.absorb3
thf(fact_4183_max_Oabsorb1,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B2 @ A )
     => ( ( ord_ma741700101516333627d_enat @ A @ B2 )
        = A ) ) ).

% max.absorb1
thf(fact_4184_max_Oabsorb1,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_eq_rat @ B2 @ A )
     => ( ( ord_max_rat @ A @ B2 )
        = A ) ) ).

% max.absorb1
thf(fact_4185_max_Oabsorb1,axiom,
    ! [B2: num,A: num] :
      ( ( ord_less_eq_num @ B2 @ A )
     => ( ( ord_max_num @ A @ B2 )
        = A ) ) ).

% max.absorb1
thf(fact_4186_max_Oabsorb1,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( ord_max_nat @ A @ B2 )
        = A ) ) ).

% max.absorb1
thf(fact_4187_max_Oabsorb1,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ( ord_max_int @ A @ B2 )
        = A ) ) ).

% max.absorb1
thf(fact_4188_max_Oabsorb2,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B2 )
     => ( ( ord_ma741700101516333627d_enat @ A @ B2 )
        = B2 ) ) ).

% max.absorb2
thf(fact_4189_max_Oabsorb2,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_max_rat @ A @ B2 )
        = B2 ) ) ).

% max.absorb2
thf(fact_4190_max_Oabsorb2,axiom,
    ! [A: num,B2: num] :
      ( ( ord_less_eq_num @ A @ B2 )
     => ( ( ord_max_num @ A @ B2 )
        = B2 ) ) ).

% max.absorb2
thf(fact_4191_max_Oabsorb2,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_max_nat @ A @ B2 )
        = B2 ) ) ).

% max.absorb2
thf(fact_4192_max_Oabsorb2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_max_int @ A @ B2 )
        = B2 ) ) ).

% max.absorb2
thf(fact_4193_max_Obounded__iff,axiom,
    ! [B2: extended_enat,C: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B2 @ C ) @ A )
      = ( ( ord_le2932123472753598470d_enat @ B2 @ A )
        & ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_4194_max_Obounded__iff,axiom,
    ! [B2: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B2 @ C ) @ A )
      = ( ( ord_less_eq_rat @ B2 @ A )
        & ( ord_less_eq_rat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_4195_max_Obounded__iff,axiom,
    ! [B2: num,C: num,A: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B2 @ C ) @ A )
      = ( ( ord_less_eq_num @ B2 @ A )
        & ( ord_less_eq_num @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_4196_max_Obounded__iff,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B2 @ C ) @ A )
      = ( ( ord_less_eq_nat @ B2 @ A )
        & ( ord_less_eq_nat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_4197_max_Obounded__iff,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B2 @ C ) @ A )
      = ( ( ord_less_eq_int @ B2 @ A )
        & ( ord_less_eq_int @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_4198_dbl__simps_I3_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_4199_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_4200_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_4201_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_4202_max__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q2 )
      = Q2 ) ).

% max_enat_simps(3)
thf(fact_4203_max__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ Q2 @ zero_z5237406670263579293d_enat )
      = Q2 ) ).

% max_enat_simps(2)
thf(fact_4204_dbl__simps_I2_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% dbl_simps(2)
thf(fact_4205_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_real @ zero_zero_real )
    = zero_zero_real ) ).

% dbl_simps(2)
thf(fact_4206_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% dbl_simps(2)
thf(fact_4207_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_int @ zero_zero_int )
    = zero_zero_int ) ).

% dbl_simps(2)
thf(fact_4208_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
      = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_4209_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_4210_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_4211_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_4212_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_4213_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
thf(fact_4214_dbl__def,axiom,
    ( neg_numeral_dbl_real
    = ( ^ [X: real] : ( plus_plus_real @ X @ X ) ) ) ).

% dbl_def
thf(fact_4215_dbl__def,axiom,
    ( neg_numeral_dbl_rat
    = ( ^ [X: rat] : ( plus_plus_rat @ X @ X ) ) ) ).

% dbl_def
thf(fact_4216_dbl__def,axiom,
    ( neg_numeral_dbl_int
    = ( ^ [X: int] : ( plus_plus_int @ X @ X ) ) ) ).

% dbl_def
thf(fact_4217_max_OcoboundedI2,axiom,
    ! [C: extended_enat,B2: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ C @ B2 )
     => ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B2 ) ) ) ).

% max.coboundedI2
thf(fact_4218_max_OcoboundedI2,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( ord_less_eq_rat @ C @ B2 )
     => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B2 ) ) ) ).

% max.coboundedI2
thf(fact_4219_max_OcoboundedI2,axiom,
    ! [C: num,B2: num,A: num] :
      ( ( ord_less_eq_num @ C @ B2 )
     => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B2 ) ) ) ).

% max.coboundedI2
thf(fact_4220_max_OcoboundedI2,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ B2 )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B2 ) ) ) ).

% max.coboundedI2
thf(fact_4221_max_OcoboundedI2,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( ord_less_eq_int @ C @ B2 )
     => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B2 ) ) ) ).

% max.coboundedI2
thf(fact_4222_max_OcoboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ C @ A )
     => ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B2 ) ) ) ).

% max.coboundedI1
thf(fact_4223_max_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B2 ) ) ) ).

% max.coboundedI1
thf(fact_4224_max_OcoboundedI1,axiom,
    ! [C: num,A: num,B2: num] :
      ( ( ord_less_eq_num @ C @ A )
     => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B2 ) ) ) ).

% max.coboundedI1
thf(fact_4225_max_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B2 ) ) ) ).

% max.coboundedI1
thf(fact_4226_max_OcoboundedI1,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B2 ) ) ) ).

% max.coboundedI1
thf(fact_4227_max_Oabsorb__iff2,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [A5: extended_enat,B4: extended_enat] :
          ( ( ord_ma741700101516333627d_enat @ A5 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_4228_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A5: rat,B4: rat] :
          ( ( ord_max_rat @ A5 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_4229_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_num
    = ( ^ [A5: num,B4: num] :
          ( ( ord_max_num @ A5 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_4230_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_max_nat @ A5 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_4231_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B4: int] :
          ( ( ord_max_int @ A5 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_4232_max_Oabsorb__iff1,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [B4: extended_enat,A5: extended_enat] :
          ( ( ord_ma741700101516333627d_enat @ A5 @ B4 )
          = A5 ) ) ) ).

% max.absorb_iff1
thf(fact_4233_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A5: rat] :
          ( ( ord_max_rat @ A5 @ B4 )
          = A5 ) ) ) ).

% max.absorb_iff1
thf(fact_4234_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_num
    = ( ^ [B4: num,A5: num] :
          ( ( ord_max_num @ A5 @ B4 )
          = A5 ) ) ) ).

% max.absorb_iff1
thf(fact_4235_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( ord_max_nat @ A5 @ B4 )
          = A5 ) ) ) ).

% max.absorb_iff1
thf(fact_4236_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A5: int] :
          ( ( ord_max_int @ A5 @ B4 )
          = A5 ) ) ) ).

% max.absorb_iff1
thf(fact_4237_le__max__iff__disj,axiom,
    ! [Z: extended_enat,X3: extended_enat,Y3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X3 @ Y3 ) )
      = ( ( ord_le2932123472753598470d_enat @ Z @ X3 )
        | ( ord_le2932123472753598470d_enat @ Z @ Y3 ) ) ) ).

% le_max_iff_disj
thf(fact_4238_le__max__iff__disj,axiom,
    ! [Z: rat,X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ Z @ ( ord_max_rat @ X3 @ Y3 ) )
      = ( ( ord_less_eq_rat @ Z @ X3 )
        | ( ord_less_eq_rat @ Z @ Y3 ) ) ) ).

% le_max_iff_disj
thf(fact_4239_le__max__iff__disj,axiom,
    ! [Z: num,X3: num,Y3: num] :
      ( ( ord_less_eq_num @ Z @ ( ord_max_num @ X3 @ Y3 ) )
      = ( ( ord_less_eq_num @ Z @ X3 )
        | ( ord_less_eq_num @ Z @ Y3 ) ) ) ).

% le_max_iff_disj
thf(fact_4240_le__max__iff__disj,axiom,
    ! [Z: nat,X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X3 @ Y3 ) )
      = ( ( ord_less_eq_nat @ Z @ X3 )
        | ( ord_less_eq_nat @ Z @ Y3 ) ) ) ).

% le_max_iff_disj
thf(fact_4241_le__max__iff__disj,axiom,
    ! [Z: int,X3: int,Y3: int] :
      ( ( ord_less_eq_int @ Z @ ( ord_max_int @ X3 @ Y3 ) )
      = ( ( ord_less_eq_int @ Z @ X3 )
        | ( ord_less_eq_int @ Z @ Y3 ) ) ) ).

% le_max_iff_disj
thf(fact_4242_max_Ocobounded2,axiom,
    ! [B2: extended_enat,A: extended_enat] : ( ord_le2932123472753598470d_enat @ B2 @ ( ord_ma741700101516333627d_enat @ A @ B2 ) ) ).

% max.cobounded2
thf(fact_4243_max_Ocobounded2,axiom,
    ! [B2: rat,A: rat] : ( ord_less_eq_rat @ B2 @ ( ord_max_rat @ A @ B2 ) ) ).

% max.cobounded2
thf(fact_4244_max_Ocobounded2,axiom,
    ! [B2: num,A: num] : ( ord_less_eq_num @ B2 @ ( ord_max_num @ A @ B2 ) ) ).

% max.cobounded2
thf(fact_4245_max_Ocobounded2,axiom,
    ! [B2: nat,A: nat] : ( ord_less_eq_nat @ B2 @ ( ord_max_nat @ A @ B2 ) ) ).

% max.cobounded2
thf(fact_4246_max_Ocobounded2,axiom,
    ! [B2: int,A: int] : ( ord_less_eq_int @ B2 @ ( ord_max_int @ A @ B2 ) ) ).

% max.cobounded2
thf(fact_4247_max_Ocobounded1,axiom,
    ! [A: extended_enat,B2: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ ( ord_ma741700101516333627d_enat @ A @ B2 ) ) ).

% max.cobounded1
thf(fact_4248_max_Ocobounded1,axiom,
    ! [A: rat,B2: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B2 ) ) ).

% max.cobounded1
thf(fact_4249_max_Ocobounded1,axiom,
    ! [A: num,B2: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B2 ) ) ).

% max.cobounded1
thf(fact_4250_max_Ocobounded1,axiom,
    ! [A: nat,B2: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B2 ) ) ).

% max.cobounded1
thf(fact_4251_max_Ocobounded1,axiom,
    ! [A: int,B2: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B2 ) ) ).

% max.cobounded1
thf(fact_4252_max_Oorder__iff,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [B4: extended_enat,A5: extended_enat] :
          ( A5
          = ( ord_ma741700101516333627d_enat @ A5 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_4253_max_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A5: rat] :
          ( A5
          = ( ord_max_rat @ A5 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_4254_max_Oorder__iff,axiom,
    ( ord_less_eq_num
    = ( ^ [B4: num,A5: num] :
          ( A5
          = ( ord_max_num @ A5 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_4255_max_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A5: nat] :
          ( A5
          = ( ord_max_nat @ A5 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_4256_max_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A5: int] :
          ( A5
          = ( ord_max_int @ A5 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_4257_max_OboundedI,axiom,
    ! [B2: extended_enat,A: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B2 @ A )
     => ( ( ord_le2932123472753598470d_enat @ C @ A )
       => ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B2 @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_4258_max_OboundedI,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B2 @ A )
     => ( ( ord_less_eq_rat @ C @ A )
       => ( ord_less_eq_rat @ ( ord_max_rat @ B2 @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_4259_max_OboundedI,axiom,
    ! [B2: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B2 @ A )
     => ( ( ord_less_eq_num @ C @ A )
       => ( ord_less_eq_num @ ( ord_max_num @ B2 @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_4260_max_OboundedI,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_eq_nat @ ( ord_max_nat @ B2 @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_4261_max_OboundedI,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( ( ord_less_eq_int @ C @ A )
       => ( ord_less_eq_int @ ( ord_max_int @ B2 @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_4262_max_OboundedE,axiom,
    ! [B2: extended_enat,C: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B2 @ C ) @ A )
     => ~ ( ( ord_le2932123472753598470d_enat @ B2 @ A )
         => ~ ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_4263_max_OboundedE,axiom,
    ! [B2: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B2 @ C ) @ A )
     => ~ ( ( ord_less_eq_rat @ B2 @ A )
         => ~ ( ord_less_eq_rat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_4264_max_OboundedE,axiom,
    ! [B2: num,C: num,A: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B2 @ C ) @ A )
     => ~ ( ( ord_less_eq_num @ B2 @ A )
         => ~ ( ord_less_eq_num @ C @ A ) ) ) ).

% max.boundedE
thf(fact_4265_max_OboundedE,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B2 @ C ) @ A )
     => ~ ( ( ord_less_eq_nat @ B2 @ A )
         => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_4266_max_OboundedE,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B2 @ C ) @ A )
     => ~ ( ( ord_less_eq_int @ B2 @ A )
         => ~ ( ord_less_eq_int @ C @ A ) ) ) ).

% max.boundedE
thf(fact_4267_max_OorderI,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( A
        = ( ord_ma741700101516333627d_enat @ A @ B2 ) )
     => ( ord_le2932123472753598470d_enat @ B2 @ A ) ) ).

% max.orderI
thf(fact_4268_max_OorderI,axiom,
    ! [A: rat,B2: rat] :
      ( ( A
        = ( ord_max_rat @ A @ B2 ) )
     => ( ord_less_eq_rat @ B2 @ A ) ) ).

% max.orderI
thf(fact_4269_max_OorderI,axiom,
    ! [A: num,B2: num] :
      ( ( A
        = ( ord_max_num @ A @ B2 ) )
     => ( ord_less_eq_num @ B2 @ A ) ) ).

% max.orderI
thf(fact_4270_max_OorderI,axiom,
    ! [A: nat,B2: nat] :
      ( ( A
        = ( ord_max_nat @ A @ B2 ) )
     => ( ord_less_eq_nat @ B2 @ A ) ) ).

% max.orderI
thf(fact_4271_max_OorderI,axiom,
    ! [A: int,B2: int] :
      ( ( A
        = ( ord_max_int @ A @ B2 ) )
     => ( ord_less_eq_int @ B2 @ A ) ) ).

% max.orderI
thf(fact_4272_max_OorderE,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B2 @ A )
     => ( A
        = ( ord_ma741700101516333627d_enat @ A @ B2 ) ) ) ).

% max.orderE
thf(fact_4273_max_OorderE,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_eq_rat @ B2 @ A )
     => ( A
        = ( ord_max_rat @ A @ B2 ) ) ) ).

% max.orderE
thf(fact_4274_max_OorderE,axiom,
    ! [B2: num,A: num] :
      ( ( ord_less_eq_num @ B2 @ A )
     => ( A
        = ( ord_max_num @ A @ B2 ) ) ) ).

% max.orderE
thf(fact_4275_max_OorderE,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( A
        = ( ord_max_nat @ A @ B2 ) ) ) ).

% max.orderE
thf(fact_4276_max_OorderE,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_eq_int @ B2 @ A )
     => ( A
        = ( ord_max_int @ A @ B2 ) ) ) ).

% max.orderE
thf(fact_4277_max_Omono,axiom,
    ! [C: extended_enat,A: extended_enat,D: extended_enat,B2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ C @ A )
     => ( ( ord_le2932123472753598470d_enat @ D @ B2 )
       => ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ C @ D ) @ ( ord_ma741700101516333627d_enat @ A @ B2 ) ) ) ) ).

% max.mono
thf(fact_4278_max_Omono,axiom,
    ! [C: rat,A: rat,D: rat,B2: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ( ord_less_eq_rat @ D @ B2 )
       => ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A @ B2 ) ) ) ) ).

% max.mono
thf(fact_4279_max_Omono,axiom,
    ! [C: num,A: num,D: num,B2: num] :
      ( ( ord_less_eq_num @ C @ A )
     => ( ( ord_less_eq_num @ D @ B2 )
       => ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A @ B2 ) ) ) ) ).

% max.mono
thf(fact_4280_max_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ( ord_less_eq_nat @ D @ B2 )
       => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B2 ) ) ) ) ).

% max.mono
thf(fact_4281_max_Omono,axiom,
    ! [C: int,A: int,D: int,B2: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ( ord_less_eq_int @ D @ B2 )
       => ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A @ B2 ) ) ) ) ).

% max.mono
thf(fact_4282_less__max__iff__disj,axiom,
    ! [Z: extended_enat,X3: extended_enat,Y3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X3 @ Y3 ) )
      = ( ( ord_le72135733267957522d_enat @ Z @ X3 )
        | ( ord_le72135733267957522d_enat @ Z @ Y3 ) ) ) ).

% less_max_iff_disj
thf(fact_4283_less__max__iff__disj,axiom,
    ! [Z: real,X3: real,Y3: real] :
      ( ( ord_less_real @ Z @ ( ord_max_real @ X3 @ Y3 ) )
      = ( ( ord_less_real @ Z @ X3 )
        | ( ord_less_real @ Z @ Y3 ) ) ) ).

% less_max_iff_disj
thf(fact_4284_less__max__iff__disj,axiom,
    ! [Z: rat,X3: rat,Y3: rat] :
      ( ( ord_less_rat @ Z @ ( ord_max_rat @ X3 @ Y3 ) )
      = ( ( ord_less_rat @ Z @ X3 )
        | ( ord_less_rat @ Z @ Y3 ) ) ) ).

% less_max_iff_disj
thf(fact_4285_less__max__iff__disj,axiom,
    ! [Z: num,X3: num,Y3: num] :
      ( ( ord_less_num @ Z @ ( ord_max_num @ X3 @ Y3 ) )
      = ( ( ord_less_num @ Z @ X3 )
        | ( ord_less_num @ Z @ Y3 ) ) ) ).

% less_max_iff_disj
thf(fact_4286_less__max__iff__disj,axiom,
    ! [Z: nat,X3: nat,Y3: nat] :
      ( ( ord_less_nat @ Z @ ( ord_max_nat @ X3 @ Y3 ) )
      = ( ( ord_less_nat @ Z @ X3 )
        | ( ord_less_nat @ Z @ Y3 ) ) ) ).

% less_max_iff_disj
thf(fact_4287_less__max__iff__disj,axiom,
    ! [Z: int,X3: int,Y3: int] :
      ( ( ord_less_int @ Z @ ( ord_max_int @ X3 @ Y3 ) )
      = ( ( ord_less_int @ Z @ X3 )
        | ( ord_less_int @ Z @ Y3 ) ) ) ).

% less_max_iff_disj
thf(fact_4288_max_Ostrict__boundedE,axiom,
    ! [B2: extended_enat,C: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ B2 @ C ) @ A )
     => ~ ( ( ord_le72135733267957522d_enat @ B2 @ A )
         => ~ ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_4289_max_Ostrict__boundedE,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( ord_less_real @ ( ord_max_real @ B2 @ C ) @ A )
     => ~ ( ( ord_less_real @ B2 @ A )
         => ~ ( ord_less_real @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_4290_max_Ostrict__boundedE,axiom,
    ! [B2: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ B2 @ C ) @ A )
     => ~ ( ( ord_less_rat @ B2 @ A )
         => ~ ( ord_less_rat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_4291_max_Ostrict__boundedE,axiom,
    ! [B2: num,C: num,A: num] :
      ( ( ord_less_num @ ( ord_max_num @ B2 @ C ) @ A )
     => ~ ( ( ord_less_num @ B2 @ A )
         => ~ ( ord_less_num @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_4292_max_Ostrict__boundedE,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ B2 @ C ) @ A )
     => ~ ( ( ord_less_nat @ B2 @ A )
         => ~ ( ord_less_nat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_4293_max_Ostrict__boundedE,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( ord_less_int @ ( ord_max_int @ B2 @ C ) @ A )
     => ~ ( ( ord_less_int @ B2 @ A )
         => ~ ( ord_less_int @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_4294_max_Ostrict__order__iff,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [B4: extended_enat,A5: extended_enat] :
          ( ( A5
            = ( ord_ma741700101516333627d_enat @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4295_max_Ostrict__order__iff,axiom,
    ( ord_less_real
    = ( ^ [B4: real,A5: real] :
          ( ( A5
            = ( ord_max_real @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4296_max_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [B4: rat,A5: rat] :
          ( ( A5
            = ( ord_max_rat @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4297_max_Ostrict__order__iff,axiom,
    ( ord_less_num
    = ( ^ [B4: num,A5: num] :
          ( ( A5
            = ( ord_max_num @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4298_max_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( A5
            = ( ord_max_nat @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4299_max_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B4: int,A5: int] :
          ( ( A5
            = ( ord_max_int @ A5 @ B4 ) )
          & ( A5 != B4 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4300_max_Ostrict__coboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ C @ A )
     => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B2 ) ) ) ).

% max.strict_coboundedI1
thf(fact_4301_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ C @ A )
     => ( ord_less_real @ C @ ( ord_max_real @ A @ B2 ) ) ) ).

% max.strict_coboundedI1
thf(fact_4302_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ C @ A )
     => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B2 ) ) ) ).

% max.strict_coboundedI1
thf(fact_4303_max_Ostrict__coboundedI1,axiom,
    ! [C: num,A: num,B2: num] :
      ( ( ord_less_num @ C @ A )
     => ( ord_less_num @ C @ ( ord_max_num @ A @ B2 ) ) ) ).

% max.strict_coboundedI1
thf(fact_4304_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( ord_less_nat @ C @ A )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B2 ) ) ) ).

% max.strict_coboundedI1
thf(fact_4305_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( ord_less_int @ C @ A )
     => ( ord_less_int @ C @ ( ord_max_int @ A @ B2 ) ) ) ).

% max.strict_coboundedI1
thf(fact_4306_max_Ostrict__coboundedI2,axiom,
    ! [C: extended_enat,B2: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ C @ B2 )
     => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B2 ) ) ) ).

% max.strict_coboundedI2
thf(fact_4307_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_real @ C @ B2 )
     => ( ord_less_real @ C @ ( ord_max_real @ A @ B2 ) ) ) ).

% max.strict_coboundedI2
thf(fact_4308_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( ord_less_rat @ C @ B2 )
     => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B2 ) ) ) ).

% max.strict_coboundedI2
thf(fact_4309_max_Ostrict__coboundedI2,axiom,
    ! [C: num,B2: num,A: num] :
      ( ( ord_less_num @ C @ B2 )
     => ( ord_less_num @ C @ ( ord_max_num @ A @ B2 ) ) ) ).

% max.strict_coboundedI2
thf(fact_4310_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( ord_less_nat @ C @ B2 )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B2 ) ) ) ).

% max.strict_coboundedI2
thf(fact_4311_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( ord_less_int @ C @ B2 )
     => ( ord_less_int @ C @ ( ord_max_int @ A @ B2 ) ) ) ).

% max.strict_coboundedI2
thf(fact_4312_even__succ__mod__exp,axiom,
    ! [A: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
          = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_4313_even__succ__mod__exp,axiom,
    ! [A: int,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
          = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_4314_even__succ__mod__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
          = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_4315_option_Osize__gen_I2_J,axiom,
    ! [X3: nat > nat,X2: nat] :
      ( ( size_option_nat @ X3 @ ( some_nat @ X2 ) )
      = ( plus_plus_nat @ ( X3 @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_4316_option_Osize__gen_I2_J,axiom,
    ! [X3: product_prod_nat_nat > nat,X2: product_prod_nat_nat] :
      ( ( size_o8335143837870341156at_nat @ X3 @ ( some_P7363390416028606310at_nat @ X2 ) )
      = ( plus_plus_nat @ ( X3 @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_4317_option_Osize__gen_I2_J,axiom,
    ! [X3: num > nat,X2: num] :
      ( ( size_option_num @ X3 @ ( some_num @ X2 ) )
      = ( plus_plus_nat @ ( X3 @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_4318_even__succ__div__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
          = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_4319_even__succ__div__exp,axiom,
    ! [A: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
          = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_4320_even__succ__div__exp,axiom,
    ! [A: int,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
          = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_4321_signed__take__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_4322_signed__take__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_4323_infinite__growing,axiom,
    ! [X9: set_real] :
      ( ( X9 != bot_bot_set_real )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ X9 )
           => ? [Xa: real] :
                ( ( member_real @ Xa @ X9 )
                & ( ord_less_real @ X5 @ Xa ) ) )
       => ~ ( finite_finite_real @ X9 ) ) ) ).

% infinite_growing
thf(fact_4324_infinite__growing,axiom,
    ! [X9: set_rat] :
      ( ( X9 != bot_bot_set_rat )
     => ( ! [X5: rat] :
            ( ( member_rat @ X5 @ X9 )
           => ? [Xa: rat] :
                ( ( member_rat @ Xa @ X9 )
                & ( ord_less_rat @ X5 @ Xa ) ) )
       => ~ ( finite_finite_rat @ X9 ) ) ) ).

% infinite_growing
thf(fact_4325_infinite__growing,axiom,
    ! [X9: set_num] :
      ( ( X9 != bot_bot_set_num )
     => ( ! [X5: num] :
            ( ( member_num @ X5 @ X9 )
           => ? [Xa: num] :
                ( ( member_num @ Xa @ X9 )
                & ( ord_less_num @ X5 @ Xa ) ) )
       => ~ ( finite_finite_num @ X9 ) ) ) ).

% infinite_growing
thf(fact_4326_infinite__growing,axiom,
    ! [X9: set_nat] :
      ( ( X9 != bot_bot_set_nat )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ X9 )
           => ? [Xa: nat] :
                ( ( member_nat @ Xa @ X9 )
                & ( ord_less_nat @ X5 @ Xa ) ) )
       => ~ ( finite_finite_nat @ X9 ) ) ) ).

% infinite_growing
thf(fact_4327_infinite__growing,axiom,
    ! [X9: set_int] :
      ( ( X9 != bot_bot_set_int )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ X9 )
           => ? [Xa: int] :
                ( ( member_int @ Xa @ X9 )
                & ( ord_less_int @ X5 @ Xa ) ) )
       => ~ ( finite_finite_int @ X9 ) ) ) ).

% infinite_growing
thf(fact_4328_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_4329_dvd__0__left__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left_iff
thf(fact_4330_dvd__0__left__iff,axiom,
    ! [A: complex] :
      ( ( dvd_dvd_complex @ zero_zero_complex @ A )
      = ( A = zero_zero_complex ) ) ).

% dvd_0_left_iff
thf(fact_4331_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
      = ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
thf(fact_4332_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
      = ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
thf(fact_4333_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
thf(fact_4334_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
      = ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
thf(fact_4335_dvd__0__right,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).

% dvd_0_right
thf(fact_4336_dvd__0__right,axiom,
    ! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).

% dvd_0_right
thf(fact_4337_dvd__0__right,axiom,
    ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).

% dvd_0_right
thf(fact_4338_dvd__0__right,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).

% dvd_0_right
thf(fact_4339_dvd__0__right,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% dvd_0_right
thf(fact_4340_dvd__0__right,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).

% dvd_0_right
thf(fact_4341_dvd__add__triv__left__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
      = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).

% dvd_add_triv_left_iff
thf(fact_4342_dvd__add__triv__left__iff,axiom,
    ! [A: real,B2: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B2 ) )
      = ( dvd_dvd_real @ A @ B2 ) ) ).

% dvd_add_triv_left_iff
thf(fact_4343_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B2 ) )
      = ( dvd_dvd_rat @ A @ B2 ) ) ).

% dvd_add_triv_left_iff
thf(fact_4344_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
      = ( dvd_dvd_nat @ A @ B2 ) ) ).

% dvd_add_triv_left_iff
thf(fact_4345_dvd__add__triv__left__iff,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B2 ) )
      = ( dvd_dvd_int @ A @ B2 ) ) ).

% dvd_add_triv_left_iff
thf(fact_4346_dvd__add__triv__right__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ A ) )
      = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).

% dvd_add_triv_right_iff
thf(fact_4347_dvd__add__triv__right__iff,axiom,
    ! [A: real,B2: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ A ) )
      = ( dvd_dvd_real @ A @ B2 ) ) ).

% dvd_add_triv_right_iff
thf(fact_4348_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ A ) )
      = ( dvd_dvd_rat @ A @ B2 ) ) ).

% dvd_add_triv_right_iff
thf(fact_4349_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
      = ( dvd_dvd_nat @ A @ B2 ) ) ).

% dvd_add_triv_right_iff
thf(fact_4350_dvd__add__triv__right__iff,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ A ) )
      = ( dvd_dvd_int @ A @ B2 ) ) ).

% dvd_add_triv_right_iff
thf(fact_4351_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_4352_dvd__1__left,axiom,
    ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).

% dvd_1_left
thf(fact_4353_div__dvd__div,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B2 )
     => ( ( dvd_dvd_Code_integer @ A @ C )
       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B2 @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
          = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ) ).

% div_dvd_div
thf(fact_4354_div__dvd__div,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( dvd_dvd_nat @ A @ C )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B2 @ A ) @ ( divide_divide_nat @ C @ A ) )
          = ( dvd_dvd_nat @ B2 @ C ) ) ) ) ).

% div_dvd_div
thf(fact_4355_div__dvd__div,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( dvd_dvd_int @ A @ C )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ B2 @ A ) @ ( divide_divide_int @ C @ A ) )
          = ( dvd_dvd_int @ B2 @ C ) ) ) ) ).

% div_dvd_div
thf(fact_4356_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_4357_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_4358_dvd__times__right__cancel__iff,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B2 @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
        = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_4359_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ A ) @ ( times_times_nat @ C @ A ) )
        = ( dvd_dvd_nat @ B2 @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_4360_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ B2 @ A ) @ ( times_times_int @ C @ A ) )
        = ( dvd_dvd_int @ B2 @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_4361_dvd__times__left__cancel__iff,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ ( times_3573771949741848930nteger @ A @ C ) )
        = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_4362_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) )
        = ( dvd_dvd_nat @ B2 @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_4363_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) )
        = ( dvd_dvd_int @ B2 @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_4364_dvd__mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B2 @ C ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( dvd_dvd_Code_integer @ A @ B2 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_4365_dvd__mult__cancel__right,axiom,
    ! [A: complex,C: complex,B2: complex] :
      ( ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B2 @ C ) )
      = ( ( C = zero_zero_complex )
        | ( dvd_dvd_complex @ A @ B2 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_4366_dvd__mult__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
      = ( ( C = zero_zero_real )
        | ( dvd_dvd_real @ A @ B2 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_4367_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
      = ( ( C = zero_zero_rat )
        | ( dvd_dvd_rat @ A @ B2 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_4368_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A @ B2 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_4369_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B2 ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( dvd_dvd_Code_integer @ A @ B2 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_4370_dvd__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( dvd_dvd_complex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B2 ) )
      = ( ( C = zero_zero_complex )
        | ( dvd_dvd_complex @ A @ B2 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_4371_dvd__mult__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
      = ( ( C = zero_zero_real )
        | ( dvd_dvd_real @ A @ B2 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_4372_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
      = ( ( C = zero_zero_rat )
        | ( dvd_dvd_rat @ A @ B2 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_4373_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A @ B2 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_4374_unit__prod,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ one_one_Code_integer ) ) ) ).

% unit_prod
thf(fact_4375_unit__prod,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
       => ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ one_one_nat ) ) ) ).

% unit_prod
thf(fact_4376_unit__prod,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ B2 @ one_one_int )
       => ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ one_one_int ) ) ) ).

% unit_prod
thf(fact_4377_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ ( times_3573771949741848930nteger @ C @ A ) ) )
      = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_4378_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ ( times_times_real @ C @ A ) ) )
      = ( dvd_dvd_real @ A @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_4379_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ ( times_times_rat @ C @ A ) ) )
      = ( dvd_dvd_rat @ A @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_4380_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ ( times_times_nat @ C @ A ) ) )
      = ( dvd_dvd_nat @ A @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_4381_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ ( times_times_int @ C @ A ) ) )
      = ( dvd_dvd_int @ A @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_4382_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B2 ) )
      = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_4383_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B2 ) )
      = ( dvd_dvd_real @ A @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_4384_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B2 ) )
      = ( dvd_dvd_rat @ A @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_4385_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B2 ) )
      = ( dvd_dvd_nat @ A @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_4386_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B2 ) )
      = ( dvd_dvd_int @ A @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_4387_dvd__mult__div__cancel,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B2 )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B2 @ A ) )
        = B2 ) ) ).

% dvd_mult_div_cancel
thf(fact_4388_dvd__mult__div__cancel,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ A ) )
        = B2 ) ) ).

% dvd_mult_div_cancel
thf(fact_4389_dvd__mult__div__cancel,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ A ) )
        = B2 ) ) ).

% dvd_mult_div_cancel
thf(fact_4390_dvd__div__mult__self,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B2 )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B2 @ A ) @ A )
        = B2 ) ) ).

% dvd_div_mult_self
thf(fact_4391_dvd__div__mult__self,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ A ) @ A )
        = B2 ) ) ).

% dvd_div_mult_self
thf(fact_4392_dvd__div__mult__self,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( times_times_int @ ( divide_divide_int @ B2 @ A ) @ A )
        = B2 ) ) ).

% dvd_div_mult_self
thf(fact_4393_unit__div__1__div__1,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_4394_unit__div__1__div__1,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_4395_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_4396_unit__div__1__unit,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).

% unit_div_1_unit
thf(fact_4397_unit__div__1__unit,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).

% unit_div_1_unit
thf(fact_4398_unit__div__1__unit,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).

% unit_div_1_unit
thf(fact_4399_unit__div,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B2 ) @ one_one_Code_integer ) ) ) ).

% unit_div
thf(fact_4400_unit__div,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
       => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).

% unit_div
thf(fact_4401_unit__div,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ B2 @ one_one_int )
       => ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).

% unit_div
thf(fact_4402_div__add,axiom,
    ! [C: code_integer,A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( dvd_dvd_Code_integer @ C @ B2 )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ) ) ).

% div_add
thf(fact_4403_div__add,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( dvd_dvd_nat @ C @ B2 )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
          = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ) ).

% div_add
thf(fact_4404_div__add,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B2 )
       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
          = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ) ).

% div_add
thf(fact_4405_div__diff,axiom,
    ! [C: code_integer,A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( dvd_dvd_Code_integer @ C @ B2 )
       => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C )
          = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ) ) ).

% div_diff
thf(fact_4406_div__diff,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B2 )
       => ( ( divide_divide_int @ ( minus_minus_int @ A @ B2 ) @ C )
          = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ) ).

% div_diff
thf(fact_4407_dvd__imp__mod__0,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( modulo_modulo_nat @ B2 @ A )
        = zero_zero_nat ) ) ).

% dvd_imp_mod_0
thf(fact_4408_dvd__imp__mod__0,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( modulo_modulo_int @ B2 @ A )
        = zero_zero_int ) ) ).

% dvd_imp_mod_0
thf(fact_4409_dvd__imp__mod__0,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B2 )
     => ( ( modulo364778990260209775nteger @ B2 @ A )
        = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
thf(fact_4410_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_4411_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_4412_unit__mult__div__div,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
        = ( divide6298287555418463151nteger @ B2 @ A ) ) ) ).

% unit_mult_div_div
thf(fact_4413_unit__mult__div__div,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( times_times_nat @ B2 @ ( divide_divide_nat @ one_one_nat @ A ) )
        = ( divide_divide_nat @ B2 @ A ) ) ) ).

% unit_mult_div_div
thf(fact_4414_unit__mult__div__div,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( times_times_int @ B2 @ ( divide_divide_int @ one_one_int @ A ) )
        = ( divide_divide_int @ B2 @ A ) ) ) ).

% unit_mult_div_div
thf(fact_4415_unit__div__mult__self,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B2 @ A ) @ A )
        = B2 ) ) ).

% unit_div_mult_self
thf(fact_4416_unit__div__mult__self,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ A ) @ A )
        = B2 ) ) ).

% unit_div_mult_self
thf(fact_4417_unit__div__mult__self,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ B2 @ A ) @ A )
        = B2 ) ) ).

% unit_div_mult_self
thf(fact_4418_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_4419_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_4420_pow__divides__pow__iff,axiom,
    ! [N: nat,A: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
        = ( dvd_dvd_nat @ A @ B2 ) ) ) ).

% pow_divides_pow_iff
thf(fact_4421_pow__divides__pow__iff,axiom,
    ! [N: nat,A: int,B2: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
        = ( dvd_dvd_int @ A @ B2 ) ) ) ).

% pow_divides_pow_iff
thf(fact_4422_even__mult__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B2 ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_mult_iff
thf(fact_4423_even__mult__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_mult_iff
thf(fact_4424_even__mult__iff,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B2 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_mult_iff
thf(fact_4425_even__add,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_add
thf(fact_4426_even__add,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_add
thf(fact_4427_even__add,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_add
thf(fact_4428_odd__add,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) ) )
      = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).

% odd_add
thf(fact_4429_odd__add,axiom,
    ! [A: nat,B2: nat] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) ) )
      = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).

% odd_add
thf(fact_4430_odd__add,axiom,
    ! [A: int,B2: int] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) ) )
      = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).

% odd_add
thf(fact_4431_even__mod__2__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ).

% even_mod_2_iff
thf(fact_4432_even__mod__2__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).

% even_mod_2_iff
thf(fact_4433_even__mod__2__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).

% even_mod_2_iff
thf(fact_4434_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_4435_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_4436_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_4437_zero__le__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_4438_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_4439_zero__le__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_4440_power__less__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_4441_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_4442_power__less__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_4443_power__less__zero__eq,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        & ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% power_less_zero_eq
thf(fact_4444_power__less__zero__eq,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% power_less_zero_eq
thf(fact_4445_power__less__zero__eq,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        & ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% power_less_zero_eq
thf(fact_4446_even__plus__one__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
      = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_4447_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_4448_even__plus__one__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_4449_even__diff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B2 ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) ) ) ).

% even_diff
thf(fact_4450_even__diff,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B2 ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) ) ) ).

% even_diff
thf(fact_4451_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_4452_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_4453_zero__less__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A != zero_zero_real ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_4454_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_4455_zero__less__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_4456_even__succ__div__2,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_4457_even__succ__div__2,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_4458_even__succ__div__2,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_4459_even__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_4460_even__succ__div__two,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_4461_even__succ__div__two,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_4462_odd__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% odd_succ_div_two
thf(fact_4463_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).

% odd_succ_div_two
thf(fact_4464_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% odd_succ_div_two
thf(fact_4465_even__power,axiom,
    ! [A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% even_power
thf(fact_4466_even__power,axiom,
    ! [A: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% even_power
thf(fact_4467_even__power,axiom,
    ! [A: int,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% even_power
thf(fact_4468_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_4469_odd__two__times__div__two__succ,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_4470_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_4471_odd__two__times__div__two__succ,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_4472_power__le__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( ord_less_eq_real @ A @ zero_zero_real ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_4473_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_4474_power__le__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( ord_less_eq_int @ A @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_4475_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_4476_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_4477_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_4478_dvd__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( dvd_dvd_nat @ B2 @ C )
       => ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_trans
thf(fact_4479_dvd__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( dvd_dvd_int @ B2 @ C )
       => ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_trans
thf(fact_4480_dvd__trans,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B2 )
     => ( ( dvd_dvd_Code_integer @ B2 @ C )
       => ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_trans
thf(fact_4481_dvd__refl,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).

% dvd_refl
thf(fact_4482_dvd__refl,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ A ) ).

% dvd_refl
thf(fact_4483_dvd__refl,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ A ) ).

% dvd_refl
thf(fact_4484_dvd__0__left,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
     => ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left
thf(fact_4485_dvd__0__left,axiom,
    ! [A: complex] :
      ( ( dvd_dvd_complex @ zero_zero_complex @ A )
     => ( A = zero_zero_complex ) ) ).

% dvd_0_left
thf(fact_4486_dvd__0__left,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
thf(fact_4487_dvd__0__left,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
thf(fact_4488_dvd__0__left,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
thf(fact_4489_dvd__0__left,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
thf(fact_4490_dvd__productE,axiom,
    ! [P4: nat,A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ P4 @ ( times_times_nat @ A @ B2 ) )
     => ~ ! [X5: nat,Y4: nat] :
            ( ( P4
              = ( times_times_nat @ X5 @ Y4 ) )
           => ( ( dvd_dvd_nat @ X5 @ A )
             => ~ ( dvd_dvd_nat @ Y4 @ B2 ) ) ) ) ).

% dvd_productE
thf(fact_4491_dvd__productE,axiom,
    ! [P4: int,A: int,B2: int] :
      ( ( dvd_dvd_int @ P4 @ ( times_times_int @ A @ B2 ) )
     => ~ ! [X5: int,Y4: int] :
            ( ( P4
              = ( times_times_int @ X5 @ Y4 ) )
           => ( ( dvd_dvd_int @ X5 @ A )
             => ~ ( dvd_dvd_int @ Y4 @ B2 ) ) ) ) ).

% dvd_productE
thf(fact_4492_division__decomp,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) )
     => ? [B8: nat,C5: nat] :
          ( ( A
            = ( times_times_nat @ B8 @ C5 ) )
          & ( dvd_dvd_nat @ B8 @ B2 )
          & ( dvd_dvd_nat @ C5 @ C ) ) ) ).

% division_decomp
thf(fact_4493_division__decomp,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) )
     => ? [B8: int,C5: int] :
          ( ( A
            = ( times_times_int @ B8 @ C5 ) )
          & ( dvd_dvd_int @ B8 @ B2 )
          & ( dvd_dvd_int @ C5 @ C ) ) ) ).

% division_decomp
thf(fact_4494_dvd__triv__right,axiom,
    ! [A: code_integer,B2: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B2 @ A ) ) ).

% dvd_triv_right
thf(fact_4495_dvd__triv__right,axiom,
    ! [A: real,B2: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B2 @ A ) ) ).

% dvd_triv_right
thf(fact_4496_dvd__triv__right,axiom,
    ! [A: rat,B2: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B2 @ A ) ) ).

% dvd_triv_right
thf(fact_4497_dvd__triv__right,axiom,
    ! [A: nat,B2: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ A ) ) ).

% dvd_triv_right
thf(fact_4498_dvd__triv__right,axiom,
    ! [A: int,B2: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ A ) ) ).

% dvd_triv_right
thf(fact_4499_dvd__mult__right,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
     => ( dvd_dvd_Code_integer @ B2 @ C ) ) ).

% dvd_mult_right
thf(fact_4500_dvd__mult__right,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ B2 ) @ C )
     => ( dvd_dvd_real @ B2 @ C ) ) ).

% dvd_mult_right
thf(fact_4501_dvd__mult__right,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B2 ) @ C )
     => ( dvd_dvd_rat @ B2 @ C ) ) ).

% dvd_mult_right
thf(fact_4502_dvd__mult__right,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
     => ( dvd_dvd_nat @ B2 @ C ) ) ).

% dvd_mult_right
thf(fact_4503_dvd__mult__right,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
     => ( dvd_dvd_int @ B2 @ C ) ) ).

% dvd_mult_right
thf(fact_4504_mult__dvd__mono,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer,D: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B2 )
     => ( ( dvd_dvd_Code_integer @ C @ D )
       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B2 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_4505_mult__dvd__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( dvd_dvd_real @ A @ B2 )
     => ( ( dvd_dvd_real @ C @ D )
       => ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_4506_mult__dvd__mono,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( dvd_dvd_rat @ A @ B2 )
     => ( ( dvd_dvd_rat @ C @ D )
       => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_4507_mult__dvd__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( dvd_dvd_nat @ C @ D )
       => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_4508_mult__dvd__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( dvd_dvd_int @ C @ D )
       => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_4509_dvd__triv__left,axiom,
    ! [A: code_integer,B2: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B2 ) ) ).

% dvd_triv_left
thf(fact_4510_dvd__triv__left,axiom,
    ! [A: real,B2: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B2 ) ) ).

% dvd_triv_left
thf(fact_4511_dvd__triv__left,axiom,
    ! [A: rat,B2: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B2 ) ) ).

% dvd_triv_left
thf(fact_4512_dvd__triv__left,axiom,
    ! [A: nat,B2: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B2 ) ) ).

% dvd_triv_left
thf(fact_4513_dvd__triv__left,axiom,
    ! [A: int,B2: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B2 ) ) ).

% dvd_triv_left
thf(fact_4514_dvd__mult__left,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
     => ( dvd_dvd_Code_integer @ A @ C ) ) ).

% dvd_mult_left
thf(fact_4515_dvd__mult__left,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ B2 ) @ C )
     => ( dvd_dvd_real @ A @ C ) ) ).

% dvd_mult_left
thf(fact_4516_dvd__mult__left,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B2 ) @ C )
     => ( dvd_dvd_rat @ A @ C ) ) ).

% dvd_mult_left
thf(fact_4517_dvd__mult__left,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
     => ( dvd_dvd_nat @ A @ C ) ) ).

% dvd_mult_left
thf(fact_4518_dvd__mult__left,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
     => ( dvd_dvd_int @ A @ C ) ) ).

% dvd_mult_left
thf(fact_4519_dvd__mult2,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B2 )
     => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B2 @ C ) ) ) ).

% dvd_mult2
thf(fact_4520_dvd__mult2,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( dvd_dvd_real @ A @ B2 )
     => ( dvd_dvd_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).

% dvd_mult2
thf(fact_4521_dvd__mult2,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B2 )
     => ( dvd_dvd_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).

% dvd_mult2
thf(fact_4522_dvd__mult2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).

% dvd_mult2
thf(fact_4523_dvd__mult2,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).

% dvd_mult2
thf(fact_4524_dvd__mult,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ C )
     => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B2 @ C ) ) ) ).

% dvd_mult
thf(fact_4525_dvd__mult,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( dvd_dvd_real @ A @ C )
     => ( dvd_dvd_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).

% dvd_mult
thf(fact_4526_dvd__mult,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( dvd_dvd_rat @ A @ C )
     => ( dvd_dvd_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).

% dvd_mult
thf(fact_4527_dvd__mult,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).

% dvd_mult
thf(fact_4528_dvd__mult,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( dvd_dvd_int @ A @ C )
     => ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).

% dvd_mult
thf(fact_4529_dvd__def,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [B4: code_integer,A5: code_integer] :
        ? [K2: code_integer] :
          ( A5
          = ( times_3573771949741848930nteger @ B4 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_4530_dvd__def,axiom,
    ( dvd_dvd_real
    = ( ^ [B4: real,A5: real] :
        ? [K2: real] :
          ( A5
          = ( times_times_real @ B4 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_4531_dvd__def,axiom,
    ( dvd_dvd_rat
    = ( ^ [B4: rat,A5: rat] :
        ? [K2: rat] :
          ( A5
          = ( times_times_rat @ B4 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_4532_dvd__def,axiom,
    ( dvd_dvd_nat
    = ( ^ [B4: nat,A5: nat] :
        ? [K2: nat] :
          ( A5
          = ( times_times_nat @ B4 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_4533_dvd__def,axiom,
    ( dvd_dvd_int
    = ( ^ [B4: int,A5: int] :
        ? [K2: int] :
          ( A5
          = ( times_times_int @ B4 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_4534_dvdI,axiom,
    ! [A: code_integer,B2: code_integer,K: code_integer] :
      ( ( A
        = ( times_3573771949741848930nteger @ B2 @ K ) )
     => ( dvd_dvd_Code_integer @ B2 @ A ) ) ).

% dvdI
thf(fact_4535_dvdI,axiom,
    ! [A: real,B2: real,K: real] :
      ( ( A
        = ( times_times_real @ B2 @ K ) )
     => ( dvd_dvd_real @ B2 @ A ) ) ).

% dvdI
thf(fact_4536_dvdI,axiom,
    ! [A: rat,B2: rat,K: rat] :
      ( ( A
        = ( times_times_rat @ B2 @ K ) )
     => ( dvd_dvd_rat @ B2 @ A ) ) ).

% dvdI
thf(fact_4537_dvdI,axiom,
    ! [A: nat,B2: nat,K: nat] :
      ( ( A
        = ( times_times_nat @ B2 @ K ) )
     => ( dvd_dvd_nat @ B2 @ A ) ) ).

% dvdI
thf(fact_4538_dvdI,axiom,
    ! [A: int,B2: int,K: int] :
      ( ( A
        = ( times_times_int @ B2 @ K ) )
     => ( dvd_dvd_int @ B2 @ A ) ) ).

% dvdI
thf(fact_4539_dvdE,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ A )
     => ~ ! [K3: code_integer] :
            ( A
           != ( times_3573771949741848930nteger @ B2 @ K3 ) ) ) ).

% dvdE
thf(fact_4540_dvdE,axiom,
    ! [B2: real,A: real] :
      ( ( dvd_dvd_real @ B2 @ A )
     => ~ ! [K3: real] :
            ( A
           != ( times_times_real @ B2 @ K3 ) ) ) ).

% dvdE
thf(fact_4541_dvdE,axiom,
    ! [B2: rat,A: rat] :
      ( ( dvd_dvd_rat @ B2 @ A )
     => ~ ! [K3: rat] :
            ( A
           != ( times_times_rat @ B2 @ K3 ) ) ) ).

% dvdE
thf(fact_4542_dvdE,axiom,
    ! [B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ B2 @ A )
     => ~ ! [K3: nat] :
            ( A
           != ( times_times_nat @ B2 @ K3 ) ) ) ).

% dvdE
thf(fact_4543_dvdE,axiom,
    ! [B2: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ A )
     => ~ ! [K3: int] :
            ( A
           != ( times_times_int @ B2 @ K3 ) ) ) ).

% dvdE
thf(fact_4544_dvd__unit__imp__unit,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B2 )
     => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).

% dvd_unit_imp_unit
thf(fact_4545_dvd__unit__imp__unit,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
       => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).

% dvd_unit_imp_unit
thf(fact_4546_dvd__unit__imp__unit,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( dvd_dvd_int @ B2 @ one_one_int )
       => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).

% dvd_unit_imp_unit
thf(fact_4547_unit__imp__dvd,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( dvd_dvd_Code_integer @ B2 @ A ) ) ).

% unit_imp_dvd
thf(fact_4548_unit__imp__dvd,axiom,
    ! [B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( dvd_dvd_nat @ B2 @ A ) ) ).

% unit_imp_dvd
thf(fact_4549_unit__imp__dvd,axiom,
    ! [B2: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( dvd_dvd_int @ B2 @ A ) ) ).

% unit_imp_dvd
thf(fact_4550_one__dvd,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).

% one_dvd
thf(fact_4551_one__dvd,axiom,
    ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).

% one_dvd
thf(fact_4552_one__dvd,axiom,
    ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).

% one_dvd
thf(fact_4553_one__dvd,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).

% one_dvd
thf(fact_4554_one__dvd,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).

% one_dvd
thf(fact_4555_one__dvd,axiom,
    ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).

% one_dvd
thf(fact_4556_dvd__add,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B2 )
     => ( ( dvd_dvd_Code_integer @ A @ C )
       => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ C ) ) ) ) ).

% dvd_add
thf(fact_4557_dvd__add,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( dvd_dvd_real @ A @ B2 )
     => ( ( dvd_dvd_real @ A @ C )
       => ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ) ).

% dvd_add
thf(fact_4558_dvd__add,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B2 )
     => ( ( dvd_dvd_rat @ A @ C )
       => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ) ).

% dvd_add
thf(fact_4559_dvd__add,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( dvd_dvd_nat @ A @ C )
       => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ) ).

% dvd_add
thf(fact_4560_dvd__add,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( dvd_dvd_int @ A @ C )
       => ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ) ).

% dvd_add
thf(fact_4561_dvd__add__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ C )
     => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ C ) )
        = ( dvd_dvd_Code_integer @ A @ B2 ) ) ) ).

% dvd_add_left_iff
thf(fact_4562_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( dvd_dvd_real @ A @ C )
     => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ C ) )
        = ( dvd_dvd_real @ A @ B2 ) ) ) ).

% dvd_add_left_iff
thf(fact_4563_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( dvd_dvd_rat @ A @ C )
     => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
        = ( dvd_dvd_rat @ A @ B2 ) ) ) ).

% dvd_add_left_iff
thf(fact_4564_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
        = ( dvd_dvd_nat @ A @ B2 ) ) ) ).

% dvd_add_left_iff
thf(fact_4565_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( dvd_dvd_int @ A @ C )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ C ) )
        = ( dvd_dvd_int @ A @ B2 ) ) ) ).

% dvd_add_left_iff
thf(fact_4566_dvd__add__right__iff,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B2 )
     => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ C ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_4567_dvd__add__right__iff,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( dvd_dvd_real @ A @ B2 )
     => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ C ) )
        = ( dvd_dvd_real @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_4568_dvd__add__right__iff,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B2 )
     => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
        = ( dvd_dvd_rat @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_4569_dvd__add__right__iff,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B2 )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_4570_dvd__add__right__iff,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ B2 )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ C ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_4571_dvd__diff,axiom,
    ! [X3: code_integer,Y3: code_integer,Z: code_integer] :
      ( ( dvd_dvd_Code_integer @ X3 @ Y3 )
     => ( ( dvd_dvd_Code_integer @ X3 @ Z )
       => ( dvd_dvd_Code_integer @ X3 @ ( minus_8373710615458151222nteger @ Y3 @ Z ) ) ) ) ).

% dvd_diff
thf(fact_4572_dvd__diff,axiom,
    ! [X3: complex,Y3: complex,Z: complex] :
      ( ( dvd_dvd_complex @ X3 @ Y3 )
     => ( ( dvd_dvd_complex @ X3 @ Z )
       => ( dvd_dvd_complex @ X3 @ ( minus_minus_complex @ Y3 @ Z ) ) ) ) ).

% dvd_diff
thf(fact_4573_dvd__diff,axiom,
    ! [X3: real,Y3: real,Z: real] :
      ( ( dvd_dvd_real @ X3 @ Y3 )
     => ( ( dvd_dvd_real @ X3 @ Z )
       => ( dvd_dvd_real @ X3 @ ( minus_minus_real @ Y3 @ Z ) ) ) ) ).

% dvd_diff
thf(fact_4574_dvd__diff,axiom,
    ! [X3: rat,Y3: rat,Z: rat] :
      ( ( dvd_dvd_rat @ X3 @ Y3 )
     => ( ( dvd_dvd_rat @ X3 @ Z )
       => ( dvd_dvd_rat @ X3 @ ( minus_minus_rat @ Y3 @ Z ) ) ) ) ).

% dvd_diff
thf(fact_4575_dvd__diff,axiom,
    ! [X3: int,Y3: int,Z: int] :
      ( ( dvd_dvd_int @ X3 @ Y3 )
     => ( ( dvd_dvd_int @ X3 @ Z )
       => ( dvd_dvd_int @ X3 @ ( minus_minus_int @ Y3 @ Z ) ) ) ) ).

% dvd_diff
thf(fact_4576_dvd__diff__commute,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ C @ B2 ) )
      = ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ B2 @ C ) ) ) ).

% dvd_diff_commute
thf(fact_4577_dvd__diff__commute,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( dvd_dvd_int @ A @ ( minus_minus_int @ C @ B2 ) )
      = ( dvd_dvd_int @ A @ ( minus_minus_int @ B2 @ C ) ) ) ).

% dvd_diff_commute
thf(fact_4578_div__div__div__same,axiom,
    ! [D: code_integer,B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ B2 )
     => ( ( dvd_dvd_Code_integer @ B2 @ A )
       => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B2 @ D ) )
          = ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ).

% div_div_div_same
thf(fact_4579_div__div__div__same,axiom,
    ! [D: nat,B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ D @ B2 )
     => ( ( dvd_dvd_nat @ B2 @ A )
       => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B2 @ D ) )
          = ( divide_divide_nat @ A @ B2 ) ) ) ) ).

% div_div_div_same
thf(fact_4580_div__div__div__same,axiom,
    ! [D: int,B2: int,A: int] :
      ( ( dvd_dvd_int @ D @ B2 )
     => ( ( dvd_dvd_int @ B2 @ A )
       => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B2 @ D ) )
          = ( divide_divide_int @ A @ B2 ) ) ) ) ).

% div_div_div_same
thf(fact_4581_dvd__div__eq__cancel,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( ( divide6298287555418463151nteger @ A @ C )
        = ( divide6298287555418463151nteger @ B2 @ C ) )
     => ( ( dvd_dvd_Code_integer @ C @ A )
       => ( ( dvd_dvd_Code_integer @ C @ B2 )
         => ( A = B2 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_4582_dvd__div__eq__cancel,axiom,
    ! [A: complex,C: complex,B2: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ C )
        = ( divide1717551699836669952omplex @ B2 @ C ) )
     => ( ( dvd_dvd_complex @ C @ A )
       => ( ( dvd_dvd_complex @ C @ B2 )
         => ( A = B2 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_4583_dvd__div__eq__cancel,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( ( divide_divide_real @ A @ C )
        = ( divide_divide_real @ B2 @ C ) )
     => ( ( dvd_dvd_real @ C @ A )
       => ( ( dvd_dvd_real @ C @ B2 )
         => ( A = B2 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_4584_dvd__div__eq__cancel,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( ( divide_divide_rat @ A @ C )
        = ( divide_divide_rat @ B2 @ C ) )
     => ( ( dvd_dvd_rat @ C @ A )
       => ( ( dvd_dvd_rat @ C @ B2 )
         => ( A = B2 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_4585_dvd__div__eq__cancel,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( ( divide_divide_nat @ A @ C )
        = ( divide_divide_nat @ B2 @ C ) )
     => ( ( dvd_dvd_nat @ C @ A )
       => ( ( dvd_dvd_nat @ C @ B2 )
         => ( A = B2 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_4586_dvd__div__eq__cancel,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ( divide_divide_int @ A @ C )
        = ( divide_divide_int @ B2 @ C ) )
     => ( ( dvd_dvd_int @ C @ A )
       => ( ( dvd_dvd_int @ C @ B2 )
         => ( A = B2 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_4587_dvd__div__eq__iff,axiom,
    ! [C: code_integer,A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( dvd_dvd_Code_integer @ C @ B2 )
       => ( ( ( divide6298287555418463151nteger @ A @ C )
            = ( divide6298287555418463151nteger @ B2 @ C ) )
          = ( A = B2 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_4588_dvd__div__eq__iff,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( dvd_dvd_complex @ C @ A )
     => ( ( dvd_dvd_complex @ C @ B2 )
       => ( ( ( divide1717551699836669952omplex @ A @ C )
            = ( divide1717551699836669952omplex @ B2 @ C ) )
          = ( A = B2 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_4589_dvd__div__eq__iff,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( dvd_dvd_real @ C @ A )
     => ( ( dvd_dvd_real @ C @ B2 )
       => ( ( ( divide_divide_real @ A @ C )
            = ( divide_divide_real @ B2 @ C ) )
          = ( A = B2 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_4590_dvd__div__eq__iff,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( dvd_dvd_rat @ C @ A )
     => ( ( dvd_dvd_rat @ C @ B2 )
       => ( ( ( divide_divide_rat @ A @ C )
            = ( divide_divide_rat @ B2 @ C ) )
          = ( A = B2 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_4591_dvd__div__eq__iff,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( dvd_dvd_nat @ C @ B2 )
       => ( ( ( divide_divide_nat @ A @ C )
            = ( divide_divide_nat @ B2 @ C ) )
          = ( A = B2 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_4592_dvd__div__eq__iff,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B2 )
       => ( ( ( divide_divide_int @ A @ C )
            = ( divide_divide_int @ B2 @ C ) )
          = ( A = B2 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_4593_dvd__power__same,axiom,
    ! [X3: code_integer,Y3: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ X3 @ Y3 )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X3 @ N ) @ ( power_8256067586552552935nteger @ Y3 @ N ) ) ) ).

% dvd_power_same
thf(fact_4594_dvd__power__same,axiom,
    ! [X3: nat,Y3: nat,N: nat] :
      ( ( dvd_dvd_nat @ X3 @ Y3 )
     => ( dvd_dvd_nat @ ( power_power_nat @ X3 @ N ) @ ( power_power_nat @ Y3 @ N ) ) ) ).

% dvd_power_same
thf(fact_4595_dvd__power__same,axiom,
    ! [X3: real,Y3: real,N: nat] :
      ( ( dvd_dvd_real @ X3 @ Y3 )
     => ( dvd_dvd_real @ ( power_power_real @ X3 @ N ) @ ( power_power_real @ Y3 @ N ) ) ) ).

% dvd_power_same
thf(fact_4596_dvd__power__same,axiom,
    ! [X3: int,Y3: int,N: nat] :
      ( ( dvd_dvd_int @ X3 @ Y3 )
     => ( dvd_dvd_int @ ( power_power_int @ X3 @ N ) @ ( power_power_int @ Y3 @ N ) ) ) ).

% dvd_power_same
thf(fact_4597_dvd__power__same,axiom,
    ! [X3: complex,Y3: complex,N: nat] :
      ( ( dvd_dvd_complex @ X3 @ Y3 )
     => ( dvd_dvd_complex @ ( power_power_complex @ X3 @ N ) @ ( power_power_complex @ Y3 @ N ) ) ) ).

% dvd_power_same
thf(fact_4598_mod__mod__cancel,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B2 )
     => ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B2 ) @ C )
        = ( modulo_modulo_nat @ A @ C ) ) ) ).

% mod_mod_cancel
thf(fact_4599_mod__mod__cancel,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( dvd_dvd_int @ C @ B2 )
     => ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B2 ) @ C )
        = ( modulo_modulo_int @ A @ C ) ) ) ).

% mod_mod_cancel
thf(fact_4600_mod__mod__cancel,axiom,
    ! [C: code_integer,B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B2 )
     => ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ C )
        = ( modulo364778990260209775nteger @ A @ C ) ) ) ).

% mod_mod_cancel
thf(fact_4601_dvd__mod,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ M )
     => ( ( dvd_dvd_nat @ K @ N )
       => ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N ) ) ) ) ).

% dvd_mod
thf(fact_4602_dvd__mod,axiom,
    ! [K: int,M: int,N: int] :
      ( ( dvd_dvd_int @ K @ M )
     => ( ( dvd_dvd_int @ K @ N )
       => ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N ) ) ) ) ).

% dvd_mod
thf(fact_4603_dvd__mod,axiom,
    ! [K: code_integer,M: code_integer,N: code_integer] :
      ( ( dvd_dvd_Code_integer @ K @ M )
     => ( ( dvd_dvd_Code_integer @ K @ N )
       => ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N ) ) ) ) ).

% dvd_mod
thf(fact_4604_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B2 ) )
     => ( ( dvd_dvd_nat @ C @ B2 )
       => ( dvd_dvd_nat @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_4605_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B2 ) )
     => ( ( dvd_dvd_int @ C @ B2 )
       => ( dvd_dvd_int @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_4606_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B2 ) )
     => ( ( dvd_dvd_Code_integer @ C @ B2 )
       => ( dvd_dvd_Code_integer @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_4607_dvd__mod__iff,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B2 )
     => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B2 ) )
        = ( dvd_dvd_nat @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_4608_dvd__mod__iff,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( dvd_dvd_int @ C @ B2 )
     => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B2 ) )
        = ( dvd_dvd_int @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_4609_dvd__mod__iff,axiom,
    ! [C: code_integer,B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B2 )
     => ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B2 ) )
        = ( dvd_dvd_Code_integer @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_4610_signed__take__bit__mult,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
      = ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ K @ L ) ) ) ).

% signed_take_bit_mult
thf(fact_4611_signed__take__bit__add,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
      = ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K @ L ) ) ) ).

% signed_take_bit_add
thf(fact_4612_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_4613_signed__take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
      = ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ K @ L ) ) ) ).

% signed_take_bit_diff
thf(fact_4614_subset__divisors__dvd,axiom,
    ! [A: complex,B2: complex] :
      ( ( ord_le211207098394363844omplex
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ A ) )
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ B2 ) ) )
      = ( dvd_dvd_complex @ A @ B2 ) ) ).

% subset_divisors_dvd
thf(fact_4615_subset__divisors__dvd,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_set_real
        @ ( collect_real
          @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
        @ ( collect_real
          @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B2 ) ) )
      = ( dvd_dvd_real @ A @ B2 ) ) ).

% subset_divisors_dvd
thf(fact_4616_subset__divisors__dvd,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_set_nat
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B2 ) ) )
      = ( dvd_dvd_nat @ A @ B2 ) ) ).

% subset_divisors_dvd
thf(fact_4617_subset__divisors__dvd,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le7084787975880047091nteger
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A ) )
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B2 ) ) )
      = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).

% subset_divisors_dvd
thf(fact_4618_subset__divisors__dvd,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_set_int
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B2 ) ) )
      = ( dvd_dvd_int @ A @ B2 ) ) ).

% subset_divisors_dvd
thf(fact_4619_strict__subset__divisors__dvd,axiom,
    ! [A: complex,B2: complex] :
      ( ( ord_less_set_complex
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ A ) )
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ B2 ) ) )
      = ( ( dvd_dvd_complex @ A @ B2 )
        & ~ ( dvd_dvd_complex @ B2 @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_4620_strict__subset__divisors__dvd,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_set_real
        @ ( collect_real
          @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
        @ ( collect_real
          @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B2 ) ) )
      = ( ( dvd_dvd_real @ A @ B2 )
        & ~ ( dvd_dvd_real @ B2 @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_4621_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_set_nat
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B2 ) ) )
      = ( ( dvd_dvd_nat @ A @ B2 )
        & ~ ( dvd_dvd_nat @ B2 @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_4622_strict__subset__divisors__dvd,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_set_int
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B2 ) ) )
      = ( ( dvd_dvd_int @ A @ B2 )
        & ~ ( dvd_dvd_int @ B2 @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_4623_strict__subset__divisors__dvd,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le1307284697595431911nteger
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A ) )
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B2 ) ) )
      = ( ( dvd_dvd_Code_integer @ A @ B2 )
        & ~ ( dvd_dvd_Code_integer @ B2 @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_4624_even__signed__take__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ M @ A ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).

% even_signed_take_bit_iff
thf(fact_4625_even__signed__take__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M @ A ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).

% even_signed_take_bit_iff
thf(fact_4626_not__is__unit__0,axiom,
    ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).

% not_is_unit_0
thf(fact_4627_not__is__unit__0,axiom,
    ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).

% not_is_unit_0
thf(fact_4628_not__is__unit__0,axiom,
    ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).

% not_is_unit_0
thf(fact_4629_pinf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z4: code_integer] :
    ! [X6: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z4 @ X6 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S ) ) ) ) ).

% pinf(9)
thf(fact_4630_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z4: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z4 @ X6 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S ) ) ) ) ).

% pinf(9)
thf(fact_4631_pinf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z4: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z4 @ X6 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S ) ) ) ) ).

% pinf(9)
thf(fact_4632_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z4 @ X6 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S ) ) ) ) ).

% pinf(9)
thf(fact_4633_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z4 @ X6 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S ) ) ) ) ).

% pinf(9)
thf(fact_4634_pinf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z4: code_integer] :
    ! [X6: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z4 @ X6 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_4635_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z4: real] :
    ! [X6: real] :
      ( ( ord_less_real @ Z4 @ X6 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_4636_pinf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z4: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ Z4 @ X6 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_4637_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ Z4 @ X6 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_4638_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ Z4 @ X6 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_4639_minf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z4: code_integer] :
    ! [X6: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X6 @ Z4 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S ) ) ) ) ).

% minf(9)
thf(fact_4640_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z4: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z4 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S ) ) ) ) ).

% minf(9)
thf(fact_4641_minf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z4: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z4 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S ) ) ) ) ).

% minf(9)
thf(fact_4642_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z4 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S ) ) ) ) ).

% minf(9)
thf(fact_4643_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z4 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S ) ) ) ) ).

% minf(9)
thf(fact_4644_minf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z4: code_integer] :
    ! [X6: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X6 @ Z4 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_4645_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z4: real] :
    ! [X6: real] :
      ( ( ord_less_real @ X6 @ Z4 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_4646_minf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z4: rat] :
    ! [X6: rat] :
      ( ( ord_less_rat @ X6 @ Z4 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_4647_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z4 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_4648_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z4 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_4649_dvd__div__eq__0__iff,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ A )
     => ( ( ( divide6298287555418463151nteger @ A @ B2 )
          = zero_z3403309356797280102nteger )
        = ( A = zero_z3403309356797280102nteger ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_4650_dvd__div__eq__0__iff,axiom,
    ! [B2: complex,A: complex] :
      ( ( dvd_dvd_complex @ B2 @ A )
     => ( ( ( divide1717551699836669952omplex @ A @ B2 )
          = zero_zero_complex )
        = ( A = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_4651_dvd__div__eq__0__iff,axiom,
    ! [B2: real,A: real] :
      ( ( dvd_dvd_real @ B2 @ A )
     => ( ( ( divide_divide_real @ A @ B2 )
          = zero_zero_real )
        = ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_4652_dvd__div__eq__0__iff,axiom,
    ! [B2: rat,A: rat] :
      ( ( dvd_dvd_rat @ B2 @ A )
     => ( ( ( divide_divide_rat @ A @ B2 )
          = zero_zero_rat )
        = ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_4653_dvd__div__eq__0__iff,axiom,
    ! [B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ B2 @ A )
     => ( ( ( divide_divide_nat @ A @ B2 )
          = zero_zero_nat )
        = ( A = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_4654_dvd__div__eq__0__iff,axiom,
    ! [B2: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ A )
     => ( ( ( divide_divide_int @ A @ B2 )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_4655_unit__mult__right__cancel,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( times_3573771949741848930nteger @ B2 @ A )
          = ( times_3573771949741848930nteger @ C @ A ) )
        = ( B2 = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_4656_unit__mult__right__cancel,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( times_times_nat @ B2 @ A )
          = ( times_times_nat @ C @ A ) )
        = ( B2 = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_4657_unit__mult__right__cancel,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( times_times_int @ B2 @ A )
          = ( times_times_int @ C @ A ) )
        = ( B2 = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_4658_unit__mult__left__cancel,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( times_3573771949741848930nteger @ A @ B2 )
          = ( times_3573771949741848930nteger @ A @ C ) )
        = ( B2 = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_4659_unit__mult__left__cancel,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( times_times_nat @ A @ B2 )
          = ( times_times_nat @ A @ C ) )
        = ( B2 = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_4660_unit__mult__left__cancel,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( times_times_int @ A @ B2 )
          = ( times_times_int @ A @ C ) )
        = ( B2 = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_4661_mult__unit__dvd__iff_H,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
        = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_4662_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
        = ( dvd_dvd_nat @ B2 @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_4663_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
        = ( dvd_dvd_int @ B2 @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_4664_dvd__mult__unit__iff_H,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_4665_dvd__mult__unit__iff_H,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_4666_dvd__mult__unit__iff_H,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_4667_mult__unit__dvd__iff,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_4668_mult__unit__dvd__iff,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_4669_mult__unit__dvd__iff,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_4670_dvd__mult__unit__iff,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B2 ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_4671_dvd__mult__unit__iff,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B2 ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_4672_dvd__mult__unit__iff,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B2 ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_4673_is__unit__mult__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
        & ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer ) ) ) ).

% is_unit_mult_iff
thf(fact_4674_is__unit__mult__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A @ one_one_nat )
        & ( dvd_dvd_nat @ B2 @ one_one_nat ) ) ) ).

% is_unit_mult_iff
thf(fact_4675_is__unit__mult__iff,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ one_one_int )
      = ( ( dvd_dvd_int @ A @ one_one_int )
        & ( dvd_dvd_int @ B2 @ one_one_int ) ) ) ).

% is_unit_mult_iff
thf(fact_4676_div__mult__div__if__dvd,axiom,
    ! [B2: code_integer,A: code_integer,D: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ A )
     => ( ( dvd_dvd_Code_integer @ D @ C )
       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ ( divide6298287555418463151nteger @ C @ D ) )
          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B2 @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_4677_div__mult__div__if__dvd,axiom,
    ! [B2: nat,A: nat,D: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ A )
     => ( ( dvd_dvd_nat @ D @ C )
       => ( ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ ( divide_divide_nat @ C @ D ) )
          = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_4678_div__mult__div__if__dvd,axiom,
    ! [B2: int,A: int,D: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ A )
     => ( ( dvd_dvd_int @ D @ C )
       => ( ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ C @ D ) )
          = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_4679_dvd__mult__imp__div,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B2 )
     => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_4680_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B2 )
     => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B2 @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_4681_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B2 )
     => ( dvd_dvd_int @ A @ ( divide_divide_int @ B2 @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_4682_dvd__div__mult2__eq,axiom,
    ! [B2: code_integer,C: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B2 @ C ) @ A )
     => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
        = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_4683_dvd__div__mult2__eq,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ C ) @ A )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_4684_dvd__div__mult2__eq,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ B2 @ C ) @ A )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_4685_div__div__eq__right,axiom,
    ! [C: code_integer,B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B2 )
     => ( ( dvd_dvd_Code_integer @ B2 @ A )
       => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B2 @ C ) )
          = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_4686_div__div__eq__right,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B2 )
     => ( ( dvd_dvd_nat @ B2 @ A )
       => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
          = ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_4687_div__div__eq__right,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( dvd_dvd_int @ C @ B2 )
     => ( ( dvd_dvd_int @ B2 @ A )
       => ( ( divide_divide_int @ A @ ( divide_divide_int @ B2 @ C ) )
          = ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_4688_div__mult__swap,axiom,
    ! [C: code_integer,B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B2 )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B2 @ C ) )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ) ).

% div_mult_swap
thf(fact_4689_div__mult__swap,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B2 )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ) ).

% div_mult_swap
thf(fact_4690_div__mult__swap,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( dvd_dvd_int @ C @ B2 )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ C ) ) ) ).

% div_mult_swap
thf(fact_4691_dvd__div__mult,axiom,
    ! [C: code_integer,B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B2 )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B2 @ C ) @ A )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B2 @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_4692_dvd__div__mult,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B2 )
     => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ C ) @ A )
        = ( divide_divide_nat @ ( times_times_nat @ B2 @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_4693_dvd__div__mult,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( dvd_dvd_int @ C @ B2 )
     => ( ( times_times_int @ ( divide_divide_int @ B2 @ C ) @ A )
        = ( divide_divide_int @ ( times_times_int @ B2 @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_4694_dvd__div__unit__iff,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B2 ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_4695_dvd__div__unit__iff,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B2 ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_4696_dvd__div__unit__iff,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B2 ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_4697_div__unit__dvd__iff,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B2 ) @ C )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_4698_div__unit__dvd__iff,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_4699_div__unit__dvd__iff,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ C )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_4700_unit__div__cancel,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ B2 @ A )
          = ( divide6298287555418463151nteger @ C @ A ) )
        = ( B2 = C ) ) ) ).

% unit_div_cancel
thf(fact_4701_unit__div__cancel,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( divide_divide_nat @ B2 @ A )
          = ( divide_divide_nat @ C @ A ) )
        = ( B2 = C ) ) ) ).

% unit_div_cancel
thf(fact_4702_unit__div__cancel,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( divide_divide_int @ B2 @ A )
          = ( divide_divide_int @ C @ A ) )
        = ( B2 = C ) ) ) ).

% unit_div_cancel
thf(fact_4703_div__plus__div__distrib__dvd__left,axiom,
    ! [C: code_integer,A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_4704_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_4705_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
        = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_4706_div__plus__div__distrib__dvd__right,axiom,
    ! [C: code_integer,B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B2 )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_4707_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B2 )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_4708_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( dvd_dvd_int @ C @ B2 )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
        = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_4709_div__power,axiom,
    ! [B2: code_integer,A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ B2 @ A )
     => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ N )
        = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B2 @ N ) ) ) ) ).

% div_power
thf(fact_4710_div__power,axiom,
    ! [B2: nat,A: nat,N: nat] :
      ( ( dvd_dvd_nat @ B2 @ A )
     => ( ( power_power_nat @ ( divide_divide_nat @ A @ B2 ) @ N )
        = ( divide_divide_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ).

% div_power
thf(fact_4711_div__power,axiom,
    ! [B2: int,A: int,N: nat] :
      ( ( dvd_dvd_int @ B2 @ A )
     => ( ( power_power_int @ ( divide_divide_int @ A @ B2 ) @ N )
        = ( divide_divide_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).

% div_power
thf(fact_4712_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( modulo_modulo_nat @ A @ B2 )
        = zero_zero_nat )
      = ( dvd_dvd_nat @ B2 @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_4713_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B2: int] :
      ( ( ( modulo_modulo_int @ A @ B2 )
        = zero_zero_int )
      = ( dvd_dvd_int @ B2 @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_4714_mod__eq__0__iff__dvd,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B2 )
        = zero_z3403309356797280102nteger )
      = ( dvd_dvd_Code_integer @ B2 @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_4715_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( modulo_modulo_nat @ B4 @ A5 )
          = zero_zero_nat ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_4716_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_int
    = ( ^ [A5: int,B4: int] :
          ( ( modulo_modulo_int @ B4 @ A5 )
          = zero_zero_int ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_4717_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [A5: code_integer,B4: code_integer] :
          ( ( modulo364778990260209775nteger @ B4 @ A5 )
          = zero_z3403309356797280102nteger ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_4718_mod__0__imp__dvd,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( modulo_modulo_nat @ A @ B2 )
        = zero_zero_nat )
     => ( dvd_dvd_nat @ B2 @ A ) ) ).

% mod_0_imp_dvd
thf(fact_4719_mod__0__imp__dvd,axiom,
    ! [A: int,B2: int] :
      ( ( ( modulo_modulo_int @ A @ B2 )
        = zero_zero_int )
     => ( dvd_dvd_int @ B2 @ A ) ) ).

% mod_0_imp_dvd
thf(fact_4720_mod__0__imp__dvd,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B2 )
        = zero_z3403309356797280102nteger )
     => ( dvd_dvd_Code_integer @ B2 @ A ) ) ).

% mod_0_imp_dvd
thf(fact_4721_dvd__power__le,axiom,
    ! [X3: code_integer,Y3: code_integer,N: nat,M: nat] :
      ( ( dvd_dvd_Code_integer @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X3 @ N ) @ ( power_8256067586552552935nteger @ Y3 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4722_dvd__power__le,axiom,
    ! [X3: nat,Y3: nat,N: nat,M: nat] :
      ( ( dvd_dvd_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_nat @ ( power_power_nat @ X3 @ N ) @ ( power_power_nat @ Y3 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4723_dvd__power__le,axiom,
    ! [X3: real,Y3: real,N: nat,M: nat] :
      ( ( dvd_dvd_real @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_real @ ( power_power_real @ X3 @ N ) @ ( power_power_real @ Y3 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4724_dvd__power__le,axiom,
    ! [X3: int,Y3: int,N: nat,M: nat] :
      ( ( dvd_dvd_int @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_int @ ( power_power_int @ X3 @ N ) @ ( power_power_int @ Y3 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4725_dvd__power__le,axiom,
    ! [X3: complex,Y3: complex,N: nat,M: nat] :
      ( ( dvd_dvd_complex @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_complex @ ( power_power_complex @ X3 @ N ) @ ( power_power_complex @ Y3 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_4726_power__le__dvd,axiom,
    ! [A: code_integer,N: nat,B2: code_integer,M: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B2 )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B2 ) ) ) ).

% power_le_dvd
thf(fact_4727_power__le__dvd,axiom,
    ! [A: nat,N: nat,B2: nat,M: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B2 )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B2 ) ) ) ).

% power_le_dvd
thf(fact_4728_power__le__dvd,axiom,
    ! [A: real,N: nat,B2: real,M: nat] :
      ( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B2 )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B2 ) ) ) ).

% power_le_dvd
thf(fact_4729_power__le__dvd,axiom,
    ! [A: int,N: nat,B2: int,M: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B2 )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B2 ) ) ) ).

% power_le_dvd
thf(fact_4730_power__le__dvd,axiom,
    ! [A: complex,N: nat,B2: complex,M: nat] :
      ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B2 )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B2 ) ) ) ).

% power_le_dvd
thf(fact_4731_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_4732_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_4733_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_4734_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_4735_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: complex] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_4736_mod__eq__dvd__iff,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ B2 @ C ) )
      = ( dvd_dvd_int @ C @ ( minus_minus_int @ A @ B2 ) ) ) ).

% mod_eq_dvd_iff
thf(fact_4737_mod__eq__dvd__iff,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ B2 @ C ) )
      = ( dvd_dvd_Code_integer @ C @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ).

% mod_eq_dvd_iff
thf(fact_4738_dvd__minus__mod,axiom,
    ! [B2: nat,A: nat] : ( dvd_dvd_nat @ B2 @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B2 ) ) ) ).

% dvd_minus_mod
thf(fact_4739_dvd__minus__mod,axiom,
    ! [B2: int,A: int] : ( dvd_dvd_int @ B2 @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B2 ) ) ) ).

% dvd_minus_mod
thf(fact_4740_dvd__minus__mod,axiom,
    ! [B2: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B2 @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ).

% dvd_minus_mod
thf(fact_4741_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_4742_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_4743_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_4744_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_4745_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_4746_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_4747_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_4748_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_4749_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_4750_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_4751_bezout__add__nat,axiom,
    ! [A: nat,B2: nat] :
    ? [D3: nat,X5: nat,Y4: nat] :
      ( ( dvd_dvd_nat @ D3 @ A )
      & ( dvd_dvd_nat @ D3 @ B2 )
      & ( ( ( times_times_nat @ A @ X5 )
          = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y4 ) @ D3 ) )
        | ( ( times_times_nat @ B2 @ X5 )
          = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D3 ) ) ) ) ).

% bezout_add_nat
thf(fact_4752_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B2: nat,X3: nat,Y3: nat] :
      ( ( dvd_dvd_nat @ D @ A )
     => ( ( dvd_dvd_nat @ D @ B2 )
       => ( ( ( ( times_times_nat @ A @ X3 )
              = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y3 ) @ D ) )
            | ( ( times_times_nat @ B2 @ X3 )
              = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) )
         => ? [X5: nat,Y4: nat] :
              ( ( dvd_dvd_nat @ D @ A )
              & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B2 ) )
              & ( ( ( times_times_nat @ A @ X5 )
                  = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ Y4 ) @ D ) )
                | ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ X5 )
                  = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D ) ) ) ) ) ) ) ).

% bezout_lemma_nat
thf(fact_4753_bezout1__nat,axiom,
    ! [A: nat,B2: nat] :
    ? [D3: nat,X5: nat,Y4: nat] :
      ( ( dvd_dvd_nat @ D3 @ A )
      & ( dvd_dvd_nat @ D3 @ B2 )
      & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X5 ) @ ( times_times_nat @ B2 @ Y4 ) )
          = D3 )
        | ( ( minus_minus_nat @ ( times_times_nat @ B2 @ X5 ) @ ( times_times_nat @ A @ Y4 ) )
          = D3 ) ) ) ).

% bezout1_nat
thf(fact_4754_zdvd__period,axiom,
    ! [A: int,D: int,X3: int,T: int,C: int] :
      ( ( dvd_dvd_int @ A @ D )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X3 @ T ) )
        = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X3 @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).

% zdvd_period
thf(fact_4755_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_4756_div2__even__ext__nat,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ( divide_divide_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X3 )
          = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y3 ) )
       => ( X3 = Y3 ) ) ) ).

% div2_even_ext_nat
thf(fact_4757_unit__dvdE,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C2: code_integer] :
              ( B2
             != ( times_3573771949741848930nteger @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_4758_unit__dvdE,axiom,
    ! [A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [C2: nat] :
              ( B2
             != ( times_times_nat @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_4759_unit__dvdE,axiom,
    ! [A: int,B2: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [C2: int] :
              ( B2
             != ( times_times_int @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_4760_unity__coeff__ex,axiom,
    ! [P: code_integer > $o,L: code_integer] :
      ( ( ? [X: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X ) ) )
      = ( ? [X: code_integer] :
            ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X @ zero_z3403309356797280102nteger ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_4761_unity__coeff__ex,axiom,
    ! [P: complex > $o,L: complex] :
      ( ( ? [X: complex] : ( P @ ( times_times_complex @ L @ X ) ) )
      = ( ? [X: complex] :
            ( ( dvd_dvd_complex @ L @ ( plus_plus_complex @ X @ zero_zero_complex ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_4762_unity__coeff__ex,axiom,
    ! [P: real > $o,L: real] :
      ( ( ? [X: real] : ( P @ ( times_times_real @ L @ X ) ) )
      = ( ? [X: real] :
            ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X @ zero_zero_real ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_4763_unity__coeff__ex,axiom,
    ! [P: rat > $o,L: rat] :
      ( ( ? [X: rat] : ( P @ ( times_times_rat @ L @ X ) ) )
      = ( ? [X: rat] :
            ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X @ zero_zero_rat ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_4764_unity__coeff__ex,axiom,
    ! [P: nat > $o,L: nat] :
      ( ( ? [X: nat] : ( P @ ( times_times_nat @ L @ X ) ) )
      = ( ? [X: nat] :
            ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X @ zero_zero_nat ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_4765_unity__coeff__ex,axiom,
    ! [P: int > $o,L: int] :
      ( ( ? [X: int] : ( P @ ( times_times_int @ L @ X ) ) )
      = ( ? [X: int] :
            ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X @ zero_zero_int ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_4766_dvd__div__eq__mult,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ A @ B2 )
       => ( ( ( divide6298287555418463151nteger @ B2 @ A )
            = C )
          = ( B2
            = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_4767_dvd__div__eq__mult,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ A @ B2 )
       => ( ( ( divide_divide_nat @ B2 @ A )
            = C )
          = ( B2
            = ( times_times_nat @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_4768_dvd__div__eq__mult,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ A @ B2 )
       => ( ( ( divide_divide_int @ B2 @ A )
            = C )
          = ( B2
            = ( times_times_int @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_4769_div__dvd__iff__mult,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( ( B2 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B2 @ A )
       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B2 ) @ C )
          = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_4770_div__dvd__iff__mult,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B2 @ A )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
          = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B2 ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_4771_div__dvd__iff__mult,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( dvd_dvd_int @ B2 @ A )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ C )
          = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B2 ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_4772_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B2: code_integer,A: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ C @ B2 )
       => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B2 @ C ) )
          = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B2 ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_4773_dvd__div__iff__mult,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( ( dvd_dvd_nat @ C @ B2 )
       => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
          = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B2 ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_4774_dvd__div__iff__mult,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( C != zero_zero_int )
     => ( ( dvd_dvd_int @ C @ B2 )
       => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B2 @ C ) )
          = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B2 ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_4775_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer,D: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( ( dvd_dvd_Code_integer @ A @ B2 )
         => ( ( dvd_dvd_Code_integer @ C @ D )
           => ( ( ( divide6298287555418463151nteger @ B2 @ A )
                = ( divide6298287555418463151nteger @ D @ C ) )
              = ( ( times_3573771949741848930nteger @ B2 @ C )
                = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_4776_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B2: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( ( dvd_dvd_nat @ A @ B2 )
         => ( ( dvd_dvd_nat @ C @ D )
           => ( ( ( divide_divide_nat @ B2 @ A )
                = ( divide_divide_nat @ D @ C ) )
              = ( ( times_times_nat @ B2 @ C )
                = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_4777_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B2: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( ( dvd_dvd_int @ A @ B2 )
         => ( ( dvd_dvd_int @ C @ D )
           => ( ( ( divide_divide_int @ B2 @ A )
                = ( divide_divide_int @ D @ C ) )
              = ( ( times_times_int @ B2 @ C )
                = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_4778_unit__div__eq__0__iff,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ A @ B2 )
          = zero_z3403309356797280102nteger )
        = ( A = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
thf(fact_4779_unit__div__eq__0__iff,axiom,
    ! [B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( ( divide_divide_nat @ A @ B2 )
          = zero_zero_nat )
        = ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
thf(fact_4780_unit__div__eq__0__iff,axiom,
    ! [B2: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( ( divide_divide_int @ A @ B2 )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
thf(fact_4781_even__numeral,axiom,
    ! [N: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ).

% even_numeral
thf(fact_4782_even__numeral,axiom,
    ! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).

% even_numeral
thf(fact_4783_even__numeral,axiom,
    ! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).

% even_numeral
thf(fact_4784_inf__period_I4_J,axiom,
    ! [D: code_integer,D4: code_integer,T: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ D4 )
     => ! [X6: code_integer,K4: code_integer] :
          ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ T ) ) )
          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X6 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4785_inf__period_I4_J,axiom,
    ! [D: complex,D4: complex,T: complex] :
      ( ( dvd_dvd_complex @ D @ D4 )
     => ! [X6: complex,K4: complex] :
          ( ( ~ ( dvd_dvd_complex @ D @ ( plus_plus_complex @ X6 @ T ) ) )
          = ( ~ ( dvd_dvd_complex @ D @ ( plus_plus_complex @ ( minus_minus_complex @ X6 @ ( times_times_complex @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4786_inf__period_I4_J,axiom,
    ! [D: real,D4: real,T: real] :
      ( ( dvd_dvd_real @ D @ D4 )
     => ! [X6: real,K4: real] :
          ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ T ) ) )
          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X6 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4787_inf__period_I4_J,axiom,
    ! [D: rat,D4: rat,T: rat] :
      ( ( dvd_dvd_rat @ D @ D4 )
     => ! [X6: rat,K4: rat] :
          ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ T ) ) )
          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4788_inf__period_I4_J,axiom,
    ! [D: int,D4: int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X6: int,K4: int] :
          ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ T ) ) )
          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4789_inf__period_I3_J,axiom,
    ! [D: code_integer,D4: code_integer,T: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ D4 )
     => ! [X6: code_integer,K4: code_integer] :
          ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ T ) )
          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X6 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4790_inf__period_I3_J,axiom,
    ! [D: complex,D4: complex,T: complex] :
      ( ( dvd_dvd_complex @ D @ D4 )
     => ! [X6: complex,K4: complex] :
          ( ( dvd_dvd_complex @ D @ ( plus_plus_complex @ X6 @ T ) )
          = ( dvd_dvd_complex @ D @ ( plus_plus_complex @ ( minus_minus_complex @ X6 @ ( times_times_complex @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4791_inf__period_I3_J,axiom,
    ! [D: real,D4: real,T: real] :
      ( ( dvd_dvd_real @ D @ D4 )
     => ! [X6: real,K4: real] :
          ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ T ) )
          = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X6 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4792_inf__period_I3_J,axiom,
    ! [D: rat,D4: rat,T: rat] :
      ( ( dvd_dvd_rat @ D @ D4 )
     => ! [X6: rat,K4: rat] :
          ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ T ) )
          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4793_inf__period_I3_J,axiom,
    ! [D: int,D4: int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X6: int,K4: int] :
          ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ T ) )
          = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4794_unit__eq__div1,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ A @ B2 )
          = C )
        = ( A
          = ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ).

% unit_eq_div1
thf(fact_4795_unit__eq__div1,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( ( divide_divide_nat @ A @ B2 )
          = C )
        = ( A
          = ( times_times_nat @ C @ B2 ) ) ) ) ).

% unit_eq_div1
thf(fact_4796_unit__eq__div1,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( ( divide_divide_int @ A @ B2 )
          = C )
        = ( A
          = ( times_times_int @ C @ B2 ) ) ) ) ).

% unit_eq_div1
thf(fact_4797_unit__eq__div2,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( A
          = ( divide6298287555418463151nteger @ C @ B2 ) )
        = ( ( times_3573771949741848930nteger @ A @ B2 )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_4798_unit__eq__div2,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( A
          = ( divide_divide_nat @ C @ B2 ) )
        = ( ( times_times_nat @ A @ B2 )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_4799_unit__eq__div2,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( A
          = ( divide_divide_int @ C @ B2 ) )
        = ( ( times_times_int @ A @ B2 )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_4800_div__mult__unit2,axiom,
    ! [C: code_integer,B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B2 @ A )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_4801_div__mult__unit2,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( dvd_dvd_nat @ B2 @ A )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_4802_div__mult__unit2,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( dvd_dvd_int @ B2 @ A )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_4803_unit__div__commute,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B2 ) ) ) ).

% unit_div_commute
thf(fact_4804_unit__div__commute,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
        = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B2 ) ) ) ).

% unit_div_commute
thf(fact_4805_unit__div__commute,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ C )
        = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B2 ) ) ) ).

% unit_div_commute
thf(fact_4806_unit__div__mult__swap,axiom,
    ! [C: code_integer,A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B2 @ C ) )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_4807_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B2: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_4808_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_4809_is__unit__div__mult2__eq,axiom,
    ! [B2: code_integer,C: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_4810_is__unit__div__mult2__eq,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ C @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_4811_is__unit__div__mult2__eq,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ C @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_4812_unit__imp__mod__eq__0,axiom,
    ! [B2: nat,A: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( modulo_modulo_nat @ A @ B2 )
        = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
thf(fact_4813_unit__imp__mod__eq__0,axiom,
    ! [B2: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( modulo_modulo_int @ A @ B2 )
        = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
thf(fact_4814_unit__imp__mod__eq__0,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( modulo364778990260209775nteger @ A @ B2 )
        = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
thf(fact_4815_is__unit__power__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_4816_is__unit__power__iff,axiom,
    ! [A: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A @ one_one_nat )
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_4817_is__unit__power__iff,axiom,
    ! [A: int,N: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
      = ( ( dvd_dvd_int @ A @ one_one_int )
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_4818_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_4819_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_4820_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_4821_bezout__add__strong__nat,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ? [D3: nat,X5: nat,Y4: nat] :
          ( ( dvd_dvd_nat @ D3 @ A )
          & ( dvd_dvd_nat @ D3 @ B2 )
          & ( ( times_times_nat @ A @ X5 )
            = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y4 ) @ D3 ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_4822_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_4823_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_4824_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q2: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( ( modulo_modulo_nat @ M @ Q2 )
          = ( modulo_modulo_nat @ N @ Q2 ) )
        = ( dvd_dvd_nat @ Q2 @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% mod_eq_dvd_iff_nat
thf(fact_4825_prod__decode__aux_Ocases,axiom,
    ! [X3: product_prod_nat_nat] :
      ~ ! [K3: nat,M4: nat] :
          ( X3
         != ( product_Pair_nat_nat @ K3 @ M4 ) ) ).

% prod_decode_aux.cases
thf(fact_4826_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M ) ) ) ) ).

% finite_divisors_nat
thf(fact_4827_even__zero,axiom,
    dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).

% even_zero
thf(fact_4828_even__zero,axiom,
    dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).

% even_zero
thf(fact_4829_even__zero,axiom,
    dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).

% even_zero
thf(fact_4830_is__unitE,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B3: code_integer] :
              ( ( B3 != zero_z3403309356797280102nteger )
             => ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
               => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
                    = B3 )
                 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B3 )
                      = A )
                   => ( ( ( times_3573771949741848930nteger @ A @ B3 )
                        = one_one_Code_integer )
                     => ( ( divide6298287555418463151nteger @ C @ A )
                       != ( times_3573771949741848930nteger @ C @ B3 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_4831_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [B3: nat] :
              ( ( B3 != zero_zero_nat )
             => ( ( dvd_dvd_nat @ B3 @ one_one_nat )
               => ( ( ( divide_divide_nat @ one_one_nat @ A )
                    = B3 )
                 => ( ( ( divide_divide_nat @ one_one_nat @ B3 )
                      = A )
                   => ( ( ( times_times_nat @ A @ B3 )
                        = one_one_nat )
                     => ( ( divide_divide_nat @ C @ A )
                       != ( times_times_nat @ C @ B3 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_4832_is__unitE,axiom,
    ! [A: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [B3: int] :
              ( ( B3 != zero_zero_int )
             => ( ( dvd_dvd_int @ B3 @ one_one_int )
               => ( ( ( divide_divide_int @ one_one_int @ A )
                    = B3 )
                 => ( ( ( divide_divide_int @ one_one_int @ B3 )
                      = A )
                   => ( ( ( times_times_int @ A @ B3 )
                        = one_one_int )
                     => ( ( divide_divide_int @ C @ A )
                       != ( times_times_int @ C @ B3 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_4833_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B2 ) )
          = ( divide6298287555418463151nteger @ one_one_Code_integer @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_4834_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B2 ) )
          = ( divide_divide_nat @ one_one_nat @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_4835_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B2: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ B2 @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B2 ) )
          = ( divide_divide_int @ one_one_int @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_4836_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B2 @ A ) )
          = ( divide6298287555418463151nteger @ one_one_Code_integer @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_4837_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ A ) )
          = ( divide_divide_nat @ one_one_nat @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_4838_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B2: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ B2 @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ A ) )
          = ( divide_divide_int @ one_one_int @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_4839_evenE,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ~ ! [B3: code_integer] :
            ( A
           != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) ) ).

% evenE
thf(fact_4840_evenE,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B3: nat] :
            ( A
           != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) ).

% evenE
thf(fact_4841_evenE,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B3: int] :
            ( A
           != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) ) ).

% evenE
thf(fact_4842_odd__one,axiom,
    ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).

% odd_one
thf(fact_4843_odd__one,axiom,
    ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).

% odd_one
thf(fact_4844_odd__one,axiom,
    ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).

% odd_one
thf(fact_4845_odd__even__add,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 )
       => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) ) ) ) ).

% odd_even_add
thf(fact_4846_odd__even__add,axiom,
    ! [A: nat,B2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
       => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% odd_even_add
thf(fact_4847_odd__even__add,axiom,
    ! [A: int,B2: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 )
       => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) ) ) ) ).

% odd_even_add
thf(fact_4848_bit__eq__rec,axiom,
    ( ( ^ [Y6: code_integer,Z5: code_integer] : Y6 = Z5 )
    = ( ^ [A5: code_integer,B4: code_integer] :
          ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 )
            = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B4 ) )
          & ( ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = ( divide6298287555418463151nteger @ B4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_4849_bit__eq__rec,axiom,
    ( ( ^ [Y6: nat,Z5: nat] : Y6 = Z5 )
    = ( ^ [A5: nat,B4: nat] :
          ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 )
            = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) )
          & ( ( divide_divide_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( divide_divide_nat @ B4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_4850_bit__eq__rec,axiom,
    ( ( ^ [Y6: int,Z5: int] : Y6 = Z5 )
    = ( ^ [A5: int,B4: int] :
          ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 )
            = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) )
          & ( ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = ( divide_divide_int @ B4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_4851_dvd__power__iff,axiom,
    ! [X3: code_integer,M: nat,N: nat] :
      ( ( X3 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X3 @ M ) @ ( power_8256067586552552935nteger @ X3 @ N ) )
        = ( ( dvd_dvd_Code_integer @ X3 @ one_one_Code_integer )
          | ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_4852_dvd__power__iff,axiom,
    ! [X3: nat,M: nat,N: nat] :
      ( ( X3 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ X3 @ M ) @ ( power_power_nat @ X3 @ N ) )
        = ( ( dvd_dvd_nat @ X3 @ one_one_nat )
          | ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_4853_dvd__power__iff,axiom,
    ! [X3: int,M: nat,N: nat] :
      ( ( X3 != zero_zero_int )
     => ( ( dvd_dvd_int @ ( power_power_int @ X3 @ M ) @ ( power_power_int @ X3 @ N ) )
        = ( ( dvd_dvd_int @ X3 @ one_one_int )
          | ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_4854_dvd__power,axiom,
    ! [N: nat,X3: code_integer] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X3 = one_one_Code_integer ) )
     => ( dvd_dvd_Code_integer @ X3 @ ( power_8256067586552552935nteger @ X3 @ N ) ) ) ).

% dvd_power
thf(fact_4855_dvd__power,axiom,
    ! [N: nat,X3: rat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X3 = one_one_rat ) )
     => ( dvd_dvd_rat @ X3 @ ( power_power_rat @ X3 @ N ) ) ) ).

% dvd_power
thf(fact_4856_dvd__power,axiom,
    ! [N: nat,X3: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X3 = one_one_nat ) )
     => ( dvd_dvd_nat @ X3 @ ( power_power_nat @ X3 @ N ) ) ) ).

% dvd_power
thf(fact_4857_dvd__power,axiom,
    ! [N: nat,X3: real] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X3 = one_one_real ) )
     => ( dvd_dvd_real @ X3 @ ( power_power_real @ X3 @ N ) ) ) ).

% dvd_power
thf(fact_4858_dvd__power,axiom,
    ! [N: nat,X3: int] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X3 = one_one_int ) )
     => ( dvd_dvd_int @ X3 @ ( power_power_int @ X3 @ N ) ) ) ).

% dvd_power
thf(fact_4859_dvd__power,axiom,
    ! [N: nat,X3: complex] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X3 = one_one_complex ) )
     => ( dvd_dvd_complex @ X3 @ ( power_power_complex @ X3 @ N ) ) ) ).

% dvd_power
thf(fact_4860_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_4861_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_4862_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_4863_dvd__minus__add,axiom,
    ! [Q2: nat,N: nat,R: nat,M: nat] :
      ( ( ord_less_eq_nat @ Q2 @ N )
     => ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R @ M ) )
       => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q2 ) )
          = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R @ M ) @ Q2 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_4864_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_4865_mod__nat__eqI,axiom,
    ! [R: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ R @ N )
     => ( ( ord_less_eq_nat @ R @ M )
       => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R ) )
         => ( ( modulo_modulo_nat @ M @ N )
            = R ) ) ) ) ).

% mod_nat_eqI
thf(fact_4866_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_4867_even__two__times__div__two,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_4868_even__two__times__div__two,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_4869_even__two__times__div__two,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_4870_even__iff__mod__2__eq__zero,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_4871_even__iff__mod__2__eq__zero,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_4872_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_4873_odd__iff__mod__2__eq__one,axiom,
    ! [A: nat] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_4874_odd__iff__mod__2__eq__one,axiom,
    ! [A: int] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_4875_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_integer] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_4876_power__mono__odd,axiom,
    ! [N: nat,A: real,B2: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ A @ B2 )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ).

% power_mono_odd
thf(fact_4877_power__mono__odd,axiom,
    ! [N: nat,A: rat,B2: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_rat @ A @ B2 )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ).

% power_mono_odd
thf(fact_4878_power__mono__odd,axiom,
    ! [N: nat,A: int,B2: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_int @ A @ B2 )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).

% power_mono_odd
thf(fact_4879_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_4880_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_4881_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_4882_even__unset__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_4883_even__unset__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_4884_even__unset__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_4885_even__set__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_4886_even__set__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_4887_even__set__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_4888_even__flip__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_4889_even__flip__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_4890_even__flip__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_4891_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_4892_oddE,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ~ ! [B3: code_integer] :
            ( A
           != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) @ one_one_Code_integer ) ) ) ).

% oddE
thf(fact_4893_oddE,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B3: nat] :
            ( A
           != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) @ one_one_nat ) ) ) ).

% oddE
thf(fact_4894_oddE,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B3: int] :
            ( A
           != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ one_one_int ) ) ) ).

% oddE
thf(fact_4895_parity__cases,axiom,
    ! [A: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
         != zero_zero_nat ) )
     => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
           != one_one_nat ) ) ) ).

% parity_cases
thf(fact_4896_parity__cases,axiom,
    ! [A: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
         != zero_zero_int ) )
     => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
           != one_one_int ) ) ) ).

% parity_cases
thf(fact_4897_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
         != zero_z3403309356797280102nteger ) )
     => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
           != one_one_Code_integer ) ) ) ).

% parity_cases
thf(fact_4898_mod2__eq__if,axiom,
    ! [A: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
          = zero_zero_nat ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
          = one_one_nat ) ) ) ).

% mod2_eq_if
thf(fact_4899_mod2__eq__if,axiom,
    ! [A: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
          = zero_zero_int ) )
      & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
          = one_one_int ) ) ) ).

% mod2_eq_if
thf(fact_4900_mod2__eq__if,axiom,
    ! [A: code_integer] :
      ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          = zero_z3403309356797280102nteger ) )
      & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          = one_one_Code_integer ) ) ) ).

% mod2_eq_if
thf(fact_4901_zero__le__even__power,axiom,
    ! [N: nat,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).

% zero_le_even_power
thf(fact_4902_zero__le__even__power,axiom,
    ! [N: nat,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).

% zero_le_even_power
thf(fact_4903_zero__le__even__power,axiom,
    ! [N: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).

% zero_le_even_power
thf(fact_4904_zero__le__odd__power,axiom,
    ! [N: nat,A: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).

% zero_le_odd_power
thf(fact_4905_zero__le__odd__power,axiom,
    ! [N: nat,A: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).

% zero_le_odd_power
thf(fact_4906_zero__le__odd__power,axiom,
    ! [N: nat,A: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% zero_le_odd_power
thf(fact_4907_zero__le__power__eq,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_4908_zero__le__power__eq,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_4909_zero__le__power__eq,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_4910_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_4911_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_4912_zero__less__power__eq,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
      = ( ( N = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( A != zero_zero_real ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_4913_zero__less__power__eq,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
      = ( ( N = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( A != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_4914_zero__less__power__eq,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
      = ( ( N = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( A != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_4915_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_4916_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_4917_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_4918_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_4919_power__le__zero__eq,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
      = ( ( ord_less_nat @ zero_zero_nat @ N )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( ord_less_eq_real @ A @ zero_zero_real ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_4920_power__le__zero__eq,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
      = ( ( ord_less_nat @ zero_zero_nat @ N )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_4921_power__le__zero__eq,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ N )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( ord_less_eq_int @ A @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_4922_option_Osize__gen_I1_J,axiom,
    ! [X3: nat > nat] :
      ( ( size_option_nat @ X3 @ none_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_4923_option_Osize__gen_I1_J,axiom,
    ! [X3: product_prod_nat_nat > nat] :
      ( ( size_o8335143837870341156at_nat @ X3 @ none_P5556105721700978146at_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_4924_option_Osize__gen_I1_J,axiom,
    ! [X3: num > nat] :
      ( ( size_option_num @ X3 @ none_num )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_4925_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_4926_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_4927_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_4928_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_4929_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M )
        | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
          = zero_z3403309356797280102nteger )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_4930_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M )
        | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          = zero_zero_nat )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_4931_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M )
        | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
          = zero_zero_int )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_4932_ex__min__if__finite,axiom,
    ! [S2: set_real] :
      ( ( finite_finite_real @ S2 )
     => ( ( S2 != bot_bot_set_real )
       => ? [X5: real] :
            ( ( member_real @ X5 @ S2 )
            & ~ ? [Xa: real] :
                  ( ( member_real @ Xa @ S2 )
                  & ( ord_less_real @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_4933_ex__min__if__finite,axiom,
    ! [S2: set_rat] :
      ( ( finite_finite_rat @ S2 )
     => ( ( S2 != bot_bot_set_rat )
       => ? [X5: rat] :
            ( ( member_rat @ X5 @ S2 )
            & ~ ? [Xa: rat] :
                  ( ( member_rat @ Xa @ S2 )
                  & ( ord_less_rat @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_4934_ex__min__if__finite,axiom,
    ! [S2: set_num] :
      ( ( finite_finite_num @ S2 )
     => ( ( S2 != bot_bot_set_num )
       => ? [X5: num] :
            ( ( member_num @ X5 @ S2 )
            & ~ ? [Xa: num] :
                  ( ( member_num @ Xa @ S2 )
                  & ( ord_less_num @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_4935_ex__min__if__finite,axiom,
    ! [S2: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( S2 != bot_bot_set_nat )
       => ? [X5: nat] :
            ( ( member_nat @ X5 @ S2 )
            & ~ ? [Xa: nat] :
                  ( ( member_nat @ Xa @ S2 )
                  & ( ord_less_nat @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_4936_ex__min__if__finite,axiom,
    ! [S2: set_int] :
      ( ( finite_finite_int @ S2 )
     => ( ( S2 != bot_bot_set_int )
       => ? [X5: int] :
            ( ( member_int @ X5 @ S2 )
            & ~ ? [Xa: int] :
                  ( ( member_int @ Xa @ S2 )
                  & ( ord_less_int @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_4937_triangle__def,axiom,
    ( nat_triangle
    = ( ^ [N2: nat] : ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% triangle_def
thf(fact_4938_vebt__buildup_Oelims,axiom,
    ! [X3: nat,Y3: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X3 )
        = Y3 )
     => ( ( ( X3 = zero_zero_nat )
         => ( Y3
           != ( vEBT_Leaf @ $false @ $false ) ) )
       => ( ( ( X3
              = ( suc @ zero_zero_nat ) )
           => ( Y3
             != ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Va2: nat] :
                ( ( X3
                  = ( suc @ ( suc @ Va2 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                     => ( Y3
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                     => ( Y3
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
thf(fact_4939_signed__take__bit__rec,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N2: nat,A5: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_4940_signed__take__bit__rec,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N2: nat,A5: int] : ( if_int @ ( N2 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_4941_flip__bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
      = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_4942_flip__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
      = ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_4943_flip__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
      = ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_4944_set__decode__Suc,axiom,
    ! [N: nat,X3: nat] :
      ( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X3 ) )
      = ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_decode_Suc
thf(fact_4945_diff__shunt__var,axiom,
    ! [X3: set_nat,Y3: set_nat] :
      ( ( ( minus_minus_set_nat @ X3 @ Y3 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ X3 @ Y3 ) ) ).

% diff_shunt_var
thf(fact_4946_diff__shunt__var,axiom,
    ! [X3: set_real,Y3: set_real] :
      ( ( ( minus_minus_set_real @ X3 @ Y3 )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ X3 @ Y3 ) ) ).

% diff_shunt_var
thf(fact_4947_diff__shunt__var,axiom,
    ! [X3: set_int,Y3: set_int] :
      ( ( ( minus_minus_set_int @ X3 @ Y3 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ X3 @ Y3 ) ) ).

% diff_shunt_var
thf(fact_4948_add__scale__eq__noteq,axiom,
    ! [R: complex,A: complex,B2: complex,C: complex,D: complex] :
      ( ( R != zero_zero_complex )
     => ( ( ( A = B2 )
          & ( C != D ) )
       => ( ( plus_plus_complex @ A @ ( times_times_complex @ R @ C ) )
         != ( plus_plus_complex @ B2 @ ( times_times_complex @ R @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4949_add__scale__eq__noteq,axiom,
    ! [R: real,A: real,B2: real,C: real,D: real] :
      ( ( R != zero_zero_real )
     => ( ( ( A = B2 )
          & ( C != D ) )
       => ( ( plus_plus_real @ A @ ( times_times_real @ R @ C ) )
         != ( plus_plus_real @ B2 @ ( times_times_real @ R @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4950_add__scale__eq__noteq,axiom,
    ! [R: rat,A: rat,B2: rat,C: rat,D: rat] :
      ( ( R != zero_zero_rat )
     => ( ( ( A = B2 )
          & ( C != D ) )
       => ( ( plus_plus_rat @ A @ ( times_times_rat @ R @ C ) )
         != ( plus_plus_rat @ B2 @ ( times_times_rat @ R @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4951_add__scale__eq__noteq,axiom,
    ! [R: nat,A: nat,B2: nat,C: nat,D: nat] :
      ( ( R != zero_zero_nat )
     => ( ( ( A = B2 )
          & ( C != D ) )
       => ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C ) )
         != ( plus_plus_nat @ B2 @ ( times_times_nat @ R @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4952_add__scale__eq__noteq,axiom,
    ! [R: int,A: int,B2: int,C: int,D: int] :
      ( ( R != zero_zero_int )
     => ( ( ( A = B2 )
          & ( C != D ) )
       => ( ( plus_plus_int @ A @ ( times_times_int @ R @ C ) )
         != ( plus_plus_int @ B2 @ ( times_times_int @ R @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4953_intind,axiom,
    ! [I: nat,N: nat,P: int > $o,X3: int] :
      ( ( ord_less_nat @ I @ N )
     => ( ( P @ X3 )
       => ( P @ ( nth_int @ ( replicate_int @ N @ X3 ) @ I ) ) ) ) ).

% intind
thf(fact_4954_intind,axiom,
    ! [I: nat,N: nat,P: nat > $o,X3: nat] :
      ( ( ord_less_nat @ I @ N )
     => ( ( P @ X3 )
       => ( P @ ( nth_nat @ ( replicate_nat @ N @ X3 ) @ I ) ) ) ) ).

% intind
thf(fact_4955_intind,axiom,
    ! [I: nat,N: nat,P: vEBT_VEBT > $o,X3: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ N )
     => ( ( P @ X3 )
       => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X3 ) @ I ) ) ) ) ).

% intind
thf(fact_4956_add_Oinverse__inverse,axiom,
    ! [A: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4957_add_Oinverse__inverse,axiom,
    ! [A: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4958_add_Oinverse__inverse,axiom,
    ! [A: complex] :
      ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4959_add_Oinverse__inverse,axiom,
    ! [A: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4960_add_Oinverse__inverse,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4961_neg__equal__iff__equal,axiom,
    ! [A: real,B2: real] :
      ( ( ( uminus_uminus_real @ A )
        = ( uminus_uminus_real @ B2 ) )
      = ( A = B2 ) ) ).

% neg_equal_iff_equal
thf(fact_4962_neg__equal__iff__equal,axiom,
    ! [A: int,B2: int] :
      ( ( ( uminus_uminus_int @ A )
        = ( uminus_uminus_int @ B2 ) )
      = ( A = B2 ) ) ).

% neg_equal_iff_equal
thf(fact_4963_neg__equal__iff__equal,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = ( uminus1482373934393186551omplex @ B2 ) )
      = ( A = B2 ) ) ).

% neg_equal_iff_equal
thf(fact_4964_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = ( uminus1351360451143612070nteger @ B2 ) )
      = ( A = B2 ) ) ).

% neg_equal_iff_equal
thf(fact_4965_neg__equal__iff__equal,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = ( uminus_uminus_rat @ B2 ) )
      = ( A = B2 ) ) ).

% neg_equal_iff_equal
thf(fact_4966_verit__minus__simplify_I4_J,axiom,
    ! [B2: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ B2 ) )
      = B2 ) ).

% verit_minus_simplify(4)
thf(fact_4967_verit__minus__simplify_I4_J,axiom,
    ! [B2: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ B2 ) )
      = B2 ) ).

% verit_minus_simplify(4)
thf(fact_4968_verit__minus__simplify_I4_J,axiom,
    ! [B2: complex] :
      ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ B2 ) )
      = B2 ) ).

% verit_minus_simplify(4)
thf(fact_4969_verit__minus__simplify_I4_J,axiom,
    ! [B2: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ B2 ) )
      = B2 ) ).

% verit_minus_simplify(4)
thf(fact_4970_verit__minus__simplify_I4_J,axiom,
    ! [B2: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ B2 ) )
      = B2 ) ).

% verit_minus_simplify(4)
thf(fact_4971_Compl__subset__Compl__iff,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A4 ) @ ( uminus1532241313380277803et_int @ B5 ) )
      = ( ord_less_eq_set_int @ B5 @ A4 ) ) ).

% Compl_subset_Compl_iff
thf(fact_4972_Compl__anti__mono,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ B5 )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B5 ) @ ( uminus1532241313380277803et_int @ A4 ) ) ) ).

% Compl_anti_mono
thf(fact_4973_compl__le__compl__iff,axiom,
    ! [X3: set_int,Y3: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X3 ) @ ( uminus1532241313380277803et_int @ Y3 ) )
      = ( ord_less_eq_set_int @ Y3 @ X3 ) ) ).

% compl_le_compl_iff
thf(fact_4974_neg__le__iff__le,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ B2 ) ) ).

% neg_le_iff_le
thf(fact_4975_neg__le__iff__le,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ B2 ) ) ).

% neg_le_iff_le
thf(fact_4976_neg__le__iff__le,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ B2 ) ) ).

% neg_le_iff_le
thf(fact_4977_neg__le__iff__le,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ B2 ) ) ).

% neg_le_iff_le
thf(fact_4978_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_real @ zero_zero_real )
    = zero_zero_real ) ).

% add.inverse_neutral
thf(fact_4979_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_4980_add_Oinverse__neutral,axiom,
    ( ( uminus1482373934393186551omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% add.inverse_neutral
thf(fact_4981_add_Oinverse__neutral,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% add.inverse_neutral
thf(fact_4982_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% add.inverse_neutral
thf(fact_4983_neg__0__equal__iff__equal,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( uminus_uminus_real @ A ) )
      = ( zero_zero_real = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4984_neg__0__equal__iff__equal,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( uminus_uminus_int @ A ) )
      = ( zero_zero_int = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4985_neg__0__equal__iff__equal,axiom,
    ! [A: complex] :
      ( ( zero_zero_complex
        = ( uminus1482373934393186551omplex @ A ) )
      = ( zero_zero_complex = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4986_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( uminus1351360451143612070nteger @ A ) )
      = ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4987_neg__0__equal__iff__equal,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( uminus_uminus_rat @ A ) )
      = ( zero_zero_rat = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4988_neg__equal__0__iff__equal,axiom,
    ! [A: real] :
      ( ( ( uminus_uminus_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% neg_equal_0_iff_equal
thf(fact_4989_neg__equal__0__iff__equal,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
thf(fact_4990_neg__equal__0__iff__equal,axiom,
    ! [A: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
thf(fact_4991_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
thf(fact_4992_neg__equal__0__iff__equal,axiom,
    ! [A: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
thf(fact_4993_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A
        = ( uminus_uminus_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% equal_neg_zero
thf(fact_4994_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A
        = ( uminus_uminus_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_4995_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
thf(fact_4996_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A
        = ( uminus_uminus_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% equal_neg_zero
thf(fact_4997_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( ( uminus_uminus_real @ A )
        = A )
      = ( A = zero_zero_real ) ) ).

% neg_equal_zero
thf(fact_4998_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = A )
      = ( A = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_4999_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = A )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
thf(fact_5000_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = A )
      = ( A = zero_zero_rat ) ) ).

% neg_equal_zero
thf(fact_5001_neg__less__iff__less,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) )
      = ( ord_less_real @ A @ B2 ) ) ).

% neg_less_iff_less
thf(fact_5002_neg__less__iff__less,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ B2 ) ) ).

% neg_less_iff_less
thf(fact_5003_neg__less__iff__less,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ B2 ) ) ).

% neg_less_iff_less
thf(fact_5004_neg__less__iff__less,axiom,
    ! [B2: rat,A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ B2 ) ) ).

% neg_less_iff_less
thf(fact_5005_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_5006_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_5007_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( M = N ) ) ).

% neg_numeral_eq_iff
thf(fact_5008_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( M = N ) ) ).

% neg_numeral_eq_iff
thf(fact_5009_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_5010_mult__minus__left,axiom,
    ! [A: real,B2: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B2 )
      = ( uminus_uminus_real @ ( times_times_real @ A @ B2 ) ) ) ).

% mult_minus_left
thf(fact_5011_mult__minus__left,axiom,
    ! [A: int,B2: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B2 )
      = ( uminus_uminus_int @ ( times_times_int @ A @ B2 ) ) ) ).

% mult_minus_left
thf(fact_5012_mult__minus__left,axiom,
    ! [A: complex,B2: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B2 ) ) ) ).

% mult_minus_left
thf(fact_5013_mult__minus__left,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
      = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B2 ) ) ) ).

% mult_minus_left
thf(fact_5014_mult__minus__left,axiom,
    ! [A: rat,B2: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B2 )
      = ( uminus_uminus_rat @ ( times_times_rat @ A @ B2 ) ) ) ).

% mult_minus_left
thf(fact_5015_minus__mult__minus,axiom,
    ! [A: real,B2: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
      = ( times_times_real @ A @ B2 ) ) ).

% minus_mult_minus
thf(fact_5016_minus__mult__minus,axiom,
    ! [A: int,B2: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
      = ( times_times_int @ A @ B2 ) ) ).

% minus_mult_minus
thf(fact_5017_minus__mult__minus,axiom,
    ! [A: complex,B2: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) )
      = ( times_times_complex @ A @ B2 ) ) ).

% minus_mult_minus
thf(fact_5018_minus__mult__minus,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( times_3573771949741848930nteger @ A @ B2 ) ) ).

% minus_mult_minus
thf(fact_5019_minus__mult__minus,axiom,
    ! [A: rat,B2: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) )
      = ( times_times_rat @ A @ B2 ) ) ).

% minus_mult_minus
thf(fact_5020_mult__minus__right,axiom,
    ! [A: real,B2: real] :
      ( ( times_times_real @ A @ ( uminus_uminus_real @ B2 ) )
      = ( uminus_uminus_real @ ( times_times_real @ A @ B2 ) ) ) ).

% mult_minus_right
thf(fact_5021_mult__minus__right,axiom,
    ! [A: int,B2: int] :
      ( ( times_times_int @ A @ ( uminus_uminus_int @ B2 ) )
      = ( uminus_uminus_int @ ( times_times_int @ A @ B2 ) ) ) ).

% mult_minus_right
thf(fact_5022_mult__minus__right,axiom,
    ! [A: complex,B2: complex] :
      ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B2 ) )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B2 ) ) ) ).

% mult_minus_right
thf(fact_5023_mult__minus__right,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B2 ) ) ) ).

% mult_minus_right
thf(fact_5024_mult__minus__right,axiom,
    ! [A: rat,B2: rat] :
      ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B2 ) )
      = ( uminus_uminus_rat @ ( times_times_rat @ A @ B2 ) ) ) ).

% mult_minus_right
thf(fact_5025_minus__add__distrib,axiom,
    ! [A: real,B2: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B2 ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) ) ) ).

% minus_add_distrib
thf(fact_5026_minus__add__distrib,axiom,
    ! [A: int,B2: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B2 ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) ) ) ).

% minus_add_distrib
thf(fact_5027_minus__add__distrib,axiom,
    ! [A: complex,B2: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B2 ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) ) ) ).

% minus_add_distrib
thf(fact_5028_minus__add__distrib,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) ) ) ).

% minus_add_distrib
thf(fact_5029_minus__add__distrib,axiom,
    ! [A: rat,B2: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B2 ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) ) ) ).

% minus_add_distrib
thf(fact_5030_minus__add__cancel,axiom,
    ! [A: real,B2: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B2 ) )
      = B2 ) ).

% minus_add_cancel
thf(fact_5031_minus__add__cancel,axiom,
    ! [A: int,B2: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B2 ) )
      = B2 ) ).

% minus_add_cancel
thf(fact_5032_minus__add__cancel,axiom,
    ! [A: complex,B2: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B2 ) )
      = B2 ) ).

% minus_add_cancel
thf(fact_5033_minus__add__cancel,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
      = B2 ) ).

% minus_add_cancel
thf(fact_5034_minus__add__cancel,axiom,
    ! [A: rat,B2: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B2 ) )
      = B2 ) ).

% minus_add_cancel
thf(fact_5035_add__minus__cancel,axiom,
    ! [A: real,B2: real] :
      ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B2 ) )
      = B2 ) ).

% add_minus_cancel
thf(fact_5036_add__minus__cancel,axiom,
    ! [A: int,B2: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B2 ) )
      = B2 ) ).

% add_minus_cancel
thf(fact_5037_add__minus__cancel,axiom,
    ! [A: complex,B2: complex] :
      ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 ) )
      = B2 ) ).

% add_minus_cancel
thf(fact_5038_add__minus__cancel,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) )
      = B2 ) ).

% add_minus_cancel
thf(fact_5039_add__minus__cancel,axiom,
    ! [A: rat,B2: rat] :
      ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B2 ) )
      = B2 ) ).

% add_minus_cancel
thf(fact_5040_minus__diff__eq,axiom,
    ! [A: real,B2: real] :
      ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B2 ) )
      = ( minus_minus_real @ B2 @ A ) ) ).

% minus_diff_eq
thf(fact_5041_minus__diff__eq,axiom,
    ! [A: int,B2: int] :
      ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B2 ) )
      = ( minus_minus_int @ B2 @ A ) ) ).

% minus_diff_eq
thf(fact_5042_minus__diff__eq,axiom,
    ! [A: complex,B2: complex] :
      ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B2 ) )
      = ( minus_minus_complex @ B2 @ A ) ) ).

% minus_diff_eq
thf(fact_5043_minus__diff__eq,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) )
      = ( minus_8373710615458151222nteger @ B2 @ A ) ) ).

% minus_diff_eq
thf(fact_5044_minus__diff__eq,axiom,
    ! [A: rat,B2: rat] :
      ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B2 ) )
      = ( minus_minus_rat @ B2 @ A ) ) ).

% minus_diff_eq
thf(fact_5045_div__minus__minus,axiom,
    ! [A: int,B2: int] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
      = ( divide_divide_int @ A @ B2 ) ) ).

% div_minus_minus
thf(fact_5046_div__minus__minus,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( divide6298287555418463151nteger @ A @ B2 ) ) ).

% div_minus_minus
thf(fact_5047_dvd__minus__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( dvd_dvd_real @ X3 @ ( uminus_uminus_real @ Y3 ) )
      = ( dvd_dvd_real @ X3 @ Y3 ) ) ).

% dvd_minus_iff
thf(fact_5048_dvd__minus__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( dvd_dvd_int @ X3 @ ( uminus_uminus_int @ Y3 ) )
      = ( dvd_dvd_int @ X3 @ Y3 ) ) ).

% dvd_minus_iff
thf(fact_5049_dvd__minus__iff,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( dvd_dvd_complex @ X3 @ ( uminus1482373934393186551omplex @ Y3 ) )
      = ( dvd_dvd_complex @ X3 @ Y3 ) ) ).

% dvd_minus_iff
thf(fact_5050_dvd__minus__iff,axiom,
    ! [X3: code_integer,Y3: code_integer] :
      ( ( dvd_dvd_Code_integer @ X3 @ ( uminus1351360451143612070nteger @ Y3 ) )
      = ( dvd_dvd_Code_integer @ X3 @ Y3 ) ) ).

% dvd_minus_iff
thf(fact_5051_dvd__minus__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( dvd_dvd_rat @ X3 @ ( uminus_uminus_rat @ Y3 ) )
      = ( dvd_dvd_rat @ X3 @ Y3 ) ) ).

% dvd_minus_iff
thf(fact_5052_minus__dvd__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( dvd_dvd_real @ ( uminus_uminus_real @ X3 ) @ Y3 )
      = ( dvd_dvd_real @ X3 @ Y3 ) ) ).

% minus_dvd_iff
thf(fact_5053_minus__dvd__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( dvd_dvd_int @ ( uminus_uminus_int @ X3 ) @ Y3 )
      = ( dvd_dvd_int @ X3 @ Y3 ) ) ).

% minus_dvd_iff
thf(fact_5054_minus__dvd__iff,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X3 ) @ Y3 )
      = ( dvd_dvd_complex @ X3 @ Y3 ) ) ).

% minus_dvd_iff
thf(fact_5055_minus__dvd__iff,axiom,
    ! [X3: code_integer,Y3: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X3 ) @ Y3 )
      = ( dvd_dvd_Code_integer @ X3 @ Y3 ) ) ).

% minus_dvd_iff
thf(fact_5056_minus__dvd__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X3 ) @ Y3 )
      = ( dvd_dvd_rat @ X3 @ Y3 ) ) ).

% minus_dvd_iff
thf(fact_5057_mod__minus__minus,axiom,
    ! [A: int,B2: int] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B2 ) ) ) ).

% mod_minus_minus
thf(fact_5058_mod__minus__minus,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ).

% mod_minus_minus
thf(fact_5059_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_5060_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_5061_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_5062_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_5063_of__bool__eq_I1_J,axiom,
    ( ( zero_n1201886186963655149omplex @ $false )
    = zero_zero_complex ) ).

% of_bool_eq(1)
thf(fact_5064_of__bool__eq_I1_J,axiom,
    ( ( zero_n3304061248610475627l_real @ $false )
    = zero_zero_real ) ).

% of_bool_eq(1)
thf(fact_5065_of__bool__eq_I1_J,axiom,
    ( ( zero_n2052037380579107095ol_rat @ $false )
    = zero_zero_rat ) ).

% of_bool_eq(1)
thf(fact_5066_of__bool__eq_I1_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $false )
    = zero_zero_nat ) ).

% of_bool_eq(1)
thf(fact_5067_of__bool__eq_I1_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $false )
    = zero_zero_int ) ).

% of_bool_eq(1)
thf(fact_5068_of__bool__eq_I1_J,axiom,
    ( ( zero_n356916108424825756nteger @ $false )
    = zero_z3403309356797280102nteger ) ).

% of_bool_eq(1)
thf(fact_5069_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n1201886186963655149omplex @ P )
        = zero_zero_complex )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_5070_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n3304061248610475627l_real @ P )
        = zero_zero_real )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_5071_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2052037380579107095ol_rat @ P )
        = zero_zero_rat )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_5072_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = zero_zero_nat )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_5073_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = zero_zero_int )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_5074_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n356916108424825756nteger @ P )
        = zero_z3403309356797280102nteger )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_5075_real__add__minus__iff,axiom,
    ! [X3: real,A: real] :
      ( ( ( plus_plus_real @ X3 @ ( uminus_uminus_real @ A ) )
        = zero_zero_real )
      = ( X3 = A ) ) ).

% real_add_minus_iff
thf(fact_5076_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_5077_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_5078_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_5079_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_5080_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_5081_of__bool__eq_I2_J,axiom,
    ( ( zero_n1201886186963655149omplex @ $true )
    = one_one_complex ) ).

% of_bool_eq(2)
thf(fact_5082_of__bool__eq_I2_J,axiom,
    ( ( zero_n3304061248610475627l_real @ $true )
    = one_one_real ) ).

% of_bool_eq(2)
thf(fact_5083_of__bool__eq_I2_J,axiom,
    ( ( zero_n2052037380579107095ol_rat @ $true )
    = one_one_rat ) ).

% of_bool_eq(2)
thf(fact_5084_of__bool__eq_I2_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $true )
    = one_one_nat ) ).

% of_bool_eq(2)
thf(fact_5085_of__bool__eq_I2_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $true )
    = one_one_int ) ).

% of_bool_eq(2)
thf(fact_5086_of__bool__eq_I2_J,axiom,
    ( ( zero_n356916108424825756nteger @ $true )
    = one_one_Code_integer ) ).

% of_bool_eq(2)
thf(fact_5087_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n1201886186963655149omplex @ P )
        = one_one_complex )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_5088_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n3304061248610475627l_real @ P )
        = one_one_real )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_5089_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2052037380579107095ol_rat @ P )
        = one_one_rat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_5090_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = one_one_nat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_5091_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = one_one_int )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_5092_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n356916108424825756nteger @ P )
        = one_one_Code_integer )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_5093_length__replicate,axiom,
    ! [N: nat,X3: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N @ X3 ) )
      = N ) ).

% length_replicate
thf(fact_5094_length__replicate,axiom,
    ! [N: nat,X3: $o] :
      ( ( size_size_list_o @ ( replicate_o @ N @ X3 ) )
      = N ) ).

% length_replicate
thf(fact_5095_length__replicate,axiom,
    ! [N: nat,X3: nat] :
      ( ( size_size_list_nat @ ( replicate_nat @ N @ X3 ) )
      = N ) ).

% length_replicate
thf(fact_5096_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2687167440665602831ol_nat
        @ ( P
          | Q ) )
      = ( ord_max_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).

% of_bool_or_iff
thf(fact_5097_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2684676970156552555ol_int
        @ ( P
          | Q ) )
      = ( ord_max_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).

% of_bool_or_iff
thf(fact_5098_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n356916108424825756nteger
        @ ( P
          | Q ) )
      = ( ord_max_Code_integer @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).

% of_bool_or_iff
thf(fact_5099_neg__less__eq__nonneg,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_5100_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_5101_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_5102_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_5103_less__eq__neg__nonpos,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% less_eq_neg_nonpos
thf(fact_5104_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_eq_neg_nonpos
thf(fact_5105_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% less_eq_neg_nonpos
thf(fact_5106_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% less_eq_neg_nonpos
thf(fact_5107_neg__le__0__iff__le,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% neg_le_0_iff_le
thf(fact_5108_neg__le__0__iff__le,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_le_0_iff_le
thf(fact_5109_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% neg_le_0_iff_le
thf(fact_5110_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% neg_le_0_iff_le
thf(fact_5111_neg__0__le__iff__le,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% neg_0_le_iff_le
thf(fact_5112_neg__0__le__iff__le,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% neg_0_le_iff_le
thf(fact_5113_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% neg_0_le_iff_le
thf(fact_5114_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% neg_0_le_iff_le
thf(fact_5115_neg__less__0__iff__less,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% neg_less_0_iff_less
thf(fact_5116_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% neg_less_0_iff_less
thf(fact_5117_neg__less__0__iff__less,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_0_iff_less
thf(fact_5118_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% neg_less_0_iff_less
thf(fact_5119_neg__0__less__iff__less,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% neg_0_less_iff_less
thf(fact_5120_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% neg_0_less_iff_less
thf(fact_5121_neg__0__less__iff__less,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% neg_0_less_iff_less
thf(fact_5122_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% neg_0_less_iff_less
thf(fact_5123_neg__less__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% neg_less_pos
thf(fact_5124_neg__less__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% neg_less_pos
thf(fact_5125_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_pos
thf(fact_5126_neg__less__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% neg_less_pos
thf(fact_5127_less__neg__neg,axiom,
    ! [A: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% less_neg_neg
thf(fact_5128_less__neg__neg,axiom,
    ! [A: int] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% less_neg_neg
thf(fact_5129_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_neg_neg
thf(fact_5130_less__neg__neg,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% less_neg_neg
thf(fact_5131_ab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_left_minus
thf(fact_5132_ab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_5133_ab__left__minus,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
      = zero_zero_complex ) ).

% ab_left_minus
thf(fact_5134_ab__left__minus,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = zero_z3403309356797280102nteger ) ).

% ab_left_minus
thf(fact_5135_ab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_left_minus
thf(fact_5136_add_Oright__inverse,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
      = zero_zero_real ) ).

% add.right_inverse
thf(fact_5137_add_Oright__inverse,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_5138_add_Oright__inverse,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
      = zero_zero_complex ) ).

% add.right_inverse
thf(fact_5139_add_Oright__inverse,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = zero_z3403309356797280102nteger ) ).

% add.right_inverse
thf(fact_5140_add_Oright__inverse,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
      = zero_zero_rat ) ).

% add.right_inverse
thf(fact_5141_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_5142_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_5143_verit__minus__simplify_I3_J,axiom,
    ! [B2: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ B2 )
      = ( uminus1482373934393186551omplex @ B2 ) ) ).

% verit_minus_simplify(3)
thf(fact_5144_verit__minus__simplify_I3_J,axiom,
    ! [B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B2 )
      = ( uminus1351360451143612070nteger @ B2 ) ) ).

% verit_minus_simplify(3)
thf(fact_5145_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_5146_diff__0,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ zero_zero_real @ A )
      = ( uminus_uminus_real @ A ) ) ).

% diff_0
thf(fact_5147_diff__0,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ zero_zero_int @ A )
      = ( uminus_uminus_int @ A ) ) ).

% diff_0
thf(fact_5148_diff__0,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ A )
      = ( uminus1482373934393186551omplex @ A ) ) ).

% diff_0
thf(fact_5149_diff__0,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% diff_0
thf(fact_5150_diff__0,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ A )
      = ( uminus_uminus_rat @ A ) ) ).

% diff_0
thf(fact_5151_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_5152_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_5153_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_5154_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_5155_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_5156_mult__minus1,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
      = ( uminus_uminus_real @ Z ) ) ).

% mult_minus1
thf(fact_5157_mult__minus1,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1
thf(fact_5158_mult__minus1,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1
thf(fact_5159_mult__minus1,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
      = ( uminus1351360451143612070nteger @ Z ) ) ).

% mult_minus1
thf(fact_5160_mult__minus1,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1
thf(fact_5161_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_5162_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_5163_mult__minus1__right,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1_right
thf(fact_5164_mult__minus1__right,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ Z ) ) ).

% mult_minus1_right
thf(fact_5165_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_5166_uminus__add__conv__diff,axiom,
    ! [A: real,B2: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B2 )
      = ( minus_minus_real @ B2 @ A ) ) ).

% uminus_add_conv_diff
thf(fact_5167_uminus__add__conv__diff,axiom,
    ! [A: int,B2: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B2 )
      = ( minus_minus_int @ B2 @ A ) ) ).

% uminus_add_conv_diff
thf(fact_5168_uminus__add__conv__diff,axiom,
    ! [A: complex,B2: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 )
      = ( minus_minus_complex @ B2 @ A ) ) ).

% uminus_add_conv_diff
thf(fact_5169_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
      = ( minus_8373710615458151222nteger @ B2 @ A ) ) ).

% uminus_add_conv_diff
thf(fact_5170_uminus__add__conv__diff,axiom,
    ! [A: rat,B2: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B2 )
      = ( minus_minus_rat @ B2 @ A ) ) ).

% uminus_add_conv_diff
thf(fact_5171_diff__minus__eq__add,axiom,
    ! [A: real,B2: real] :
      ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B2 ) )
      = ( plus_plus_real @ A @ B2 ) ) ).

% diff_minus_eq_add
thf(fact_5172_diff__minus__eq__add,axiom,
    ! [A: int,B2: int] :
      ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B2 ) )
      = ( plus_plus_int @ A @ B2 ) ) ).

% diff_minus_eq_add
thf(fact_5173_diff__minus__eq__add,axiom,
    ! [A: complex,B2: complex] :
      ( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B2 ) )
      = ( plus_plus_complex @ A @ B2 ) ) ).

% diff_minus_eq_add
thf(fact_5174_diff__minus__eq__add,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( plus_p5714425477246183910nteger @ A @ B2 ) ) ).

% diff_minus_eq_add
thf(fact_5175_diff__minus__eq__add,axiom,
    ! [A: rat,B2: rat] :
      ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B2 ) )
      = ( plus_plus_rat @ A @ B2 ) ) ).

% diff_minus_eq_add
thf(fact_5176_divide__minus1,axiom,
    ! [X3: real] :
      ( ( divide_divide_real @ X3 @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ X3 ) ) ).

% divide_minus1
thf(fact_5177_divide__minus1,axiom,
    ! [X3: complex] :
      ( ( divide1717551699836669952omplex @ X3 @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ X3 ) ) ).

% divide_minus1
thf(fact_5178_divide__minus1,axiom,
    ! [X3: rat] :
      ( ( divide_divide_rat @ X3 @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ X3 ) ) ).

% divide_minus1
thf(fact_5179_div__minus1__right,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ A ) ) ).

% div_minus1_right
thf(fact_5180_div__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% div_minus1_right
thf(fact_5181_minus__mod__self1,axiom,
    ! [B2: int,A: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ B2 @ A ) @ B2 )
      = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).

% minus_mod_self1
thf(fact_5182_minus__mod__self1,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B2 @ A ) @ B2 )
      = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).

% minus_mod_self1
thf(fact_5183_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_5184_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_5185_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_5186_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_5187_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_5188_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_5189_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_5190_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_5191_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_5192_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_5193_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n1201886186963655149omplex @ ~ P )
      = ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P ) ) ) ).

% of_bool_not_iff
thf(fact_5194_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_5195_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_5196_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_5197_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_5198_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_5199_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] :
      ( ( bit_ri6519982836138164636nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% signed_take_bit_of_minus_1
thf(fact_5200_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_5201_Ball__set__replicate,axiom,
    ! [N: nat,A: int,P: int > $o] :
      ( ( ! [X: int] :
            ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
           => ( P @ X ) ) )
      = ( ( P @ A )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_5202_Ball__set__replicate,axiom,
    ! [N: nat,A: nat,P: nat > $o] :
      ( ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
           => ( P @ X ) ) )
      = ( ( P @ A )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_5203_Ball__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
           => ( P @ X ) ) )
      = ( ( P @ A )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_5204_Bex__set__replicate,axiom,
    ! [N: nat,A: int,P: int > $o] :
      ( ( ? [X: int] :
            ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
            & ( P @ X ) ) )
      = ( ( P @ A )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_5205_Bex__set__replicate,axiom,
    ! [N: nat,A: nat,P: nat > $o] :
      ( ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
            & ( P @ X ) ) )
      = ( ( P @ A )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_5206_Bex__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ? [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
            & ( P @ X ) ) )
      = ( ( P @ A )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_5207_in__set__replicate,axiom,
    ! [X3: real,N: nat,Y3: real] :
      ( ( member_real @ X3 @ ( set_real2 @ ( replicate_real @ N @ Y3 ) ) )
      = ( ( X3 = Y3 )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_5208_in__set__replicate,axiom,
    ! [X3: complex,N: nat,Y3: complex] :
      ( ( member_complex @ X3 @ ( set_complex2 @ ( replicate_complex @ N @ Y3 ) ) )
      = ( ( X3 = Y3 )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_5209_in__set__replicate,axiom,
    ! [X3: int,N: nat,Y3: int] :
      ( ( member_int @ X3 @ ( set_int2 @ ( replicate_int @ N @ Y3 ) ) )
      = ( ( X3 = Y3 )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_5210_in__set__replicate,axiom,
    ! [X3: nat,N: nat,Y3: nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N @ Y3 ) ) )
      = ( ( X3 = Y3 )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_5211_in__set__replicate,axiom,
    ! [X3: vEBT_VEBT,N: nat,Y3: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ Y3 ) ) )
      = ( ( X3 = Y3 )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_5212_nth__replicate,axiom,
    ! [I: nat,N: nat,X3: int] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_int @ ( replicate_int @ N @ X3 ) @ I )
        = X3 ) ) ).

% nth_replicate
thf(fact_5213_nth__replicate,axiom,
    ! [I: nat,N: nat,X3: nat] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_nat @ ( replicate_nat @ N @ X3 ) @ I )
        = X3 ) ) ).

% nth_replicate
thf(fact_5214_nth__replicate,axiom,
    ! [I: nat,N: nat,X3: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X3 ) @ I )
        = X3 ) ) ).

% nth_replicate
thf(fact_5215_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_5216_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_5217_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_5218_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_5219_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_5220_triangle__Suc,axiom,
    ! [N: nat] :
      ( ( nat_triangle @ ( suc @ N ) )
      = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).

% triangle_Suc
thf(fact_5221_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_5222_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_5223_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_5224_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
    = zero_z3403309356797280102nteger ) ).

% add_neg_numeral_special(8)
thf(fact_5225_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_5226_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_5227_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_5228_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_5229_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = zero_z3403309356797280102nteger ) ).

% add_neg_numeral_special(7)
thf(fact_5230_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_5231_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_5232_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_5233_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_5234_diff__numeral__special_I12_J,axiom,
    ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = zero_z3403309356797280102nteger ) ).

% diff_numeral_special(12)
thf(fact_5235_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_5236_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_5237_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_5238_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1482373934393186551omplex @ one_one_complex )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_5239_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_5240_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_5241_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_5242_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_5243_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_5244_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_5245_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_5246_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_5247_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_5248_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_5249_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_5250_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_5251_left__minus__one__mult__self,axiom,
    ! [N: nat,A: real] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_5252_left__minus__one__mult__self,axiom,
    ! [N: nat,A: int] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_5253_left__minus__one__mult__self,axiom,
    ! [N: nat,A: complex] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_5254_left__minus__one__mult__self,axiom,
    ! [N: nat,A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_5255_left__minus__one__mult__self,axiom,
    ! [N: nat,A: rat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_5256_mod__minus1__right,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = zero_zero_int ) ).

% mod_minus1_right
thf(fact_5257_mod__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = zero_z3403309356797280102nteger ) ).

% mod_minus1_right
thf(fact_5258_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
      & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
          = ( numeral_numeral_real @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_5259_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( numera6620942414471956472nteger @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_5260_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
      & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( numeral_numeral_rat @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_5261_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( numeral_numeral_int @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_5262_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
          = ( numeral_numeral_real @ V ) ) )
      & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_5263_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
          = ( numera6620942414471956472nteger @ V ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_5264_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ V ) ) )
      & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_5265_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
          = ( numeral_numeral_int @ V ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_5266_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
      & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_5267_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_5268_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
      & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_5269_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_5270_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y3: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(168)
thf(fact_5271_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y3: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(168)
thf(fact_5272_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y3: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(168)
thf(fact_5273_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y3: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(168)
thf(fact_5274_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y3: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(168)
thf(fact_5275_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_5276_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_5277_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_5278_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_5279_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_5280_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_5281_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_5282_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_5283_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_5284_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_5285_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y3: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
      = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).

% semiring_norm(172)
thf(fact_5286_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y3: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).

% semiring_norm(172)
thf(fact_5287_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y3: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).

% semiring_norm(172)
thf(fact_5288_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y3: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).

% semiring_norm(172)
thf(fact_5289_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y3: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).

% semiring_norm(172)
thf(fact_5290_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y3: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
      = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(171)
thf(fact_5291_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y3: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(171)
thf(fact_5292_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y3: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(171)
thf(fact_5293_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y3: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
      = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(171)
thf(fact_5294_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y3: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(171)
thf(fact_5295_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y3: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y3 ) )
      = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(170)
thf(fact_5296_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y3: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y3 ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(170)
thf(fact_5297_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y3: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y3 ) )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(170)
thf(fact_5298_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y3: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y3 ) )
      = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(170)
thf(fact_5299_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y3: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y3 ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).

% semiring_norm(170)
thf(fact_5300_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_5301_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_5302_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_5303_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_5304_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_5305_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_5306_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_5307_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_5308_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_5309_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_5310_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_5311_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_5312_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_5313_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_5314_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_5315_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_5316_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( ord_less_eq_num @ N @ M ) ) ).

% neg_numeral_le_iff
thf(fact_5317_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_5318_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_5319_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_5320_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_5321_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( ord_less_num @ N @ M ) ) ).

% neg_numeral_less_iff
thf(fact_5322_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_5323_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_5324_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_5325_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_5326_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_5327_divide__le__eq__numeral1_I2_J,axiom,
    ! [B2: real,W: num,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
      = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B2 ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_5328_divide__le__eq__numeral1_I2_J,axiom,
    ! [B2: rat,W: num,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
      = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B2 ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_5329_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B2: real,W: num] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_5330_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B2: rat,W: num] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_5331_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B2: real,W: num,A: real] :
      ( ( ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
        = A )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
           != zero_zero_real )
         => ( B2
            = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_5332_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B2: complex,W: num,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
        = A )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
           != zero_zero_complex )
         => ( B2
            = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_5333_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B2: rat,W: num,A: rat] :
      ( ( ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
        = A )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
           != zero_zero_rat )
         => ( B2
            = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_5334_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B2: real,W: num] :
      ( ( A
        = ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
           != zero_zero_real )
         => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
            = B2 ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_5335_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B2: complex,W: num] :
      ( ( A
        = ( divide1717551699836669952omplex @ B2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
           != zero_zero_complex )
         => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
            = B2 ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_5336_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B2: rat,W: num] :
      ( ( A
        = ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
           != zero_zero_rat )
         => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
            = B2 ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_5337_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_5338_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_5339_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_5340_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_5341_divide__less__eq__numeral1_I2_J,axiom,
    ! [B2: real,W: num,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
      = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B2 ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_5342_divide__less__eq__numeral1_I2_J,axiom,
    ! [B2: rat,W: num,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
      = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B2 ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_5343_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B2: real,W: num] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ord_less_real @ B2 @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_5344_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B2: rat,W: num] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ord_less_rat @ B2 @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_5345_power2__minus,axiom,
    ! [A: real] :
      ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_5346_power2__minus,axiom,
    ! [A: int] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_5347_power2__minus,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_5348_power2__minus,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_5349_power2__minus,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_5350_odd__of__bool__self,axiom,
    ! [P4: $o] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P4 ) ) )
      = P4 ) ).

% odd_of_bool_self
thf(fact_5351_odd__of__bool__self,axiom,
    ! [P4: $o] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P4 ) ) )
      = P4 ) ).

% odd_of_bool_self
thf(fact_5352_odd__of__bool__self,axiom,
    ! [P4: $o] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P4 ) ) )
      = P4 ) ).

% odd_of_bool_self
thf(fact_5353_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_5354_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_5355_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_5356_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_5357_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_5358_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_5359_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_5360_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_5361_diff__numeral__special_I11_J,axiom,
    ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_5362_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_5363_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_5364_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_5365_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_5366_diff__numeral__special_I10_J,axiom,
    ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_5367_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_5368_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_5369_minus__1__div__2__eq,axiom,
    ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% minus_1_div_2_eq
thf(fact_5370_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_5371_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_5372_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_5373_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_5374_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_5375_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_5376_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_5377_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_5378_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_5379_of__bool__half__eq__0,axiom,
    ! [B2: $o] :
      ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = zero_zero_nat ) ).

% of_bool_half_eq_0
thf(fact_5380_of__bool__half__eq__0,axiom,
    ! [B2: $o] :
      ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = zero_zero_int ) ).

% of_bool_half_eq_0
thf(fact_5381_of__bool__half__eq__0,axiom,
    ! [B2: $o] :
      ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B2 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = zero_z3403309356797280102nteger ) ).

% of_bool_half_eq_0
thf(fact_5382_power__minus__odd,axiom,
    ! [N: nat,A: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
        = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ).

% power_minus_odd
thf(fact_5383_power__minus__odd,axiom,
    ! [N: nat,A: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
        = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ).

% power_minus_odd
thf(fact_5384_power__minus__odd,axiom,
    ! [N: nat,A: complex] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
        = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ).

% power_minus_odd
thf(fact_5385_power__minus__odd,axiom,
    ! [N: nat,A: code_integer] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
        = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).

% power_minus_odd
thf(fact_5386_power__minus__odd,axiom,
    ! [N: nat,A: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
        = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ).

% power_minus_odd
thf(fact_5387_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
        = ( power_power_real @ A @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_5388_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
        = ( power_power_int @ A @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_5389_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: complex] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
        = ( power_power_complex @ A @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_5390_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
        = ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_5391_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
        = ( power_power_rat @ A @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_5392_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_5393_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_5394_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_5395_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_5396_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_5397_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_5398_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_5399_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_5400_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_5401_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_5402_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_5403_set__decode__0,axiom,
    ! [X3: nat] :
      ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X3 ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X3 ) ) ) ).

% set_decode_0
thf(fact_5404_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_5405_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_5406_dbl__simps_I4_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_5407_dbl__simps_I4_J,axiom,
    ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_5408_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_5409_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_5410_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_5411_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_5412_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_5413_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_5414_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_5415_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_5416_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_5417_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_5418_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_5419_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_5420_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_5421_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_5422_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_5423_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_5424_signed__take__bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% signed_take_bit_0
thf(fact_5425_signed__take__bit__0,axiom,
    ! [A: int] :
      ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% signed_take_bit_0
thf(fact_5426_one__div__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_5427_one__div__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_5428_one__div__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_5429_bits__1__div__exp,axiom,
    ! [N: nat] :
      ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_5430_bits__1__div__exp,axiom,
    ! [N: nat] :
      ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_5431_bits__1__div__exp,axiom,
    ! [N: nat] :
      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_5432_one__mod__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% one_mod_2_pow_eq
thf(fact_5433_one__mod__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% one_mod_2_pow_eq
thf(fact_5434_one__mod__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% one_mod_2_pow_eq
thf(fact_5435_signed__take__bit__minus,axiom,
    ! [N: nat,K: int] :
      ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N @ K ) ) )
      = ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).

% signed_take_bit_minus
thf(fact_5436_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ M @ N )
     => ( ( dvd_dvd_nat @ N @ M )
       => ( M = N ) ) ) ).

% dvd_antisym
thf(fact_5437_equation__minus__iff,axiom,
    ! [A: real,B2: real] :
      ( ( A
        = ( uminus_uminus_real @ B2 ) )
      = ( B2
        = ( uminus_uminus_real @ A ) ) ) ).

% equation_minus_iff
thf(fact_5438_equation__minus__iff,axiom,
    ! [A: int,B2: int] :
      ( ( A
        = ( uminus_uminus_int @ B2 ) )
      = ( B2
        = ( uminus_uminus_int @ A ) ) ) ).

% equation_minus_iff
thf(fact_5439_equation__minus__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ B2 ) )
      = ( B2
        = ( uminus1482373934393186551omplex @ A ) ) ) ).

% equation_minus_iff
thf(fact_5440_equation__minus__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ B2 ) )
      = ( B2
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% equation_minus_iff
thf(fact_5441_equation__minus__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B2 ) )
      = ( B2
        = ( uminus_uminus_rat @ A ) ) ) ).

% equation_minus_iff
thf(fact_5442_minus__equation__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ( uminus_uminus_real @ A )
        = B2 )
      = ( ( uminus_uminus_real @ B2 )
        = A ) ) ).

% minus_equation_iff
thf(fact_5443_minus__equation__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ( uminus_uminus_int @ A )
        = B2 )
      = ( ( uminus_uminus_int @ B2 )
        = A ) ) ).

% minus_equation_iff
thf(fact_5444_minus__equation__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = B2 )
      = ( ( uminus1482373934393186551omplex @ B2 )
        = A ) ) ).

% minus_equation_iff
thf(fact_5445_minus__equation__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = B2 )
      = ( ( uminus1351360451143612070nteger @ B2 )
        = A ) ) ).

% minus_equation_iff
thf(fact_5446_minus__equation__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B2 )
      = ( ( uminus_uminus_rat @ B2 )
        = A ) ) ).

% minus_equation_iff
thf(fact_5447_of__bool__eq__iff,axiom,
    ! [P4: $o,Q2: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P4 )
        = ( zero_n2687167440665602831ol_nat @ Q2 ) )
      = ( P4 = Q2 ) ) ).

% of_bool_eq_iff
thf(fact_5448_of__bool__eq__iff,axiom,
    ! [P4: $o,Q2: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P4 )
        = ( zero_n2684676970156552555ol_int @ Q2 ) )
      = ( P4 = Q2 ) ) ).

% of_bool_eq_iff
thf(fact_5449_of__bool__eq__iff,axiom,
    ! [P4: $o,Q2: $o] :
      ( ( ( zero_n356916108424825756nteger @ P4 )
        = ( zero_n356916108424825756nteger @ Q2 ) )
      = ( P4 = Q2 ) ) ).

% of_bool_eq_iff
thf(fact_5450_verit__negate__coefficient_I3_J,axiom,
    ! [A: real,B2: real] :
      ( ( A = B2 )
     => ( ( uminus_uminus_real @ A )
        = ( uminus_uminus_real @ B2 ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_5451_verit__negate__coefficient_I3_J,axiom,
    ! [A: int,B2: int] :
      ( ( A = B2 )
     => ( ( uminus_uminus_int @ A )
        = ( uminus_uminus_int @ B2 ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_5452_verit__negate__coefficient_I3_J,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( A = B2 )
     => ( ( uminus1351360451143612070nteger @ A )
        = ( uminus1351360451143612070nteger @ B2 ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_5453_verit__negate__coefficient_I3_J,axiom,
    ! [A: rat,B2: rat] :
      ( ( A = B2 )
     => ( ( uminus_uminus_rat @ A )
        = ( uminus_uminus_rat @ B2 ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_5454_compl__mono,axiom,
    ! [X3: set_int,Y3: set_int] :
      ( ( ord_less_eq_set_int @ X3 @ Y3 )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ ( uminus1532241313380277803et_int @ X3 ) ) ) ).

% compl_mono
thf(fact_5455_compl__le__swap1,axiom,
    ! [Y3: set_int,X3: set_int] :
      ( ( ord_less_eq_set_int @ Y3 @ ( uminus1532241313380277803et_int @ X3 ) )
     => ( ord_less_eq_set_int @ X3 @ ( uminus1532241313380277803et_int @ Y3 ) ) ) ).

% compl_le_swap1
thf(fact_5456_compl__le__swap2,axiom,
    ! [Y3: set_int,X3: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ X3 )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X3 ) @ Y3 ) ) ).

% compl_le_swap2
thf(fact_5457_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n3304061248610475627l_real
        @ ( P
          & Q ) )
      = ( times_times_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) ) ) ).

% of_bool_conj
thf(fact_5458_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2052037380579107095ol_rat
        @ ( P
          & Q ) )
      = ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).

% of_bool_conj
thf(fact_5459_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2687167440665602831ol_nat
        @ ( P
          & Q ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).

% of_bool_conj
thf(fact_5460_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2684676970156552555ol_int
        @ ( P
          & Q ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).

% of_bool_conj
thf(fact_5461_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n356916108424825756nteger
        @ ( P
          & Q ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).

% of_bool_conj
thf(fact_5462_le__imp__neg__le,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).

% le_imp_neg_le
thf(fact_5463_le__imp__neg__le,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B2 )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% le_imp_neg_le
thf(fact_5464_le__imp__neg__le,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) ) ) ).

% le_imp_neg_le
thf(fact_5465_le__imp__neg__le,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).

% le_imp_neg_le
thf(fact_5466_minus__le__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ A ) ) ).

% minus_le_iff
thf(fact_5467_minus__le__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
      = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B2 ) @ A ) ) ).

% minus_le_iff
thf(fact_5468_minus__le__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B2 )
      = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ A ) ) ).

% minus_le_iff
thf(fact_5469_minus__le__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ A ) ) ).

% minus_le_iff
thf(fact_5470_le__minus__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B2 ) )
      = ( ord_less_eq_real @ B2 @ ( uminus_uminus_real @ A ) ) ) ).

% le_minus_iff
thf(fact_5471_le__minus__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( ord_le3102999989581377725nteger @ B2 @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% le_minus_iff
thf(fact_5472_le__minus__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B2 ) )
      = ( ord_less_eq_rat @ B2 @ ( uminus_uminus_rat @ A ) ) ) ).

% le_minus_iff
thf(fact_5473_le__minus__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B2 ) )
      = ( ord_less_eq_int @ B2 @ ( uminus_uminus_int @ A ) ) ) ).

% le_minus_iff
thf(fact_5474_verit__negate__coefficient_I2_J,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_5475_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ B2 )
     => ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_5476_verit__negate__coefficient_I2_J,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B2 )
     => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_5477_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ B2 )
     => ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_5478_minus__less__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B2 )
      = ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ A ) ) ).

% minus_less_iff
thf(fact_5479_minus__less__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B2 )
      = ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ A ) ) ).

% minus_less_iff
thf(fact_5480_minus__less__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
      = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B2 ) @ A ) ) ).

% minus_less_iff
thf(fact_5481_minus__less__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B2 )
      = ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ A ) ) ).

% minus_less_iff
thf(fact_5482_less__minus__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ B2 ) )
      = ( ord_less_real @ B2 @ ( uminus_uminus_real @ A ) ) ) ).

% less_minus_iff
thf(fact_5483_less__minus__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ B2 ) )
      = ( ord_less_int @ B2 @ ( uminus_uminus_int @ A ) ) ) ).

% less_minus_iff
thf(fact_5484_less__minus__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( ord_le6747313008572928689nteger @ B2 @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% less_minus_iff
thf(fact_5485_less__minus__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B2 ) )
      = ( ord_less_rat @ B2 @ ( uminus_uminus_rat @ A ) ) ) ).

% less_minus_iff
thf(fact_5486_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_5487_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_5488_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
     != ( numera6690914467698888265omplex @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_5489_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
     != ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_5490_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_5491_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_5492_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_5493_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numera6690914467698888265omplex @ M )
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5494_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numera6620942414471956472nteger @ M )
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5495_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_5496_square__eq__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ( times_times_real @ A @ A )
        = ( times_times_real @ B2 @ B2 ) )
      = ( ( A = B2 )
        | ( A
          = ( uminus_uminus_real @ B2 ) ) ) ) ).

% square_eq_iff
thf(fact_5497_square__eq__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ( times_times_int @ A @ A )
        = ( times_times_int @ B2 @ B2 ) )
      = ( ( A = B2 )
        | ( A
          = ( uminus_uminus_int @ B2 ) ) ) ) ).

% square_eq_iff
thf(fact_5498_square__eq__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( times_times_complex @ A @ A )
        = ( times_times_complex @ B2 @ B2 ) )
      = ( ( A = B2 )
        | ( A
          = ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).

% square_eq_iff
thf(fact_5499_square__eq__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ A )
        = ( times_3573771949741848930nteger @ B2 @ B2 ) )
      = ( ( A = B2 )
        | ( A
          = ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).

% square_eq_iff
thf(fact_5500_square__eq__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( times_times_rat @ A @ A )
        = ( times_times_rat @ B2 @ B2 ) )
      = ( ( A = B2 )
        | ( A
          = ( uminus_uminus_rat @ B2 ) ) ) ) ).

% square_eq_iff
thf(fact_5501_minus__mult__commute,axiom,
    ! [A: real,B2: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B2 )
      = ( times_times_real @ A @ ( uminus_uminus_real @ B2 ) ) ) ).

% minus_mult_commute
thf(fact_5502_minus__mult__commute,axiom,
    ! [A: int,B2: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B2 )
      = ( times_times_int @ A @ ( uminus_uminus_int @ B2 ) ) ) ).

% minus_mult_commute
thf(fact_5503_minus__mult__commute,axiom,
    ! [A: complex,B2: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 )
      = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B2 ) ) ) ).

% minus_mult_commute
thf(fact_5504_minus__mult__commute,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
      = ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) ) ) ).

% minus_mult_commute
thf(fact_5505_minus__mult__commute,axiom,
    ! [A: rat,B2: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B2 )
      = ( times_times_rat @ A @ ( uminus_uminus_rat @ B2 ) ) ) ).

% minus_mult_commute
thf(fact_5506_one__neq__neg__one,axiom,
    ( one_one_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% one_neq_neg_one
thf(fact_5507_one__neq__neg__one,axiom,
    ( one_one_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% one_neq_neg_one
thf(fact_5508_one__neq__neg__one,axiom,
    ( one_one_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% one_neq_neg_one
thf(fact_5509_one__neq__neg__one,axiom,
    ( one_one_Code_integer
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% one_neq_neg_one
thf(fact_5510_one__neq__neg__one,axiom,
    ( one_one_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% one_neq_neg_one
thf(fact_5511_is__num__normalize_I8_J,axiom,
    ! [A: real,B2: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B2 ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_5512_is__num__normalize_I8_J,axiom,
    ! [A: int,B2: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B2 ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_5513_is__num__normalize_I8_J,axiom,
    ! [A: complex,B2: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B2 ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B2 ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_5514_is__num__normalize_I8_J,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_5515_is__num__normalize_I8_J,axiom,
    ! [A: rat,B2: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B2 ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_5516_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B2: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B2 ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_5517_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B2: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B2 ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_5518_add_Oinverse__distrib__swap,axiom,
    ! [A: complex,B2: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B2 ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B2 ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_5519_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_5520_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B2: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B2 ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_5521_group__cancel_Oneg1,axiom,
    ! [A4: real,K: real,A: real] :
      ( ( A4
        = ( plus_plus_real @ K @ A ) )
     => ( ( uminus_uminus_real @ A4 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_5522_group__cancel_Oneg1,axiom,
    ! [A4: int,K: int,A: int] :
      ( ( A4
        = ( plus_plus_int @ K @ A ) )
     => ( ( uminus_uminus_int @ A4 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_5523_group__cancel_Oneg1,axiom,
    ! [A4: complex,K: complex,A: complex] :
      ( ( A4
        = ( plus_plus_complex @ K @ A ) )
     => ( ( uminus1482373934393186551omplex @ A4 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_5524_group__cancel_Oneg1,axiom,
    ! [A4: code_integer,K: code_integer,A: code_integer] :
      ( ( A4
        = ( plus_p5714425477246183910nteger @ K @ A ) )
     => ( ( uminus1351360451143612070nteger @ A4 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_5525_group__cancel_Oneg1,axiom,
    ! [A4: rat,K: rat,A: rat] :
      ( ( A4
        = ( plus_plus_rat @ K @ A ) )
     => ( ( uminus_uminus_rat @ A4 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_5526_minus__diff__minus,axiom,
    ! [A: real,B2: real] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
      = ( uminus_uminus_real @ ( minus_minus_real @ A @ B2 ) ) ) ).

% minus_diff_minus
thf(fact_5527_minus__diff__minus,axiom,
    ! [A: int,B2: int] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
      = ( uminus_uminus_int @ ( minus_minus_int @ A @ B2 ) ) ) ).

% minus_diff_minus
thf(fact_5528_minus__diff__minus,axiom,
    ! [A: complex,B2: complex] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) )
      = ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B2 ) ) ) ).

% minus_diff_minus
thf(fact_5529_minus__diff__minus,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ).

% minus_diff_minus
thf(fact_5530_minus__diff__minus,axiom,
    ! [A: rat,B2: rat] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) )
      = ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B2 ) ) ) ).

% minus_diff_minus
thf(fact_5531_minus__diff__commute,axiom,
    ! [B2: real,A: real] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ B2 ) @ A )
      = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ).

% minus_diff_commute
thf(fact_5532_minus__diff__commute,axiom,
    ! [B2: int,A: int] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ B2 ) @ A )
      = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).

% minus_diff_commute
thf(fact_5533_minus__diff__commute,axiom,
    ! [B2: complex,A: complex] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B2 ) @ A )
      = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 ) ) ).

% minus_diff_commute
thf(fact_5534_minus__diff__commute,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B2 ) @ A )
      = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).

% minus_diff_commute
thf(fact_5535_minus__diff__commute,axiom,
    ! [B2: rat,A: rat] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ B2 ) @ A )
      = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ).

% minus_diff_commute
thf(fact_5536_minus__divide__right,axiom,
    ! [A: real,B2: real] :
      ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
      = ( divide_divide_real @ A @ ( uminus_uminus_real @ B2 ) ) ) ).

% minus_divide_right
thf(fact_5537_minus__divide__right,axiom,
    ! [A: complex,B2: complex] :
      ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
      = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B2 ) ) ) ).

% minus_divide_right
thf(fact_5538_minus__divide__right,axiom,
    ! [A: rat,B2: rat] :
      ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) )
      = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B2 ) ) ) ).

% minus_divide_right
thf(fact_5539_minus__divide__divide,axiom,
    ! [A: real,B2: real] :
      ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
      = ( divide_divide_real @ A @ B2 ) ) ).

% minus_divide_divide
thf(fact_5540_minus__divide__divide,axiom,
    ! [A: complex,B2: complex] :
      ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) )
      = ( divide1717551699836669952omplex @ A @ B2 ) ) ).

% minus_divide_divide
thf(fact_5541_minus__divide__divide,axiom,
    ! [A: rat,B2: rat] :
      ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) )
      = ( divide_divide_rat @ A @ B2 ) ) ).

% minus_divide_divide
thf(fact_5542_minus__divide__left,axiom,
    ! [A: real,B2: real] :
      ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ).

% minus_divide_left
thf(fact_5543_minus__divide__left,axiom,
    ! [A: complex,B2: complex] :
      ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
      = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B2 ) ) ).

% minus_divide_left
thf(fact_5544_minus__divide__left,axiom,
    ! [A: rat,B2: rat] :
      ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) )
      = ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ).

% minus_divide_left
thf(fact_5545_div__minus__right,axiom,
    ! [A: int,B2: int] :
      ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B2 ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).

% div_minus_right
thf(fact_5546_div__minus__right,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).

% div_minus_right
thf(fact_5547_mod__minus__eq,axiom,
    ! [A: int,B2: int] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B2 ) ) @ B2 )
      = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).

% mod_minus_eq
thf(fact_5548_mod__minus__eq,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B2 ) ) @ B2 )
      = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).

% mod_minus_eq
thf(fact_5549_mod__minus__cong,axiom,
    ! [A: int,B2: int,A2: int] :
      ( ( ( modulo_modulo_int @ A @ B2 )
        = ( modulo_modulo_int @ A2 @ B2 ) )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 )
        = ( modulo_modulo_int @ ( uminus_uminus_int @ A2 ) @ B2 ) ) ) ).

% mod_minus_cong
thf(fact_5550_mod__minus__cong,axiom,
    ! [A: code_integer,B2: code_integer,A2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B2 )
        = ( modulo364778990260209775nteger @ A2 @ B2 ) )
     => ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
        = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 ) ) ) ).

% mod_minus_cong
thf(fact_5551_mod__minus__right,axiom,
    ! [A: int,B2: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B2 ) )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).

% mod_minus_right
thf(fact_5552_mod__minus__right,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ) ).

% mod_minus_right
thf(fact_5553_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_5554_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_5555_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_5556_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_5557_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_5558_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).

% of_bool_less_eq_one
thf(fact_5559_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).

% of_bool_less_eq_one
thf(fact_5560_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).

% of_bool_less_eq_one
thf(fact_5561_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).

% of_bool_less_eq_one
thf(fact_5562_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).

% of_bool_less_eq_one
thf(fact_5563_of__bool__def,axiom,
    ( zero_n1201886186963655149omplex
    = ( ^ [P5: $o] : ( if_complex @ P5 @ one_one_complex @ zero_zero_complex ) ) ) ).

% of_bool_def
thf(fact_5564_of__bool__def,axiom,
    ( zero_n3304061248610475627l_real
    = ( ^ [P5: $o] : ( if_real @ P5 @ one_one_real @ zero_zero_real ) ) ) ).

% of_bool_def
thf(fact_5565_of__bool__def,axiom,
    ( zero_n2052037380579107095ol_rat
    = ( ^ [P5: $o] : ( if_rat @ P5 @ one_one_rat @ zero_zero_rat ) ) ) ).

% of_bool_def
thf(fact_5566_of__bool__def,axiom,
    ( zero_n2687167440665602831ol_nat
    = ( ^ [P5: $o] : ( if_nat @ P5 @ one_one_nat @ zero_zero_nat ) ) ) ).

% of_bool_def
thf(fact_5567_of__bool__def,axiom,
    ( zero_n2684676970156552555ol_int
    = ( ^ [P5: $o] : ( if_int @ P5 @ one_one_int @ zero_zero_int ) ) ) ).

% of_bool_def
thf(fact_5568_of__bool__def,axiom,
    ( zero_n356916108424825756nteger
    = ( ^ [P5: $o] : ( if_Code_integer @ P5 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).

% of_bool_def
thf(fact_5569_split__of__bool,axiom,
    ! [P: complex > $o,P4: $o] :
      ( ( P @ ( zero_n1201886186963655149omplex @ P4 ) )
      = ( ( P4
         => ( P @ one_one_complex ) )
        & ( ~ P4
         => ( P @ zero_zero_complex ) ) ) ) ).

% split_of_bool
thf(fact_5570_split__of__bool,axiom,
    ! [P: real > $o,P4: $o] :
      ( ( P @ ( zero_n3304061248610475627l_real @ P4 ) )
      = ( ( P4
         => ( P @ one_one_real ) )
        & ( ~ P4
         => ( P @ zero_zero_real ) ) ) ) ).

% split_of_bool
thf(fact_5571_split__of__bool,axiom,
    ! [P: rat > $o,P4: $o] :
      ( ( P @ ( zero_n2052037380579107095ol_rat @ P4 ) )
      = ( ( P4
         => ( P @ one_one_rat ) )
        & ( ~ P4
         => ( P @ zero_zero_rat ) ) ) ) ).

% split_of_bool
thf(fact_5572_split__of__bool,axiom,
    ! [P: nat > $o,P4: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
      = ( ( P4
         => ( P @ one_one_nat ) )
        & ( ~ P4
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_of_bool
thf(fact_5573_split__of__bool,axiom,
    ! [P: int > $o,P4: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
      = ( ( P4
         => ( P @ one_one_int ) )
        & ( ~ P4
         => ( P @ zero_zero_int ) ) ) ) ).

% split_of_bool
thf(fact_5574_split__of__bool,axiom,
    ! [P: code_integer > $o,P4: $o] :
      ( ( P @ ( zero_n356916108424825756nteger @ P4 ) )
      = ( ( P4
         => ( P @ one_one_Code_integer ) )
        & ( ~ P4
         => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).

% split_of_bool
thf(fact_5575_split__of__bool__asm,axiom,
    ! [P: complex > $o,P4: $o] :
      ( ( P @ ( zero_n1201886186963655149omplex @ P4 ) )
      = ( ~ ( ( P4
              & ~ ( P @ one_one_complex ) )
            | ( ~ P4
              & ~ ( P @ zero_zero_complex ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_5576_split__of__bool__asm,axiom,
    ! [P: real > $o,P4: $o] :
      ( ( P @ ( zero_n3304061248610475627l_real @ P4 ) )
      = ( ~ ( ( P4
              & ~ ( P @ one_one_real ) )
            | ( ~ P4
              & ~ ( P @ zero_zero_real ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_5577_split__of__bool__asm,axiom,
    ! [P: rat > $o,P4: $o] :
      ( ( P @ ( zero_n2052037380579107095ol_rat @ P4 ) )
      = ( ~ ( ( P4
              & ~ ( P @ one_one_rat ) )
            | ( ~ P4
              & ~ ( P @ zero_zero_rat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_5578_split__of__bool__asm,axiom,
    ! [P: nat > $o,P4: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
      = ( ~ ( ( P4
              & ~ ( P @ one_one_nat ) )
            | ( ~ P4
              & ~ ( P @ zero_zero_nat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_5579_split__of__bool__asm,axiom,
    ! [P: int > $o,P4: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
      = ( ~ ( ( P4
              & ~ ( P @ one_one_int ) )
            | ( ~ P4
              & ~ ( P @ zero_zero_int ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_5580_split__of__bool__asm,axiom,
    ! [P: code_integer > $o,P4: $o] :
      ( ( P @ ( zero_n356916108424825756nteger @ P4 ) )
      = ( ~ ( ( P4
              & ~ ( P @ one_one_Code_integer ) )
            | ( ~ P4
              & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_5581_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_5582_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5583_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_5584_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_5585_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_5586_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_5587_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_5588_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_5589_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5590_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5591_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5592_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_z3403309356797280102nteger
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5593_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5594_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_5595_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_5596_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5597_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_5598_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_5599_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_5600_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_5601_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_5602_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_5603_le__minus__one__simps_I2_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).

% le_minus_one_simps(2)
thf(fact_5604_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_5605_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_5606_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_5607_le__minus__one__simps_I4_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(4)
thf(fact_5608_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_5609_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_5610_zero__neq__neg__one,axiom,
    ( zero_zero_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% zero_neq_neg_one
thf(fact_5611_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_5612_zero__neq__neg__one,axiom,
    ( zero_zero_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% zero_neq_neg_one
thf(fact_5613_zero__neq__neg__one,axiom,
    ( zero_z3403309356797280102nteger
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% zero_neq_neg_one
thf(fact_5614_zero__neq__neg__one,axiom,
    ( zero_zero_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% zero_neq_neg_one
thf(fact_5615_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B2: real] :
      ( ( ( uminus_uminus_real @ A )
        = B2 )
      = ( ( plus_plus_real @ A @ B2 )
        = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_5616_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B2: int] :
      ( ( ( uminus_uminus_int @ A )
        = B2 )
      = ( ( plus_plus_int @ A @ B2 )
        = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_5617_neg__eq__iff__add__eq__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = B2 )
      = ( ( plus_plus_complex @ A @ B2 )
        = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_5618_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = B2 )
      = ( ( plus_p5714425477246183910nteger @ A @ B2 )
        = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_5619_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B2 )
      = ( ( plus_plus_rat @ A @ B2 )
        = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_5620_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B2: real] :
      ( ( A
        = ( uminus_uminus_real @ B2 ) )
      = ( ( plus_plus_real @ A @ B2 )
        = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_5621_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B2: int] :
      ( ( A
        = ( uminus_uminus_int @ B2 ) )
      = ( ( plus_plus_int @ A @ B2 )
        = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_5622_eq__neg__iff__add__eq__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ B2 ) )
      = ( ( plus_plus_complex @ A @ B2 )
        = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_5623_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ B2 ) )
      = ( ( plus_p5714425477246183910nteger @ A @ B2 )
        = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_5624_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B2: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B2 ) )
      = ( ( plus_plus_rat @ A @ B2 )
        = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_5625_add_Oinverse__unique,axiom,
    ! [A: real,B2: real] :
      ( ( ( plus_plus_real @ A @ B2 )
        = zero_zero_real )
     => ( ( uminus_uminus_real @ A )
        = B2 ) ) ).

% add.inverse_unique
thf(fact_5626_add_Oinverse__unique,axiom,
    ! [A: int,B2: int] :
      ( ( ( plus_plus_int @ A @ B2 )
        = zero_zero_int )
     => ( ( uminus_uminus_int @ A )
        = B2 ) ) ).

% add.inverse_unique
thf(fact_5627_add_Oinverse__unique,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( plus_plus_complex @ A @ B2 )
        = zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ A )
        = B2 ) ) ).

% add.inverse_unique
thf(fact_5628_add_Oinverse__unique,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B2 )
        = zero_z3403309356797280102nteger )
     => ( ( uminus1351360451143612070nteger @ A )
        = B2 ) ) ).

% add.inverse_unique
thf(fact_5629_add_Oinverse__unique,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( plus_plus_rat @ A @ B2 )
        = zero_zero_rat )
     => ( ( uminus_uminus_rat @ A )
        = B2 ) ) ).

% add.inverse_unique
thf(fact_5630_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_group_add_class.ab_left_minus
thf(fact_5631_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_group_add_class.ab_left_minus
thf(fact_5632_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
      = zero_zero_complex ) ).

% ab_group_add_class.ab_left_minus
thf(fact_5633_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = zero_z3403309356797280102nteger ) ).

% ab_group_add_class.ab_left_minus
thf(fact_5634_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_group_add_class.ab_left_minus
thf(fact_5635_add__eq__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ( plus_plus_real @ A @ B2 )
        = zero_zero_real )
      = ( B2
        = ( uminus_uminus_real @ A ) ) ) ).

% add_eq_0_iff
thf(fact_5636_add__eq__0__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ( plus_plus_int @ A @ B2 )
        = zero_zero_int )
      = ( B2
        = ( uminus_uminus_int @ A ) ) ) ).

% add_eq_0_iff
thf(fact_5637_add__eq__0__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( plus_plus_complex @ A @ B2 )
        = zero_zero_complex )
      = ( B2
        = ( uminus1482373934393186551omplex @ A ) ) ) ).

% add_eq_0_iff
thf(fact_5638_add__eq__0__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B2 )
        = zero_z3403309356797280102nteger )
      = ( B2
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% add_eq_0_iff
thf(fact_5639_add__eq__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( plus_plus_rat @ A @ B2 )
        = zero_zero_rat )
      = ( B2
        = ( uminus_uminus_rat @ A ) ) ) ).

% add_eq_0_iff
thf(fact_5640_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_5641_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_5642_less__minus__one__simps_I2_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).

% less_minus_one_simps(2)
thf(fact_5643_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_5644_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_5645_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_5646_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(4)
thf(fact_5647_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_5648_numeral__times__minus__swap,axiom,
    ! [W: num,X3: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X3 ) )
      = ( times_times_real @ X3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5649_numeral__times__minus__swap,axiom,
    ! [W: num,X3: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X3 ) )
      = ( times_times_int @ X3 @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5650_numeral__times__minus__swap,axiom,
    ! [W: num,X3: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X3 ) )
      = ( times_times_complex @ X3 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5651_numeral__times__minus__swap,axiom,
    ! [W: num,X3: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X3 ) )
      = ( times_3573771949741848930nteger @ X3 @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5652_numeral__times__minus__swap,axiom,
    ! [W: num,X3: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X3 ) )
      = ( times_times_rat @ X3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5653_nonzero__minus__divide__right,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
        = ( divide_divide_real @ A @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_5654_nonzero__minus__divide__right,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
        = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_5655_nonzero__minus__divide__right,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) )
        = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_5656_nonzero__minus__divide__divide,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
        = ( divide_divide_real @ A @ B2 ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_5657_nonzero__minus__divide__divide,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) )
        = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_5658_nonzero__minus__divide__divide,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) )
        = ( divide_divide_rat @ A @ B2 ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_5659_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_5660_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_5661_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_5662_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_Code_integer
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_5663_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_5664_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ N )
     != ( uminus_uminus_real @ one_one_real ) ) ).

% numeral_neq_neg_one
thf(fact_5665_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ N )
     != ( uminus_uminus_int @ one_one_int ) ) ).

% numeral_neq_neg_one
thf(fact_5666_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ N )
     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% numeral_neq_neg_one
thf(fact_5667_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger @ N )
     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% numeral_neq_neg_one
thf(fact_5668_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ N )
     != ( uminus_uminus_rat @ one_one_rat ) ) ).

% numeral_neq_neg_one
thf(fact_5669_square__eq__1__iff,axiom,
    ! [X3: real] :
      ( ( ( times_times_real @ X3 @ X3 )
        = one_one_real )
      = ( ( X3 = one_one_real )
        | ( X3
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% square_eq_1_iff
thf(fact_5670_square__eq__1__iff,axiom,
    ! [X3: int] :
      ( ( ( times_times_int @ X3 @ X3 )
        = one_one_int )
      = ( ( X3 = one_one_int )
        | ( X3
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% square_eq_1_iff
thf(fact_5671_square__eq__1__iff,axiom,
    ! [X3: complex] :
      ( ( ( times_times_complex @ X3 @ X3 )
        = one_one_complex )
      = ( ( X3 = one_one_complex )
        | ( X3
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% square_eq_1_iff
thf(fact_5672_square__eq__1__iff,axiom,
    ! [X3: code_integer] :
      ( ( ( times_3573771949741848930nteger @ X3 @ X3 )
        = one_one_Code_integer )
      = ( ( X3 = one_one_Code_integer )
        | ( X3
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% square_eq_1_iff
thf(fact_5673_square__eq__1__iff,axiom,
    ! [X3: rat] :
      ( ( ( times_times_rat @ X3 @ X3 )
        = one_one_rat )
      = ( ( X3 = one_one_rat )
        | ( X3
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% square_eq_1_iff
thf(fact_5674_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A5: real,B4: real] : ( plus_plus_real @ A5 @ ( uminus_uminus_real @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5675_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5676_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A5: complex,B4: complex] : ( plus_plus_complex @ A5 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5677_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A5: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A5 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5678_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ A5 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5679_diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A5: real,B4: real] : ( plus_plus_real @ A5 @ ( uminus_uminus_real @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5680_diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5681_diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A5: complex,B4: complex] : ( plus_plus_complex @ A5 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5682_diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A5: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A5 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5683_diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ A5 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5684_group__cancel_Osub2,axiom,
    ! [B5: real,K: real,B2: real,A: real] :
      ( ( B5
        = ( plus_plus_real @ K @ B2 ) )
     => ( ( minus_minus_real @ A @ B5 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B2 ) ) ) ) ).

% group_cancel.sub2
thf(fact_5685_group__cancel_Osub2,axiom,
    ! [B5: int,K: int,B2: int,A: int] :
      ( ( B5
        = ( plus_plus_int @ K @ B2 ) )
     => ( ( minus_minus_int @ A @ B5 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B2 ) ) ) ) ).

% group_cancel.sub2
thf(fact_5686_group__cancel_Osub2,axiom,
    ! [B5: complex,K: complex,B2: complex,A: complex] :
      ( ( B5
        = ( plus_plus_complex @ K @ B2 ) )
     => ( ( minus_minus_complex @ A @ B5 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B2 ) ) ) ) ).

% group_cancel.sub2
thf(fact_5687_group__cancel_Osub2,axiom,
    ! [B5: code_integer,K: code_integer,B2: code_integer,A: code_integer] :
      ( ( B5
        = ( plus_p5714425477246183910nteger @ K @ B2 ) )
     => ( ( minus_8373710615458151222nteger @ A @ B5 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ) ).

% group_cancel.sub2
thf(fact_5688_group__cancel_Osub2,axiom,
    ! [B5: rat,K: rat,B2: rat,A: rat] :
      ( ( B5
        = ( plus_plus_rat @ K @ B2 ) )
     => ( ( minus_minus_rat @ A @ B5 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B2 ) ) ) ) ).

% group_cancel.sub2
thf(fact_5689_replicate__length__same,axiom,
    ! [Xs2: list_int,X3: int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ ( set_int2 @ Xs2 ) )
         => ( X5 = X3 ) )
     => ( ( replicate_int @ ( size_size_list_int @ Xs2 ) @ X3 )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_5690_replicate__length__same,axiom,
    ! [Xs2: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( X5 = X3 ) )
     => ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ X3 )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_5691_replicate__length__same,axiom,
    ! [Xs2: list_o,X3: $o] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ ( set_o2 @ Xs2 ) )
         => ( X5 = X3 ) )
     => ( ( replicate_o @ ( size_size_list_o @ Xs2 ) @ X3 )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_5692_replicate__length__same,axiom,
    ! [Xs2: list_nat,X3: nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
         => ( X5 = X3 ) )
     => ( ( replicate_nat @ ( size_size_list_nat @ Xs2 ) @ X3 )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_5693_replicate__eqI,axiom,
    ! [Xs2: list_real,N: nat,X3: real] :
      ( ( ( size_size_list_real @ Xs2 )
        = N )
     => ( ! [Y4: real] :
            ( ( member_real @ Y4 @ ( set_real2 @ Xs2 ) )
           => ( Y4 = X3 ) )
       => ( Xs2
          = ( replicate_real @ N @ X3 ) ) ) ) ).

% replicate_eqI
thf(fact_5694_replicate__eqI,axiom,
    ! [Xs2: list_complex,N: nat,X3: complex] :
      ( ( ( size_s3451745648224563538omplex @ Xs2 )
        = N )
     => ( ! [Y4: complex] :
            ( ( member_complex @ Y4 @ ( set_complex2 @ Xs2 ) )
           => ( Y4 = X3 ) )
       => ( Xs2
          = ( replicate_complex @ N @ X3 ) ) ) ) ).

% replicate_eqI
thf(fact_5695_replicate__eqI,axiom,
    ! [Xs2: list_int,N: nat,X3: int] :
      ( ( ( size_size_list_int @ Xs2 )
        = N )
     => ( ! [Y4: int] :
            ( ( member_int @ Y4 @ ( set_int2 @ Xs2 ) )
           => ( Y4 = X3 ) )
       => ( Xs2
          = ( replicate_int @ N @ X3 ) ) ) ) ).

% replicate_eqI
thf(fact_5696_replicate__eqI,axiom,
    ! [Xs2: list_VEBT_VEBT,N: nat,X3: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = N )
     => ( ! [Y4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ Y4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( Y4 = X3 ) )
       => ( Xs2
          = ( replicate_VEBT_VEBT @ N @ X3 ) ) ) ) ).

% replicate_eqI
thf(fact_5697_replicate__eqI,axiom,
    ! [Xs2: list_o,N: nat,X3: $o] :
      ( ( ( size_size_list_o @ Xs2 )
        = N )
     => ( ! [Y4: $o] :
            ( ( member_o @ Y4 @ ( set_o2 @ Xs2 ) )
           => ( Y4 = X3 ) )
       => ( Xs2
          = ( replicate_o @ N @ X3 ) ) ) ) ).

% replicate_eqI
thf(fact_5698_replicate__eqI,axiom,
    ! [Xs2: list_nat,N: nat,X3: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = N )
     => ( ! [Y4: nat] :
            ( ( member_nat @ Y4 @ ( set_nat2 @ Xs2 ) )
           => ( Y4 = X3 ) )
       => ( Xs2
          = ( replicate_nat @ N @ X3 ) ) ) ) ).

% replicate_eqI
thf(fact_5699_dvd__neg__div,axiom,
    ! [B2: real,A: real] :
      ( ( dvd_dvd_real @ B2 @ A )
     => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B2 )
        = ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) ) ) ) ).

% dvd_neg_div
thf(fact_5700_dvd__neg__div,axiom,
    ! [B2: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ A )
     => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 )
        = ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) ) ) ).

% dvd_neg_div
thf(fact_5701_dvd__neg__div,axiom,
    ! [B2: complex,A: complex] :
      ( ( dvd_dvd_complex @ B2 @ A )
     => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B2 )
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) ) ) ) ).

% dvd_neg_div
thf(fact_5702_dvd__neg__div,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ A )
     => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
        = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ).

% dvd_neg_div
thf(fact_5703_dvd__neg__div,axiom,
    ! [B2: rat,A: rat] :
      ( ( dvd_dvd_rat @ B2 @ A )
     => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B2 )
        = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) ) ) ) ).

% dvd_neg_div
thf(fact_5704_dvd__div__neg,axiom,
    ! [B2: real,A: real] :
      ( ( dvd_dvd_real @ B2 @ A )
     => ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B2 ) )
        = ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) ) ) ) ).

% dvd_div_neg
thf(fact_5705_dvd__div__neg,axiom,
    ! [B2: int,A: int] :
      ( ( dvd_dvd_int @ B2 @ A )
     => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B2 ) )
        = ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) ) ) ).

% dvd_div_neg
thf(fact_5706_dvd__div__neg,axiom,
    ! [B2: complex,A: complex] :
      ( ( dvd_dvd_complex @ B2 @ A )
     => ( ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B2 ) )
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) ) ) ) ).

% dvd_div_neg
thf(fact_5707_dvd__div__neg,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ A )
     => ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
        = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ).

% dvd_div_neg
thf(fact_5708_dvd__div__neg,axiom,
    ! [B2: rat,A: rat] :
      ( ( dvd_dvd_rat @ B2 @ A )
     => ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B2 ) )
        = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) ) ) ) ).

% dvd_div_neg
thf(fact_5709_subset__Compl__self__eq,axiom,
    ! [A4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ A4 ) )
      = ( A4 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
thf(fact_5710_subset__Compl__self__eq,axiom,
    ! [A4: set_real] :
      ( ( ord_less_eq_set_real @ A4 @ ( uminus612125837232591019t_real @ A4 ) )
      = ( A4 = bot_bot_set_real ) ) ).

% subset_Compl_self_eq
thf(fact_5711_subset__Compl__self__eq,axiom,
    ! [A4: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ ( uminus1532241313380277803et_int @ A4 ) )
      = ( A4 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
thf(fact_5712_real__minus__mult__self__le,axiom,
    ! [U: real,X3: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X3 @ X3 ) ) ).

% real_minus_mult_self_le
thf(fact_5713_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_5714_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_5715_minus__real__def,axiom,
    ( minus_minus_real
    = ( ^ [X: real,Y: real] : ( plus_plus_real @ X @ ( uminus_uminus_real @ Y ) ) ) ) ).

% minus_real_def
thf(fact_5716_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_5717_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_5718_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_5719_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_5720_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_5721_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_le_zero
thf(fact_5722_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_5723_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_5724_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_5725_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_5726_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_5727_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_5728_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_5729_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_5730_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_less_zero
thf(fact_5731_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_5732_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_5733_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(3)
thf(fact_5734_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_5735_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_5736_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_5737_le__minus__one__simps_I1_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% le_minus_one_simps(1)
thf(fact_5738_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_5739_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_5740_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_5741_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_5742_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(3)
thf(fact_5743_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_5744_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_5745_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_5746_less__minus__one__simps_I1_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% less_minus_one_simps(1)
thf(fact_5747_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_5748_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_5749_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_5750_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_5751_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_5752_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_5753_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_le_neg_one
thf(fact_5754_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_5755_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_5756_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_5757_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% neg_numeral_le_neg_one
thf(fact_5758_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_5759_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_5760_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_5761_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_le_numeral
thf(fact_5762_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_5763_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_5764_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_5765_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_le_one
thf(fact_5766_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_5767_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_5768_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_5769_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_5770_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_5771_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_5772_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_5773_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_5774_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_5775_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_5776_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_5777_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_5778_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_less_neg_one
thf(fact_5779_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_5780_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_5781_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_5782_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_less_numeral
thf(fact_5783_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_5784_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_5785_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_5786_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_less_one
thf(fact_5787_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_5788_eq__minus__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( A
        = ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ A @ C )
            = ( uminus_uminus_real @ B2 ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_5789_eq__minus__divide__eq,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B2 @ C ) ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ A @ C )
            = ( uminus1482373934393186551omplex @ B2 ) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_5790_eq__minus__divide__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( A
        = ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A @ C )
            = ( uminus_uminus_rat @ B2 ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_5791_minus__divide__eq__eq,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) )
        = A )
      = ( ( ( C != zero_zero_real )
         => ( ( uminus_uminus_real @ B2 )
            = ( times_times_real @ A @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_5792_minus__divide__eq__eq,axiom,
    ! [B2: complex,C: complex,A: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B2 @ C ) )
        = A )
      = ( ( ( C != zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ B2 )
            = ( times_times_complex @ A @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_5793_minus__divide__eq__eq,axiom,
    ! [B2: rat,C: rat,A: rat] :
      ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) )
        = A )
      = ( ( ( C != zero_zero_rat )
         => ( ( uminus_uminus_rat @ B2 )
            = ( times_times_rat @ A @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_5794_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( B2 != zero_zero_real )
     => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
          = C )
        = ( ( uminus_uminus_real @ A )
          = ( times_times_real @ C @ B2 ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_5795_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: complex,A: complex,C: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
          = C )
        = ( ( uminus1482373934393186551omplex @ A )
          = ( times_times_complex @ C @ B2 ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_5796_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) )
          = C )
        = ( ( uminus_uminus_rat @ A )
          = ( times_times_rat @ C @ B2 ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_5797_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( C
          = ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) ) )
        = ( ( times_times_real @ C @ B2 )
          = ( uminus_uminus_real @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_5798_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: complex,C: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( C
          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
        = ( ( times_times_complex @ C @ B2 )
          = ( uminus1482373934393186551omplex @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_5799_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: rat,C: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( C
          = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) ) )
        = ( ( times_times_rat @ C @ B2 )
          = ( uminus_uminus_rat @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_5800_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_5801_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_5802_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_5803_mult__1s__ring__1_I2_J,axiom,
    ! [B2: code_integer] :
      ( ( times_3573771949741848930nteger @ B2 @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
      = ( uminus1351360451143612070nteger @ B2 ) ) ).

% mult_1s_ring_1(2)
thf(fact_5804_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_5805_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_5806_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_5807_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_5808_mult__1s__ring__1_I1_J,axiom,
    ! [B2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B2 )
      = ( uminus1351360451143612070nteger @ B2 ) ) ).

% mult_1s_ring_1(1)
thf(fact_5809_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_5810_divide__eq__minus__1__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ( divide_divide_real @ A @ B2 )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( ( B2 != zero_zero_real )
        & ( A
          = ( uminus_uminus_real @ B2 ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_5811_divide__eq__minus__1__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ B2 )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( ( B2 != zero_zero_complex )
        & ( A
          = ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_5812_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( divide_divide_rat @ A @ B2 )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( ( B2 != zero_zero_rat )
        & ( A
          = ( uminus_uminus_rat @ B2 ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_5813_uminus__numeral__One,axiom,
    ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% uminus_numeral_One
thf(fact_5814_uminus__numeral__One,axiom,
    ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% uminus_numeral_One
thf(fact_5815_uminus__numeral__One,axiom,
    ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% uminus_numeral_One
thf(fact_5816_uminus__numeral__One,axiom,
    ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% uminus_numeral_One
thf(fact_5817_uminus__numeral__One,axiom,
    ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% uminus_numeral_One
thf(fact_5818_power__minus,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).

% power_minus
thf(fact_5819_power__minus,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).

% power_minus
thf(fact_5820_power__minus,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A @ N ) ) ) ).

% power_minus
thf(fact_5821_power__minus,axiom,
    ! [A: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% power_minus
thf(fact_5822_power__minus,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A @ N ) ) ) ).

% power_minus
thf(fact_5823_power__minus__Bit0,axiom,
    ! [X3: real,K: num] :
      ( ( power_power_real @ ( uminus_uminus_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5824_power__minus__Bit0,axiom,
    ! [X3: int,K: num] :
      ( ( power_power_int @ ( uminus_uminus_int @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5825_power__minus__Bit0,axiom,
    ! [X3: complex,K: num] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5826_power__minus__Bit0,axiom,
    ! [X3: code_integer,K: num] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5827_power__minus__Bit0,axiom,
    ! [X3: rat,K: num] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5828_real__add__less__0__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X3 @ Y3 ) @ zero_zero_real )
      = ( ord_less_real @ Y3 @ ( uminus_uminus_real @ X3 ) ) ) ).

% real_add_less_0_iff
thf(fact_5829_real__0__less__add__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X3 @ Y3 ) )
      = ( ord_less_real @ ( uminus_uminus_real @ X3 ) @ Y3 ) ) ).

% real_0_less_add_iff
thf(fact_5830_real__add__le__0__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ X3 @ Y3 ) @ zero_zero_real )
      = ( ord_less_eq_real @ Y3 @ ( uminus_uminus_real @ X3 ) ) ) ).

% real_add_le_0_iff
thf(fact_5831_real__0__le__add__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X3 @ Y3 ) )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ X3 ) @ Y3 ) ) ).

% real_0_le_add_iff
thf(fact_5832_pos__minus__divide__less__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
        = ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_5833_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_5834_pos__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_5835_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_5836_neg__minus__divide__less__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_5837_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_5838_neg__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
        = ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_5839_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_5840_minus__divide__less__eq,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_5841_minus__divide__less__eq,axiom,
    ! [B2: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_5842_less__minus__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ A @ 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 @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_5843_less__minus__divide__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A @ 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 @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_5844_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_5845_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_5846_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_5847_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_5848_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_5849_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_5850_subset__decode__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% subset_decode_imp_le
thf(fact_5851_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: real,A: real,B2: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
          = B2 ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
          = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_5852_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: complex,A: complex,B2: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B2 )
          = B2 ) )
      & ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B2 )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_5853_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: rat,A: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B2 )
          = B2 ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B2 )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_5854_minus__divide__add__eq__iff,axiom,
    ! [Z: real,X3: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X3 @ Z ) ) @ Y3 )
        = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X3 ) @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_5855_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,X3: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X3 @ Z ) ) @ Y3 )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X3 ) @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_5856_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X3: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X3 @ Z ) ) @ Y3 )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X3 ) @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_5857_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: real,A: real,B2: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
          = ( uminus_uminus_real @ B2 ) ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
          = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_5858_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: complex,A: complex,B2: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B2 )
          = ( uminus1482373934393186551omplex @ B2 ) ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B2 )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_5859_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: rat,A: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B2 )
          = ( uminus_uminus_rat @ B2 ) ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B2 )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_5860_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: real,A: real,B2: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
          = ( uminus_uminus_real @ B2 ) ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
          = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_5861_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: complex,A: complex,B2: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B2 )
          = ( uminus1482373934393186551omplex @ B2 ) ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B2 )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_5862_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: rat,A: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B2 )
          = ( uminus_uminus_rat @ B2 ) ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B2 )
          = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_5863_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,X3: real,Y3: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X3 @ Z ) ) @ Y3 )
        = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X3 ) @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_5864_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,X3: complex,Y3: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X3 @ Z ) ) @ Y3 )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X3 ) @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_5865_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X3: rat,Y3: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X3 @ Z ) ) @ Y3 )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X3 ) @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_5866_even__minus,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).

% even_minus
thf(fact_5867_even__minus,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( uminus1351360451143612070nteger @ A ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).

% even_minus
thf(fact_5868_power2__eq__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X3 = Y3 )
        | ( X3
          = ( uminus_uminus_real @ Y3 ) ) ) ) ).

% power2_eq_iff
thf(fact_5869_power2__eq__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X3 = Y3 )
        | ( X3
          = ( uminus_uminus_int @ Y3 ) ) ) ) ).

% power2_eq_iff
thf(fact_5870_power2__eq__iff,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X3 = Y3 )
        | ( X3
          = ( uminus1482373934393186551omplex @ Y3 ) ) ) ) ).

% power2_eq_iff
thf(fact_5871_power2__eq__iff,axiom,
    ! [X3: code_integer,Y3: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X3 = Y3 )
        | ( X3
          = ( uminus1351360451143612070nteger @ Y3 ) ) ) ) ).

% power2_eq_iff
thf(fact_5872_power2__eq__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X3 = Y3 )
        | ( X3
          = ( uminus_uminus_rat @ Y3 ) ) ) ) ).

% power2_eq_iff
thf(fact_5873_uminus__power__if,axiom,
    ! [N: nat,A: real] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
          = ( power_power_real @ A @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
          = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_5874_uminus__power__if,axiom,
    ! [N: nat,A: int] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
          = ( power_power_int @ A @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
          = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_5875_uminus__power__if,axiom,
    ! [N: nat,A: complex] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
          = ( power_power_complex @ A @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
          = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_5876_uminus__power__if,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
          = ( power_8256067586552552935nteger @ A @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
          = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_5877_uminus__power__if,axiom,
    ! [N: nat,A: rat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
          = ( power_power_rat @ A @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
          = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_5878_verit__less__mono__div__int2,axiom,
    ! [A4: int,B5: int,N: int] :
      ( ( ord_less_eq_int @ A4 @ B5 )
     => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
       => ( ord_less_eq_int @ ( divide_divide_int @ B5 @ N ) @ ( divide_divide_int @ A4 @ N ) ) ) ) ).

% verit_less_mono_div_int2
thf(fact_5879_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_5880_of__bool__odd__eq__mod__2,axiom,
    ! [A: nat] :
      ( ( zero_n2687167440665602831ol_nat
        @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
      = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% of_bool_odd_eq_mod_2
thf(fact_5881_of__bool__odd__eq__mod__2,axiom,
    ! [A: int] :
      ( ( zero_n2684676970156552555ol_int
        @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
      = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% of_bool_odd_eq_mod_2
thf(fact_5882_of__bool__odd__eq__mod__2,axiom,
    ! [A: code_integer] :
      ( ( zero_n356916108424825756nteger
        @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% of_bool_odd_eq_mod_2
thf(fact_5883_pos__minus__divide__le__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_5884_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_5885_pos__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_5886_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_5887_neg__minus__divide__le__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_5888_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_5889_neg__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_5890_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_5891_minus__divide__le__eq,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_5892_minus__divide__le__eq,axiom,
    ! [B2: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_5893_le__minus__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A @ 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 @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_5894_le__minus__divide__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ 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 @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_5895_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_5896_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_5897_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_5898_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_5899_power2__eq__1__iff,axiom,
    ! [A: real] :
      ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
      = ( ( A = one_one_real )
        | ( A
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% power2_eq_1_iff
thf(fact_5900_power2__eq__1__iff,axiom,
    ! [A: int] :
      ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_int )
      = ( ( A = one_one_int )
        | ( A
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% power2_eq_1_iff
thf(fact_5901_power2__eq__1__iff,axiom,
    ! [A: complex] :
      ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_complex )
      = ( ( A = one_one_complex )
        | ( A
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% power2_eq_1_iff
thf(fact_5902_power2__eq__1__iff,axiom,
    ! [A: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_Code_integer )
      = ( ( A = one_one_Code_integer )
        | ( A
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% power2_eq_1_iff
thf(fact_5903_power2__eq__1__iff,axiom,
    ! [A: rat] :
      ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_rat )
      = ( ( A = one_one_rat )
        | ( A
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% power2_eq_1_iff
thf(fact_5904_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_5905_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_5906_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_5907_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_5908_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_5909_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_5910_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_5911_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_5912_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_5913_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_5914_realpow__square__minus__le,axiom,
    ! [U: real,X3: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% realpow_square_minus_le
thf(fact_5915_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_5916_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_5917_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_5918_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_5919_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_5920_zminus1__lemma,axiom,
    ! [A: int,B2: int,Q2: int,R: int] :
      ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q2 @ R ) )
     => ( ( B2 != zero_zero_int )
       => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B2 @ ( product_Pair_int_int @ ( if_int @ ( R = zero_zero_int ) @ ( uminus_uminus_int @ Q2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q2 ) @ one_one_int ) ) @ ( if_int @ ( R = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B2 @ R ) ) ) ) ) ) ).

% zminus1_lemma
thf(fact_5921_bits__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [A3: nat] :
          ( ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = A3 )
         => ( P @ A3 ) )
     => ( ! [A3: nat,B3: $o] :
            ( ( P @ A3 )
           => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                = A3 )
             => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_5922_bits__induct,axiom,
    ! [P: int > $o,A: int] :
      ( ! [A3: int] :
          ( ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = A3 )
         => ( P @ A3 ) )
     => ( ! [A3: int,B3: $o] :
            ( ( P @ A3 )
           => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = A3 )
             => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_5923_bits__induct,axiom,
    ! [P: code_integer > $o,A: code_integer] :
      ( ! [A3: code_integer] :
          ( ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = A3 )
         => ( P @ A3 ) )
     => ( ! [A3: code_integer,B3: $o] :
            ( ( P @ A3 )
           => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B3 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
                = A3 )
             => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B3 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_5924_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_5925_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_5926_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_5927_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_5928_square__le__1,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_eq_real @ X3 @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).

% square_le_1
thf(fact_5929_square__le__1,axiom,
    ! [X3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X3 )
     => ( ( ord_le3102999989581377725nteger @ X3 @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% square_le_1
thf(fact_5930_square__le__1,axiom,
    ! [X3: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X3 )
     => ( ( ord_less_eq_rat @ X3 @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).

% square_le_1
thf(fact_5931_square__le__1,axiom,
    ! [X3: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X3 )
     => ( ( ord_less_eq_int @ X3 @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% square_le_1
thf(fact_5932_minus__power__mult__self,axiom,
    ! [A: real,N: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
      = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_5933_minus__power__mult__self,axiom,
    ! [A: int,N: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
      = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_5934_minus__power__mult__self,axiom,
    ! [A: complex,N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) )
      = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_5935_minus__power__mult__self,axiom,
    ! [A: code_integer,N: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
      = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_5936_minus__power__mult__self,axiom,
    ! [A: rat,N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
      = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_5937_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_5938_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_5939_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_5940_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_5941_add__0__iff,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2
        = ( plus_plus_complex @ B2 @ A ) )
      = ( A = zero_zero_complex ) ) ).

% add_0_iff
thf(fact_5942_add__0__iff,axiom,
    ! [B2: real,A: real] :
      ( ( B2
        = ( plus_plus_real @ B2 @ A ) )
      = ( A = zero_zero_real ) ) ).

% add_0_iff
thf(fact_5943_add__0__iff,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2
        = ( plus_plus_rat @ B2 @ A ) )
      = ( A = zero_zero_rat ) ) ).

% add_0_iff
thf(fact_5944_add__0__iff,axiom,
    ! [B2: nat,A: nat] :
      ( ( B2
        = ( plus_plus_nat @ B2 @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_5945_add__0__iff,axiom,
    ! [B2: int,A: int] :
      ( ( B2
        = ( plus_plus_int @ B2 @ A ) )
      = ( A = zero_zero_int ) ) ).

% add_0_iff
thf(fact_5946_exp__mod__exp,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_5947_exp__mod__exp,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_5948_exp__mod__exp,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_5949_crossproduct__noteq,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ( A != B2 )
        & ( C != D ) )
      = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) )
       != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B2 @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_5950_crossproduct__noteq,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ( A != B2 )
        & ( C != D ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) )
       != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_5951_crossproduct__noteq,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ( A != B2 )
        & ( C != D ) )
      = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) )
       != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_5952_crossproduct__noteq,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ( A != B2 )
        & ( C != D ) )
      = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) )
       != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B2 @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_5953_crossproduct__eq,axiom,
    ! [W: real,Y3: real,X3: real,Z: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ W @ Y3 ) @ ( times_times_real @ X3 @ Z ) )
        = ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X3 @ Y3 ) ) )
      = ( ( W = X3 )
        | ( Y3 = Z ) ) ) ).

% crossproduct_eq
thf(fact_5954_crossproduct__eq,axiom,
    ! [W: rat,Y3: rat,X3: rat,Z: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y3 ) @ ( times_times_rat @ X3 @ Z ) )
        = ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X3 @ Y3 ) ) )
      = ( ( W = X3 )
        | ( Y3 = Z ) ) ) ).

% crossproduct_eq
thf(fact_5955_crossproduct__eq,axiom,
    ! [W: nat,Y3: nat,X3: nat,Z: nat] :
      ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y3 ) @ ( times_times_nat @ X3 @ Z ) )
        = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X3 @ Y3 ) ) )
      = ( ( W = X3 )
        | ( Y3 = Z ) ) ) ).

% crossproduct_eq
thf(fact_5956_crossproduct__eq,axiom,
    ! [W: int,Y3: int,X3: int,Z: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ W @ Y3 ) @ ( times_times_int @ X3 @ Z ) )
        = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X3 @ Y3 ) ) )
      = ( ( W = X3 )
        | ( Y3 = Z ) ) ) ).

% crossproduct_eq
thf(fact_5957_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_5958_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_5959_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_5960_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_5961_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_5962_int__bit__induct,axiom,
    ! [P: int > $o,K: int] :
      ( ( P @ zero_zero_int )
     => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
       => ( ! [K3: int] :
              ( ( P @ K3 )
             => ( ( K3 != zero_zero_int )
               => ( P @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
         => ( ! [K3: int] :
                ( ( P @ K3 )
               => ( ( K3
                   != ( uminus_uminus_int @ one_one_int ) )
                 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
           => ( P @ K ) ) ) ) ) ).

% int_bit_induct
thf(fact_5963_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_5964_set__decode__def,axiom,
    ( nat_set_decode
    = ( ^ [X: nat] :
          ( collect_nat
          @ ^ [N2: nat] :
              ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% set_decode_def
thf(fact_5965_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_nat
        @ ( zero_n2687167440665602831ol_nat
          @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
             != zero_zero_nat )
            & ( ord_less_eq_nat @ N @ M ) ) )
        @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_5966_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_int
        @ ( zero_n2684676970156552555ol_int
          @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
             != zero_zero_int )
            & ( ord_less_eq_nat @ N @ M ) ) )
        @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_5967_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( times_3573771949741848930nteger
        @ ( zero_n356916108424825756nteger
          @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
             != zero_z3403309356797280102nteger )
            & ( ord_less_eq_nat @ N @ M ) ) )
        @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_5968_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.simps(3)
thf(fact_5969_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R: int] :
      ( ( adjust_div @ ( product_Pair_int_int @ Q2 @ R ) )
      = ( plus_plus_int @ Q2 @ ( zero_n2684676970156552555ol_int @ ( R != zero_zero_int ) ) ) ) ).

% Divides.adjust_div_eq
thf(fact_5970_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_5971_vebt__buildup_Opelims,axiom,
    ! [X3: nat,Y3: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X3 )
        = Y3 )
     => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X3 )
       => ( ( ( X3 = zero_zero_nat )
           => ( ( Y3
                = ( vEBT_Leaf @ $false @ $false ) )
             => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
         => ( ( ( X3
                = ( suc @ zero_zero_nat ) )
             => ( ( Y3
                  = ( vEBT_Leaf @ $false @ $false ) )
               => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va2: nat] :
                  ( ( X3
                    = ( suc @ ( suc @ Va2 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                       => ( Y3
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                       => ( Y3
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.pelims
thf(fact_5972_ln__one__minus__pos__lower__bound,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X3 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ).

% ln_one_minus_pos_lower_bound
thf(fact_5973_artanh__def,axiom,
    ( artanh_real
    = ( ^ [X: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X ) @ ( minus_minus_real @ one_one_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% artanh_def
thf(fact_5974_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_eq_int @ M @ N )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X: int] : X
          @ ( 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_5975_divmod__step__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L2: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_def
thf(fact_5976_divmod__step__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L2: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_def
thf(fact_5977_divmod__step__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L2: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_def
thf(fact_5978_semiring__norm_I90_J,axiom,
    ! [M: num,N: num] :
      ( ( ( bit1 @ M )
        = ( bit1 @ N ) )
      = ( M = N ) ) ).

% semiring_norm(90)
thf(fact_5979_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( ( bit1 @ X32 )
        = ( bit1 @ Y32 ) )
      = ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
thf(fact_5980_semiring__norm_I88_J,axiom,
    ! [M: num,N: num] :
      ( ( bit0 @ M )
     != ( bit1 @ N ) ) ).

% semiring_norm(88)
thf(fact_5981_semiring__norm_I89_J,axiom,
    ! [M: num,N: num] :
      ( ( bit1 @ M )
     != ( bit0 @ N ) ) ).

% semiring_norm(89)
thf(fact_5982_semiring__norm_I84_J,axiom,
    ! [N: num] :
      ( one
     != ( bit1 @ N ) ) ).

% semiring_norm(84)
thf(fact_5983_semiring__norm_I86_J,axiom,
    ! [M: num] :
      ( ( bit1 @ M )
     != one ) ).

% semiring_norm(86)
thf(fact_5984_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_5985_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_5986_case__prod__conv,axiom,
    ! [F: nat > nat > nat,A: nat,B2: nat] :
      ( ( produc6842872674320459806at_nat @ F @ ( product_Pair_nat_nat @ A @ B2 ) )
      = ( F @ A @ B2 ) ) ).

% case_prod_conv
thf(fact_5987_case__prod__conv,axiom,
    ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,A: nat,B2: nat] :
      ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ A @ B2 ) )
      = ( F @ A @ B2 ) ) ).

% case_prod_conv
thf(fact_5988_case__prod__conv,axiom,
    ! [F: nat > nat > product_prod_nat_nat > $o,A: nat,B2: nat] :
      ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ A @ B2 ) )
      = ( F @ A @ B2 ) ) ).

% case_prod_conv
thf(fact_5989_case__prod__conv,axiom,
    ! [F: int > int > product_prod_int_int,A: int,B2: int] :
      ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ A @ B2 ) )
      = ( F @ A @ B2 ) ) ).

% case_prod_conv
thf(fact_5990_case__prod__conv,axiom,
    ! [F: int > int > $o,A: int,B2: int] :
      ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B2 ) )
      = ( F @ A @ B2 ) ) ).

% case_prod_conv
thf(fact_5991_ln__inj__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ( ( ln_ln_real @ X3 )
            = ( ln_ln_real @ Y3 ) )
          = ( X3 = Y3 ) ) ) ) ).

% ln_inj_iff
thf(fact_5992_ln__less__cancel__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) )
          = ( ord_less_real @ X3 @ Y3 ) ) ) ) ).

% ln_less_cancel_iff
thf(fact_5993_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_5994_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_5995_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_5996_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_5997_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_5998_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_5999_semiring__norm_I70_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).

% semiring_norm(70)
thf(fact_6000_semiring__norm_I77_J,axiom,
    ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).

% semiring_norm(77)
thf(fact_6001_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_6002_semiring__norm_I3_J,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bit0 @ N ) )
      = ( bit1 @ N ) ) ).

% semiring_norm(3)
thf(fact_6003_semiring__norm_I4_J,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bit1 @ N ) )
      = ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).

% semiring_norm(4)
thf(fact_6004_semiring__norm_I5_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ one )
      = ( bit1 @ M ) ) ).

% semiring_norm(5)
thf(fact_6005_semiring__norm_I8_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ one )
      = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).

% semiring_norm(8)
thf(fact_6006_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_6007_ln__le__cancel__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) )
          = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ).

% ln_le_cancel_iff
thf(fact_6008_ln__less__zero__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ ( ln_ln_real @ X3 ) @ zero_zero_real )
        = ( ord_less_real @ X3 @ one_one_real ) ) ) ).

% ln_less_zero_iff
thf(fact_6009_ln__gt__zero__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
        = ( ord_less_real @ one_one_real @ X3 ) ) ) ).

% ln_gt_zero_iff
thf(fact_6010_ln__eq__zero__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ( ln_ln_real @ X3 )
          = zero_zero_real )
        = ( X3 = one_one_real ) ) ) ).

% ln_eq_zero_iff
thf(fact_6011_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_6012_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_6013_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_6014_ln__ge__zero__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
        = ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).

% ln_ge_zero_iff
thf(fact_6015_ln__le__zero__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ zero_zero_real )
        = ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ).

% ln_le_zero_iff
thf(fact_6016_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_6017_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_6018_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_6019_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_6020_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_6021_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_6022_verit__eq__simplify_I14_J,axiom,
    ! [X2: num,X32: num] :
      ( ( bit0 @ X2 )
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(14)
thf(fact_6023_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] :
      ( one
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(12)
thf(fact_6024_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > nat,X1: nat,X2: nat] :
      ( ( produc6842872674320459806at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_6025_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X1: nat,X2: nat] :
      ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_6026_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > product_prod_nat_nat > $o,X1: nat,X2: nat] :
      ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_6027_old_Oprod_Ocase,axiom,
    ! [F: int > int > product_prod_int_int,X1: int,X2: int] :
      ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_6028_old_Oprod_Ocase,axiom,
    ! [F: int > int > $o,X1: int,X2: int] :
      ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_6029_sum__mono,axiom,
    ! [K5: set_nat,F: nat > rat,G: nat > rat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K5 ) @ ( groups2906978787729119204at_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6030_sum__mono,axiom,
    ! [K5: set_real,F: real > rat,G: real > rat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K5 ) @ ( groups1300246762558778688al_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6031_sum__mono,axiom,
    ! [K5: set_int,F: int > rat,G: int > rat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K5 ) @ ( groups3906332499630173760nt_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6032_sum__mono,axiom,
    ! [K5: set_complex,F: complex > rat,G: complex > rat] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ K5 ) @ ( groups5058264527183730370ex_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6033_sum__mono,axiom,
    ! [K5: set_real,F: real > nat,G: real > nat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K5 ) @ ( groups1935376822645274424al_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6034_sum__mono,axiom,
    ! [K5: set_int,F: int > nat,G: int > nat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6035_sum__mono,axiom,
    ! [K5: set_complex,F: complex > nat,G: complex > nat] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ K5 ) @ ( groups5693394587270226106ex_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6036_sum__mono,axiom,
    ! [K5: set_nat,F: nat > int,G: nat > int] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ K5 )
         => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K5 ) @ ( groups3539618377306564664at_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6037_sum__mono,axiom,
    ! [K5: set_real,F: real > int,G: real > int] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ K5 )
         => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K5 ) @ ( groups1932886352136224148al_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6038_sum__mono,axiom,
    ! [K5: set_complex,F: complex > int,G: complex > int] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ K5 )
         => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ K5 ) @ ( groups5690904116761175830ex_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_6039_sum__product,axiom,
    ! [F: int > int,A4: set_int,G: int > int,B5: set_int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ G @ B5 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [I4: int] :
            ( groups4538972089207619220nt_int
            @ ^ [J3: int] : ( times_times_int @ ( F @ I4 ) @ ( G @ J3 ) )
            @ B5 )
        @ A4 ) ) ).

% sum_product
thf(fact_6040_sum__product,axiom,
    ! [F: complex > complex,A4: set_complex,G: complex > complex,B5: set_complex] :
      ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A4 ) @ ( groups7754918857620584856omplex @ G @ B5 ) )
      = ( groups7754918857620584856omplex
        @ ^ [I4: complex] :
            ( groups7754918857620584856omplex
            @ ^ [J3: complex] : ( times_times_complex @ ( F @ I4 ) @ ( G @ J3 ) )
            @ B5 )
        @ A4 ) ) ).

% sum_product
thf(fact_6041_sum__product,axiom,
    ! [F: nat > nat,A4: set_nat,G: nat > nat,B5: set_nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ G @ B5 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( times_times_nat @ ( F @ I4 ) @ ( G @ J3 ) )
            @ B5 )
        @ A4 ) ) ).

% sum_product
thf(fact_6042_sum__product,axiom,
    ! [F: nat > real,A4: set_nat,G: nat > real,B5: set_nat] :
      ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ ( groups6591440286371151544t_real @ G @ B5 ) )
      = ( groups6591440286371151544t_real
        @ ^ [I4: nat] :
            ( groups6591440286371151544t_real
            @ ^ [J3: nat] : ( times_times_real @ ( F @ I4 ) @ ( G @ J3 ) )
            @ B5 )
        @ A4 ) ) ).

% sum_product
thf(fact_6043_sum__distrib__right,axiom,
    ! [F: int > int,A4: set_int,R: int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ R )
      = ( groups4538972089207619220nt_int
        @ ^ [N2: int] : ( times_times_int @ ( F @ N2 ) @ R )
        @ A4 ) ) ).

% sum_distrib_right
thf(fact_6044_sum__distrib__right,axiom,
    ! [F: complex > complex,A4: set_complex,R: complex] :
      ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A4 ) @ R )
      = ( groups7754918857620584856omplex
        @ ^ [N2: complex] : ( times_times_complex @ ( F @ N2 ) @ R )
        @ A4 ) ) ).

% sum_distrib_right
thf(fact_6045_sum__distrib__right,axiom,
    ! [F: nat > nat,A4: set_nat,R: nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ R )
      = ( groups3542108847815614940at_nat
        @ ^ [N2: nat] : ( times_times_nat @ ( F @ N2 ) @ R )
        @ A4 ) ) ).

% sum_distrib_right
thf(fact_6046_sum__distrib__right,axiom,
    ! [F: nat > real,A4: set_nat,R: real] :
      ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ R )
      = ( groups6591440286371151544t_real
        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ R )
        @ A4 ) ) ).

% sum_distrib_right
thf(fact_6047_sum__distrib__left,axiom,
    ! [R: int,F: int > int,A4: set_int] :
      ( ( times_times_int @ R @ ( groups4538972089207619220nt_int @ F @ A4 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [N2: int] : ( times_times_int @ R @ ( F @ N2 ) )
        @ A4 ) ) ).

% sum_distrib_left
thf(fact_6048_sum__distrib__left,axiom,
    ! [R: complex,F: complex > complex,A4: set_complex] :
      ( ( times_times_complex @ R @ ( groups7754918857620584856omplex @ F @ A4 ) )
      = ( groups7754918857620584856omplex
        @ ^ [N2: complex] : ( times_times_complex @ R @ ( F @ N2 ) )
        @ A4 ) ) ).

% sum_distrib_left
thf(fact_6049_sum__distrib__left,axiom,
    ! [R: nat,F: nat > nat,A4: set_nat] :
      ( ( times_times_nat @ R @ ( groups3542108847815614940at_nat @ F @ A4 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [N2: nat] : ( times_times_nat @ R @ ( F @ N2 ) )
        @ A4 ) ) ).

% sum_distrib_left
thf(fact_6050_sum__distrib__left,axiom,
    ! [R: real,F: nat > real,A4: set_nat] :
      ( ( times_times_real @ R @ ( groups6591440286371151544t_real @ F @ A4 ) )
      = ( groups6591440286371151544t_real
        @ ^ [N2: nat] : ( times_times_real @ R @ ( F @ N2 ) )
        @ A4 ) ) ).

% sum_distrib_left
thf(fact_6051_sum_Odistrib,axiom,
    ! [G: int > int,H2: int > int,A4: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X: int] : ( plus_plus_int @ ( G @ X ) @ ( H2 @ X ) )
        @ A4 )
      = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A4 ) @ ( groups4538972089207619220nt_int @ H2 @ A4 ) ) ) ).

% sum.distrib
thf(fact_6052_sum_Odistrib,axiom,
    ! [G: complex > complex,H2: complex > complex,A4: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X: complex] : ( plus_plus_complex @ ( G @ X ) @ ( H2 @ X ) )
        @ A4 )
      = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A4 ) @ ( groups7754918857620584856omplex @ H2 @ A4 ) ) ) ).

% sum.distrib
thf(fact_6053_sum_Odistrib,axiom,
    ! [G: nat > nat,H2: nat > nat,A4: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : ( plus_plus_nat @ ( G @ X ) @ ( H2 @ X ) )
        @ A4 )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A4 ) @ ( groups3542108847815614940at_nat @ H2 @ A4 ) ) ) ).

% sum.distrib
thf(fact_6054_sum_Odistrib,axiom,
    ! [G: nat > real,H2: nat > real,A4: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X: nat] : ( plus_plus_real @ ( G @ X ) @ ( H2 @ X ) )
        @ A4 )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A4 ) @ ( groups6591440286371151544t_real @ H2 @ A4 ) ) ) ).

% sum.distrib
thf(fact_6055_sum__subtractf,axiom,
    ! [F: int > int,G: int > int,A4: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X: int] : ( minus_minus_int @ ( F @ X ) @ ( G @ X ) )
        @ A4 )
      = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ G @ A4 ) ) ) ).

% sum_subtractf
thf(fact_6056_sum__subtractf,axiom,
    ! [F: complex > complex,G: complex > complex,A4: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X: complex] : ( minus_minus_complex @ ( F @ X ) @ ( G @ X ) )
        @ A4 )
      = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A4 ) @ ( groups7754918857620584856omplex @ G @ A4 ) ) ) ).

% sum_subtractf
thf(fact_6057_sum__subtractf,axiom,
    ! [F: nat > real,G: nat > real,A4: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X: nat] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) )
        @ A4 )
      = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ ( groups6591440286371151544t_real @ G @ A4 ) ) ) ).

% sum_subtractf
thf(fact_6058_sum__divide__distrib,axiom,
    ! [F: complex > complex,A4: set_complex,R: complex] :
      ( ( divide1717551699836669952omplex @ ( groups7754918857620584856omplex @ F @ A4 ) @ R )
      = ( groups7754918857620584856omplex
        @ ^ [N2: complex] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ R )
        @ A4 ) ) ).

% sum_divide_distrib
thf(fact_6059_sum__divide__distrib,axiom,
    ! [F: nat > real,A4: set_nat,R: real] :
      ( ( divide_divide_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ R )
      = ( groups6591440286371151544t_real
        @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ R )
        @ A4 ) ) ).

% sum_divide_distrib
thf(fact_6060_mod__sum__eq,axiom,
    ! [F: int > int,A: int,A4: set_int] :
      ( ( modulo_modulo_int
        @ ( groups4538972089207619220nt_int
          @ ^ [I4: int] : ( modulo_modulo_int @ ( F @ I4 ) @ A )
          @ A4 )
        @ A )
      = ( modulo_modulo_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ A ) ) ).

% mod_sum_eq
thf(fact_6061_mod__sum__eq,axiom,
    ! [F: nat > nat,A: nat,A4: set_nat] :
      ( ( modulo_modulo_nat
        @ ( groups3542108847815614940at_nat
          @ ^ [I4: nat] : ( modulo_modulo_nat @ ( F @ I4 ) @ A )
          @ A4 )
        @ A )
      = ( modulo_modulo_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ A ) ) ).

% mod_sum_eq
thf(fact_6062_cond__case__prod__eta,axiom,
    ! [F: nat > nat > nat,G: product_prod_nat_nat > nat] :
      ( ! [X5: nat,Y4: nat] :
          ( ( F @ X5 @ Y4 )
          = ( G @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) )
     => ( ( produc6842872674320459806at_nat @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_6063_cond__case__prod__eta,axiom,
    ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ! [X5: nat,Y4: nat] :
          ( ( F @ X5 @ Y4 )
          = ( G @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) )
     => ( ( produc27273713700761075at_nat @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_6064_cond__case__prod__eta,axiom,
    ! [F: nat > nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ! [X5: nat,Y4: nat] :
          ( ( F @ X5 @ Y4 )
          = ( G @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) )
     => ( ( produc8739625826339149834_nat_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_6065_cond__case__prod__eta,axiom,
    ! [F: int > int > product_prod_int_int,G: product_prod_int_int > product_prod_int_int] :
      ( ! [X5: int,Y4: int] :
          ( ( F @ X5 @ Y4 )
          = ( G @ ( product_Pair_int_int @ X5 @ Y4 ) ) )
     => ( ( produc4245557441103728435nt_int @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_6066_cond__case__prod__eta,axiom,
    ! [F: int > int > $o,G: product_prod_int_int > $o] :
      ( ! [X5: int,Y4: int] :
          ( ( F @ X5 @ Y4 )
          = ( G @ ( product_Pair_int_int @ X5 @ Y4 ) ) )
     => ( ( produc4947309494688390418_int_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_6067_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > nat] :
      ( ( produc6842872674320459806at_nat
        @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_6068_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( produc27273713700761075at_nat
        @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_6069_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( produc8739625826339149834_nat_o
        @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_6070_case__prod__eta,axiom,
    ! [F: product_prod_int_int > product_prod_int_int] :
      ( ( produc4245557441103728435nt_int
        @ ^ [X: int,Y: int] : ( F @ ( product_Pair_int_int @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_6071_case__prod__eta,axiom,
    ! [F: product_prod_int_int > $o] :
      ( ( produc4947309494688390418_int_o
        @ ^ [X: int,Y: int] : ( F @ ( product_Pair_int_int @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_6072_case__prodE2,axiom,
    ! [Q: nat > $o,P: nat > nat > nat,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc6842872674320459806at_nat @ P @ Z ) )
     => ~ ! [X5: nat,Y4: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X5 @ Y4 ) )
           => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_6073_case__prodE2,axiom,
    ! [Q: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,P: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc27273713700761075at_nat @ P @ Z ) )
     => ~ ! [X5: nat,Y4: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X5 @ Y4 ) )
           => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_6074_case__prodE2,axiom,
    ! [Q: ( product_prod_nat_nat > $o ) > $o,P: nat > nat > product_prod_nat_nat > $o,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc8739625826339149834_nat_o @ P @ Z ) )
     => ~ ! [X5: nat,Y4: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X5 @ Y4 ) )
           => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_6075_case__prodE2,axiom,
    ! [Q: product_prod_int_int > $o,P: int > int > product_prod_int_int,Z: product_prod_int_int] :
      ( ( Q @ ( produc4245557441103728435nt_int @ P @ Z ) )
     => ~ ! [X5: int,Y4: int] :
            ( ( Z
              = ( product_Pair_int_int @ X5 @ Y4 ) )
           => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_6076_case__prodE2,axiom,
    ! [Q: $o > $o,P: int > int > $o,Z: product_prod_int_int] :
      ( ( Q @ ( produc4947309494688390418_int_o @ P @ Z ) )
     => ~ ! [X5: int,Y4: int] :
            ( ( Z
              = ( product_Pair_int_int @ X5 @ Y4 ) )
           => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_6077_sum__nonneg,axiom,
    ! [A4: set_real,F: real > real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_6078_sum__nonneg,axiom,
    ! [A4: set_int,F: int > real] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_6079_sum__nonneg,axiom,
    ! [A4: set_complex,F: complex > real] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A4 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_6080_sum__nonneg,axiom,
    ! [A4: set_nat,F: nat > rat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_6081_sum__nonneg,axiom,
    ! [A4: set_real,F: real > rat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_6082_sum__nonneg,axiom,
    ! [A4: set_int,F: int > rat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_6083_sum__nonneg,axiom,
    ! [A4: set_complex,F: complex > rat] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_6084_sum__nonneg,axiom,
    ! [A4: set_real,F: real > nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_6085_sum__nonneg,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_6086_sum__nonneg,axiom,
    ! [A4: set_complex,F: complex > nat] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A4 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_6087_sum__nonpos,axiom,
    ! [A4: set_real,F: real > real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_6088_sum__nonpos,axiom,
    ! [A4: set_int,F: int > real] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_6089_sum__nonpos,axiom,
    ! [A4: set_complex,F: complex > real] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A4 )
         => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_6090_sum__nonpos,axiom,
    ! [A4: set_nat,F: nat > rat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_6091_sum__nonpos,axiom,
    ! [A4: set_real,F: real > rat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_6092_sum__nonpos,axiom,
    ! [A4: set_int,F: int > rat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_6093_sum__nonpos,axiom,
    ! [A4: set_complex,F: complex > rat] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A4 )
         => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_6094_sum__nonpos,axiom,
    ! [A4: set_real,F: real > nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_6095_sum__nonpos,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_6096_sum__nonpos,axiom,
    ! [A4: set_complex,F: complex > nat] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A4 )
         => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_6097_sum__mono__inv,axiom,
    ! [F: real > rat,I5: set_real,G: real > rat,I: real] :
      ( ( ( groups1300246762558778688al_rat @ F @ I5 )
        = ( groups1300246762558778688al_rat @ G @ I5 ) )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_real @ I @ I5 )
         => ( ( finite_finite_real @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6098_sum__mono__inv,axiom,
    ! [F: nat > rat,I5: set_nat,G: nat > rat,I: nat] :
      ( ( ( groups2906978787729119204at_rat @ F @ I5 )
        = ( groups2906978787729119204at_rat @ G @ I5 ) )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( finite_finite_nat @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6099_sum__mono__inv,axiom,
    ! [F: int > rat,I5: set_int,G: int > rat,I: int] :
      ( ( ( groups3906332499630173760nt_rat @ F @ I5 )
        = ( groups3906332499630173760nt_rat @ G @ I5 ) )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_int @ I @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6100_sum__mono__inv,axiom,
    ! [F: complex > rat,I5: set_complex,G: complex > rat,I: complex] :
      ( ( ( groups5058264527183730370ex_rat @ F @ I5 )
        = ( groups5058264527183730370ex_rat @ G @ I5 ) )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_complex @ I @ I5 )
         => ( ( finite3207457112153483333omplex @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6101_sum__mono__inv,axiom,
    ! [F: real > nat,I5: set_real,G: real > nat,I: real] :
      ( ( ( groups1935376822645274424al_nat @ F @ I5 )
        = ( groups1935376822645274424al_nat @ G @ I5 ) )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_real @ I @ I5 )
         => ( ( finite_finite_real @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6102_sum__mono__inv,axiom,
    ! [F: int > nat,I5: set_int,G: int > nat,I: int] :
      ( ( ( groups4541462559716669496nt_nat @ F @ I5 )
        = ( groups4541462559716669496nt_nat @ G @ I5 ) )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_int @ I @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6103_sum__mono__inv,axiom,
    ! [F: complex > nat,I5: set_complex,G: complex > nat,I: complex] :
      ( ( ( groups5693394587270226106ex_nat @ F @ I5 )
        = ( groups5693394587270226106ex_nat @ G @ I5 ) )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_complex @ I @ I5 )
         => ( ( finite3207457112153483333omplex @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6104_sum__mono__inv,axiom,
    ! [F: real > int,I5: set_real,G: real > int,I: real] :
      ( ( ( groups1932886352136224148al_int @ F @ I5 )
        = ( groups1932886352136224148al_int @ G @ I5 ) )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ I5 )
           => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_real @ I @ I5 )
         => ( ( finite_finite_real @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6105_sum__mono__inv,axiom,
    ! [F: nat > int,I5: set_nat,G: nat > int,I: nat] :
      ( ( ( groups3539618377306564664at_int @ F @ I5 )
        = ( groups3539618377306564664at_int @ G @ I5 ) )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ I5 )
           => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( finite_finite_nat @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6106_sum__mono__inv,axiom,
    ! [F: complex > int,I5: set_complex,G: complex > int,I: complex] :
      ( ( ( groups5690904116761175830ex_int @ F @ I5 )
        = ( groups5690904116761175830ex_int @ G @ I5 ) )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ I5 )
           => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_complex @ I @ I5 )
         => ( ( finite3207457112153483333omplex @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_6107_ln__less__self,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ord_less_real @ ( ln_ln_real @ X3 ) @ X3 ) ) ).

% ln_less_self
thf(fact_6108_xor__num_Ocases,axiom,
    ! [X3: product_prod_num_num] :
      ( ( X3
       != ( product_Pair_num_num @ one @ one ) )
     => ( ! [N3: num] :
            ( X3
           != ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) )
       => ( ! [N3: num] :
              ( X3
             != ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) )
         => ( ! [M4: num] :
                ( X3
               != ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) )
           => ( ! [M4: num,N3: num] :
                  ( X3
                 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N3 ) ) )
             => ( ! [M4: num,N3: num] :
                    ( X3
                   != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N3 ) ) )
               => ( ! [M4: num] :
                      ( X3
                     != ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) )
                 => ( ! [M4: num,N3: num] :
                        ( X3
                       != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N3 ) ) )
                   => ~ ! [M4: num,N3: num] :
                          ( X3
                         != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.cases
thf(fact_6109_num_Oexhaust,axiom,
    ! [Y3: num] :
      ( ( Y3 != one )
     => ( ! [X22: num] :
            ( Y3
           != ( bit0 @ X22 ) )
       => ~ ! [X33: num] :
              ( Y3
             != ( bit1 @ X33 ) ) ) ) ).

% num.exhaust
thf(fact_6110_sum__le__included,axiom,
    ! [S: set_int,T: set_int,G: int > real,I: int > int,F: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X5 ) ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6111_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G: complex > real,I: complex > int,F: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X5 ) ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6112_sum__le__included,axiom,
    ! [S: set_complex,T: set_int,G: int > real,I: int > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X5 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6113_sum__le__included,axiom,
    ! [S: set_complex,T: set_complex,G: complex > real,I: complex > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X5 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6114_sum__le__included,axiom,
    ! [S: set_nat,T: set_nat,G: nat > rat,I: nat > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6115_sum__le__included,axiom,
    ! [S: set_nat,T: set_int,G: int > rat,I: int > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6116_sum__le__included,axiom,
    ! [S: set_nat,T: set_complex,G: complex > rat,I: complex > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6117_sum__le__included,axiom,
    ! [S: set_int,T: set_nat,G: nat > rat,I: nat > int,F: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6118_sum__le__included,axiom,
    ! [S: set_int,T: set_int,G: int > rat,I: int > int,F: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6119_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G: complex > rat,I: complex > int,F: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_6120_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ A4 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ A4 )
            = zero_zero_real )
          = ( ! [X: real] :
                ( ( member_real @ X @ A4 )
               => ( ( F @ X )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6121_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A4 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ A4 )
            = zero_zero_real )
          = ( ! [X: int] :
                ( ( member_int @ X @ A4 )
               => ( ( F @ X )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6122_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A4 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ A4 )
            = zero_zero_real )
          = ( ! [X: complex] :
                ( ( member_complex @ X @ A4 )
               => ( ( F @ X )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6123_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ A4 )
            = zero_zero_rat )
          = ( ! [X: real] :
                ( ( member_real @ X @ A4 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6124_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ A4 )
            = zero_zero_rat )
          = ( ! [X: nat] :
                ( ( member_nat @ X @ A4 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6125_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ A4 )
            = zero_zero_rat )
          = ( ! [X: int] :
                ( ( member_int @ X @ A4 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6126_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ A4 )
            = zero_zero_rat )
          = ( ! [X: complex] :
                ( ( member_complex @ X @ A4 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6127_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F: real > nat] :
      ( ( finite_finite_real @ A4 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ A4 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F @ A4 )
            = zero_zero_nat )
          = ( ! [X: real] :
                ( ( member_real @ X @ A4 )
               => ( ( F @ X )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6128_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A4 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F @ A4 )
            = zero_zero_nat )
          = ( ! [X: int] :
                ( ( member_int @ X @ A4 )
               => ( ( F @ X )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6129_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A4 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
       => ( ( ( groups5693394587270226106ex_nat @ F @ A4 )
            = zero_zero_nat )
          = ( ! [X: complex] :
                ( ( member_complex @ X @ A4 )
               => ( ( F @ X )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_6130_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A4 )
           => ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X6: int] :
              ( ( member_int @ X6 @ A4 )
              & ( ord_less_real @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6131_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A4 )
           => ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X6: complex] :
              ( ( member_complex @ X6 @ A4 )
              & ( ord_less_real @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6132_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ A4 )
           => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X6: nat] :
              ( ( member_nat @ X6 @ A4 )
              & ( ord_less_rat @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6133_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A4 )
           => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X6: int] :
              ( ( member_int @ X6 @ A4 )
              & ( ord_less_rat @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6134_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A4 )
           => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X6: complex] :
              ( ( member_complex @ X6 @ A4 )
              & ( ord_less_rat @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6135_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A4 )
           => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X6: int] :
              ( ( member_int @ X6 @ A4 )
              & ( ord_less_nat @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6136_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A4 )
           => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X6: complex] :
              ( ( member_complex @ X6 @ A4 )
              & ( ord_less_nat @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6137_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F: nat > int,G: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ A4 )
           => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X6: nat] :
              ( ( member_nat @ X6 @ A4 )
              & ( ord_less_int @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6138_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: complex > int,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A4 )
           => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X6: complex] :
              ( ( member_complex @ X6 @ A4 )
              & ( ord_less_int @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6139_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: int > int,G: int > int] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A4 )
           => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X6: int] :
              ( ( member_int @ X6 @ A4 )
              & ( ord_less_int @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_6140_sum_Orelated,axiom,
    ! [R2: complex > complex > $o,S2: set_nat,H2: nat > complex,G: nat > complex] :
      ( ( R2 @ zero_zero_complex @ zero_zero_complex )
     => ( ! [X15: complex,Y15: complex,X22: complex,Y22: complex] :
            ( ( ( R2 @ X15 @ X22 )
              & ( R2 @ Y15 @ Y22 ) )
           => ( R2 @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X22 @ Y22 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S2 )
               => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R2 @ ( groups2073611262835488442omplex @ H2 @ S2 ) @ ( groups2073611262835488442omplex @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_6141_sum_Orelated,axiom,
    ! [R2: complex > complex > $o,S2: set_int,H2: int > complex,G: int > complex] :
      ( ( R2 @ zero_zero_complex @ zero_zero_complex )
     => ( ! [X15: complex,Y15: complex,X22: complex,Y22: complex] :
            ( ( ( R2 @ X15 @ X22 )
              & ( R2 @ Y15 @ Y22 ) )
           => ( R2 @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X22 @ Y22 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S2 )
               => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R2 @ ( groups3049146728041665814omplex @ H2 @ S2 ) @ ( groups3049146728041665814omplex @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_6142_sum_Orelated,axiom,
    ! [R2: real > real > $o,S2: set_int,H2: int > real,G: int > real] :
      ( ( R2 @ zero_zero_real @ zero_zero_real )
     => ( ! [X15: real,Y15: real,X22: real,Y22: real] :
            ( ( ( R2 @ X15 @ X22 )
              & ( R2 @ Y15 @ Y22 ) )
           => ( R2 @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X22 @ Y22 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S2 )
               => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R2 @ ( groups8778361861064173332t_real @ H2 @ S2 ) @ ( groups8778361861064173332t_real @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_6143_sum_Orelated,axiom,
    ! [R2: real > real > $o,S2: set_complex,H2: complex > real,G: complex > real] :
      ( ( R2 @ zero_zero_real @ zero_zero_real )
     => ( ! [X15: real,Y15: real,X22: real,Y22: real] :
            ( ( ( R2 @ X15 @ X22 )
              & ( R2 @ Y15 @ Y22 ) )
           => ( R2 @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X22 @ Y22 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R2 @ ( groups5808333547571424918x_real @ H2 @ S2 ) @ ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_6144_sum_Orelated,axiom,
    ! [R2: rat > rat > $o,S2: set_nat,H2: nat > rat,G: nat > rat] :
      ( ( R2 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X15: rat,Y15: rat,X22: rat,Y22: rat] :
            ( ( ( R2 @ X15 @ X22 )
              & ( R2 @ Y15 @ Y22 ) )
           => ( R2 @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X22 @ Y22 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S2 )
               => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R2 @ ( groups2906978787729119204at_rat @ H2 @ S2 ) @ ( groups2906978787729119204at_rat @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_6145_sum_Orelated,axiom,
    ! [R2: rat > rat > $o,S2: set_int,H2: int > rat,G: int > rat] :
      ( ( R2 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X15: rat,Y15: rat,X22: rat,Y22: rat] :
            ( ( ( R2 @ X15 @ X22 )
              & ( R2 @ Y15 @ Y22 ) )
           => ( R2 @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X22 @ Y22 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S2 )
               => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R2 @ ( groups3906332499630173760nt_rat @ H2 @ S2 ) @ ( groups3906332499630173760nt_rat @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_6146_sum_Orelated,axiom,
    ! [R2: rat > rat > $o,S2: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( R2 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X15: rat,Y15: rat,X22: rat,Y22: rat] :
            ( ( ( R2 @ X15 @ X22 )
              & ( R2 @ Y15 @ Y22 ) )
           => ( R2 @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X22 @ Y22 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R2 @ ( groups5058264527183730370ex_rat @ H2 @ S2 ) @ ( groups5058264527183730370ex_rat @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_6147_sum_Orelated,axiom,
    ! [R2: nat > nat > $o,S2: set_int,H2: int > nat,G: int > nat] :
      ( ( R2 @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X15: nat,Y15: nat,X22: nat,Y22: nat] :
            ( ( ( R2 @ X15 @ X22 )
              & ( R2 @ Y15 @ Y22 ) )
           => ( R2 @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X22 @ Y22 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S2 )
               => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R2 @ ( groups4541462559716669496nt_nat @ H2 @ S2 ) @ ( groups4541462559716669496nt_nat @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_6148_sum_Orelated,axiom,
    ! [R2: nat > nat > $o,S2: set_complex,H2: complex > nat,G: complex > nat] :
      ( ( R2 @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X15: nat,Y15: nat,X22: nat,Y22: nat] :
            ( ( ( R2 @ X15 @ X22 )
              & ( R2 @ Y15 @ Y22 ) )
           => ( R2 @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X22 @ Y22 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R2 @ ( groups5693394587270226106ex_nat @ H2 @ S2 ) @ ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_6149_sum_Orelated,axiom,
    ! [R2: int > int > $o,S2: set_nat,H2: nat > int,G: nat > int] :
      ( ( R2 @ zero_zero_int @ zero_zero_int )
     => ( ! [X15: int,Y15: int,X22: int,Y22: int] :
            ( ( ( R2 @ X15 @ X22 )
              & ( R2 @ Y15 @ Y22 ) )
           => ( R2 @ ( plus_plus_int @ X15 @ Y15 ) @ ( plus_plus_int @ X22 @ Y22 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S2 )
               => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R2 @ ( groups3539618377306564664at_int @ H2 @ S2 ) @ ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_6150_sum__strict__mono,axiom,
    ! [A4: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( A4 != bot_bot_set_complex )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ A4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6151_sum__strict__mono,axiom,
    ! [A4: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ A4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6152_sum__strict__mono,axiom,
    ! [A4: set_real,F: real > real,G: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ A4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6153_sum__strict__mono,axiom,
    ! [A4: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( A4 != bot_bot_set_complex )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ A4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6154_sum__strict__mono,axiom,
    ! [A4: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ A4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6155_sum__strict__mono,axiom,
    ! [A4: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ A4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6156_sum__strict__mono,axiom,
    ! [A4: set_real,F: real > rat,G: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ A4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6157_sum__strict__mono,axiom,
    ! [A4: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( A4 != bot_bot_set_complex )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ A4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6158_sum__strict__mono,axiom,
    ! [A4: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ A4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6159_sum__strict__mono,axiom,
    ! [A4: set_real,F: real > nat,G: real > nat] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ A4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_6160_ln__bound,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ X3 ) ) ).

% ln_bound
thf(fact_6161_ln__gt__zero__imp__gt__one,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ord_less_real @ one_one_real @ X3 ) ) ) ).

% ln_gt_zero_imp_gt_one
thf(fact_6162_ln__less__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ one_one_real )
       => ( ord_less_real @ ( ln_ln_real @ X3 ) @ zero_zero_real ) ) ) ).

% ln_less_zero
thf(fact_6163_ln__gt__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ one_one_real @ X3 )
     => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) ) ) ).

% ln_gt_zero
thf(fact_6164_ln__ge__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ one_one_real @ X3 )
     => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) ) ) ).

% ln_ge_zero
thf(fact_6165_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).

% numeral_Bit1
thf(fact_6166_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_6167_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_6168_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_6169_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_6170_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_6171_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_6172_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_6173_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_6174_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_6175_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_6176_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_6177_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_6178_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_6179_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_6180_power__minus__Bit1,axiom,
    ! [X3: real,K: num] :
      ( ( power_power_real @ ( uminus_uminus_real @ X3 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6181_power__minus__Bit1,axiom,
    ! [X3: int,K: num] :
      ( ( power_power_int @ ( uminus_uminus_int @ X3 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6182_power__minus__Bit1,axiom,
    ! [X3: complex,K: num] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X3 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6183_power__minus__Bit1,axiom,
    ! [X3: code_integer,K: num] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X3 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6184_power__minus__Bit1,axiom,
    ! [X3: rat,K: num] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ X3 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6185_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > real,B5: real,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S )
            = B5 )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6186_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > real,B5: real,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S )
            = B5 )
         => ( ( member_int @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6187_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > real,B5: real,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S )
            = B5 )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6188_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > rat,B5: rat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S )
            = B5 )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6189_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: nat > rat,B5: rat,I: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S )
            = B5 )
         => ( ( member_nat @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6190_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > rat,B5: rat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S )
            = B5 )
         => ( ( member_int @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6191_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > rat,B5: rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S )
            = B5 )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6192_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > nat,B5: nat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F @ S )
            = B5 )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_nat @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6193_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > nat,B5: nat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F @ S )
            = B5 )
         => ( ( member_int @ I @ S )
           => ( ord_less_eq_nat @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6194_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > nat,B5: nat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
       => ( ( ( groups5693394587270226106ex_nat @ F @ S )
            = B5 )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_nat @ ( F @ I ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_6195_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > real,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S )
            = zero_zero_real )
         => ( ( member_real @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6196_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > real,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S )
            = zero_zero_real )
         => ( ( member_int @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6197_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > real,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S )
            = zero_zero_real )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6198_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > rat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_real @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6199_sum__nonneg__0,axiom,
    ! [S: set_nat,F: nat > rat,I: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_nat @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6200_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > rat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_int @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6201_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6202_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > nat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F @ S )
            = zero_zero_nat )
         => ( ( member_real @ I @ S )
           => ( ( F @ I )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6203_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > nat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F @ S )
            = zero_zero_nat )
         => ( ( member_int @ I @ S )
           => ( ( F @ I )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6204_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > nat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
       => ( ( ( groups5693394587270226106ex_nat @ F @ S )
            = zero_zero_nat )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_6205_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).

% numeral_code(3)
thf(fact_6206_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_6207_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_6208_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_6209_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_6210_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_6211_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_6212_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_6213_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_6214_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_6215_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I3: real] :
                ( ( member_real @ I3 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6216_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6217_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6218_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I3: real] :
                ( ( member_real @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6219_sum__pos2,axiom,
    ! [I5: set_nat,I: nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6220_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6221_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6222_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: real > nat] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ! [I3: real] :
                ( ( member_real @ I3 @ I5 )
               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
           => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6223_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: int > nat] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ I5 )
               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
           => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6224_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ! [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
           => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_6225_sum__pos,axiom,
    ! [I5: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6226_sum__pos,axiom,
    ! [I5: set_int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6227_sum__pos,axiom,
    ! [I5: set_real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6228_sum__pos,axiom,
    ! [I5: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6229_sum__pos,axiom,
    ! [I5: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6230_sum__pos,axiom,
    ! [I5: set_int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6231_sum__pos,axiom,
    ! [I5: set_real,F: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6232_sum__pos,axiom,
    ! [I5: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6233_sum__pos,axiom,
    ! [I5: set_int,F: int > nat] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6234_sum__pos,axiom,
    ! [I5: set_real,F: real > nat] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ I5 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_6235_ln__ge__zero__imp__ge__one,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).

% ln_ge_zero_imp_ge_one
thf(fact_6236_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_complex ) )
             => ( ( ( groups5754745047067104278omplex @ G @ A4 )
                  = ( groups5754745047067104278omplex @ H2 @ B5 ) )
                = ( ( groups5754745047067104278omplex @ G @ C4 )
                  = ( groups5754745047067104278omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6237_sum_Osame__carrier,axiom,
    ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > complex,H2: nat > complex] :
      ( ( finite_finite_nat @ C4 )
     => ( ( ord_less_eq_set_nat @ A4 @ C4 )
       => ( ( ord_less_eq_set_nat @ B5 @ C4 )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_complex ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_complex ) )
             => ( ( ( groups2073611262835488442omplex @ G @ A4 )
                  = ( groups2073611262835488442omplex @ H2 @ B5 ) )
                = ( ( groups2073611262835488442omplex @ G @ C4 )
                  = ( groups2073611262835488442omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6238_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ A4 )
                  = ( groups8097168146408367636l_real @ H2 @ B5 ) )
                = ( ( groups8097168146408367636l_real @ G @ C4 )
                  = ( groups8097168146408367636l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6239_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ A4 )
                  = ( groups5808333547571424918x_real @ H2 @ B5 ) )
                = ( ( groups5808333547571424918x_real @ G @ C4 )
                  = ( groups5808333547571424918x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6240_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G @ A4 )
                  = ( groups1300246762558778688al_rat @ H2 @ B5 ) )
                = ( ( groups1300246762558778688al_rat @ G @ C4 )
                  = ( groups1300246762558778688al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6241_sum_Osame__carrier,axiom,
    ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > rat,H2: nat > rat] :
      ( ( finite_finite_nat @ C4 )
     => ( ( ord_less_eq_set_nat @ A4 @ C4 )
       => ( ( ord_less_eq_set_nat @ B5 @ C4 )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups2906978787729119204at_rat @ G @ A4 )
                  = ( groups2906978787729119204at_rat @ H2 @ B5 ) )
                = ( ( groups2906978787729119204at_rat @ G @ C4 )
                  = ( groups2906978787729119204at_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6242_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups5058264527183730370ex_rat @ G @ A4 )
                  = ( groups5058264527183730370ex_rat @ H2 @ B5 ) )
                = ( ( groups5058264527183730370ex_rat @ G @ C4 )
                  = ( groups5058264527183730370ex_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6243_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_nat ) )
             => ( ( ( groups1935376822645274424al_nat @ G @ A4 )
                  = ( groups1935376822645274424al_nat @ H2 @ B5 ) )
                = ( ( groups1935376822645274424al_nat @ G @ C4 )
                  = ( groups1935376822645274424al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6244_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > nat,H2: complex > nat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_nat ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_nat ) )
             => ( ( ( groups5693394587270226106ex_nat @ G @ A4 )
                  = ( groups5693394587270226106ex_nat @ H2 @ B5 ) )
                = ( ( groups5693394587270226106ex_nat @ G @ C4 )
                  = ( groups5693394587270226106ex_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6245_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G: real > int,H2: real > int] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_int ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_int ) )
             => ( ( ( groups1932886352136224148al_int @ G @ A4 )
                  = ( groups1932886352136224148al_int @ H2 @ B5 ) )
                = ( ( groups1932886352136224148al_int @ G @ C4 )
                  = ( groups1932886352136224148al_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_6246_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_complex ) )
             => ( ( ( groups5754745047067104278omplex @ G @ C4 )
                  = ( groups5754745047067104278omplex @ H2 @ C4 ) )
               => ( ( groups5754745047067104278omplex @ G @ A4 )
                  = ( groups5754745047067104278omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6247_sum_Osame__carrierI,axiom,
    ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > complex,H2: nat > complex] :
      ( ( finite_finite_nat @ C4 )
     => ( ( ord_less_eq_set_nat @ A4 @ C4 )
       => ( ( ord_less_eq_set_nat @ B5 @ C4 )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_complex ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_complex ) )
             => ( ( ( groups2073611262835488442omplex @ G @ C4 )
                  = ( groups2073611262835488442omplex @ H2 @ C4 ) )
               => ( ( groups2073611262835488442omplex @ G @ A4 )
                  = ( groups2073611262835488442omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6248_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ C4 )
                  = ( groups8097168146408367636l_real @ H2 @ C4 ) )
               => ( ( groups8097168146408367636l_real @ G @ A4 )
                  = ( groups8097168146408367636l_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6249_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ C4 )
                  = ( groups5808333547571424918x_real @ H2 @ C4 ) )
               => ( ( groups5808333547571424918x_real @ G @ A4 )
                  = ( groups5808333547571424918x_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6250_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G @ C4 )
                  = ( groups1300246762558778688al_rat @ H2 @ C4 ) )
               => ( ( groups1300246762558778688al_rat @ G @ A4 )
                  = ( groups1300246762558778688al_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6251_sum_Osame__carrierI,axiom,
    ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > rat,H2: nat > rat] :
      ( ( finite_finite_nat @ C4 )
     => ( ( ord_less_eq_set_nat @ A4 @ C4 )
       => ( ( ord_less_eq_set_nat @ B5 @ C4 )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups2906978787729119204at_rat @ G @ C4 )
                  = ( groups2906978787729119204at_rat @ H2 @ C4 ) )
               => ( ( groups2906978787729119204at_rat @ G @ A4 )
                  = ( groups2906978787729119204at_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6252_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups5058264527183730370ex_rat @ G @ C4 )
                  = ( groups5058264527183730370ex_rat @ H2 @ C4 ) )
               => ( ( groups5058264527183730370ex_rat @ G @ A4 )
                  = ( groups5058264527183730370ex_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6253_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_nat ) )
             => ( ( ( groups1935376822645274424al_nat @ G @ C4 )
                  = ( groups1935376822645274424al_nat @ H2 @ C4 ) )
               => ( ( groups1935376822645274424al_nat @ G @ A4 )
                  = ( groups1935376822645274424al_nat @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6254_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > nat,H2: complex > nat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_nat ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_nat ) )
             => ( ( ( groups5693394587270226106ex_nat @ G @ C4 )
                  = ( groups5693394587270226106ex_nat @ H2 @ C4 ) )
               => ( ( groups5693394587270226106ex_nat @ G @ A4 )
                  = ( groups5693394587270226106ex_nat @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6255_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G: real > int,H2: real > int] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_int ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = zero_zero_int ) )
             => ( ( ( groups1932886352136224148al_int @ G @ C4 )
                  = ( groups1932886352136224148al_int @ H2 @ C4 ) )
               => ( ( groups1932886352136224148al_int @ G @ A4 )
                  = ( groups1932886352136224148al_int @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_6256_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ( groups2073611262835488442omplex @ G @ S2 )
            = ( groups2073611262835488442omplex @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6257_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ S2 )
            = ( groups5808333547571424918x_real @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6258_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ( groups2906978787729119204at_rat @ G @ S2 )
            = ( groups2906978787729119204at_rat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6259_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ S2 )
            = ( groups5058264527183730370ex_rat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6260_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ S2 )
            = ( groups5693394587270226106ex_nat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6261_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups3539618377306564664at_int @ G @ S2 )
            = ( groups3539618377306564664at_int @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6262_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ S2 )
            = ( groups5690904116761175830ex_int @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6263_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S2: set_int,G: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S2 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ( groups3049146728041665814omplex @ G @ S2 )
            = ( groups3049146728041665814omplex @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6264_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S2: set_int,G: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S2 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ( groups8778361861064173332t_real @ G @ S2 )
            = ( groups8778361861064173332t_real @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6265_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S2: set_int,G: int > rat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S2 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ( groups3906332499630173760nt_rat @ G @ S2 )
            = ( groups3906332499630173760nt_rat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_6266_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ( groups2073611262835488442omplex @ G @ T3 )
            = ( groups2073611262835488442omplex @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6267_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ T3 )
            = ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6268_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ( groups2906978787729119204at_rat @ G @ T3 )
            = ( groups2906978787729119204at_rat @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6269_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ T3 )
            = ( groups5058264527183730370ex_rat @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6270_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ T3 )
            = ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6271_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups3539618377306564664at_int @ G @ T3 )
            = ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6272_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ T3 )
            = ( groups5690904116761175830ex_int @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6273_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S2: set_int,G: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S2 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ( groups3049146728041665814omplex @ G @ T3 )
            = ( groups3049146728041665814omplex @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6274_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S2: set_int,G: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S2 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ( groups8778361861064173332t_real @ G @ T3 )
            = ( groups8778361861064173332t_real @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6275_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S2: set_int,G: int > rat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S2 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ( groups3906332499630173760nt_rat @ G @ T3 )
            = ( groups3906332499630173760nt_rat @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_6276_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H2: real > complex,G: real > complex] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = zero_zero_complex ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5754745047067104278omplex @ G @ S2 )
              = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6277_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_nat,S2: set_nat,H2: nat > complex,G: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = zero_zero_complex ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups2073611262835488442omplex @ G @ S2 )
              = ( groups2073611262835488442omplex @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6278_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H2: real > real,G: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ S2 )
              = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6279_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H2: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ S2 )
              = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6280_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H2: real > rat,G: real > rat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1300246762558778688al_rat @ G @ S2 )
              = ( groups1300246762558778688al_rat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6281_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_nat,S2: set_nat,H2: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups2906978787729119204at_rat @ G @ S2 )
              = ( groups2906978787729119204at_rat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6282_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5058264527183730370ex_rat @ G @ S2 )
              = ( groups5058264527183730370ex_rat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6283_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H2: real > nat,G: real > nat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1935376822645274424al_nat @ G @ S2 )
              = ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6284_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H2: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5693394587270226106ex_nat @ G @ S2 )
              = ( groups5693394587270226106ex_nat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6285_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H2: real > int,G: real > int] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = zero_zero_int ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1932886352136224148al_int @ G @ S2 )
              = ( groups1932886352136224148al_int @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_6286_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5754745047067104278omplex @ G @ T3 )
              = ( groups5754745047067104278omplex @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6287_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > complex,H2: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups2073611262835488442omplex @ G @ T3 )
              = ( groups2073611262835488442omplex @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6288_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ T3 )
              = ( groups8097168146408367636l_real @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6289_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ T3 )
              = ( groups5808333547571424918x_real @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6290_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1300246762558778688al_rat @ G @ T3 )
              = ( groups1300246762558778688al_rat @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6291_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > rat,H2: nat > rat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups2906978787729119204at_rat @ G @ T3 )
              = ( groups2906978787729119204at_rat @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6292_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5058264527183730370ex_rat @ G @ T3 )
              = ( groups5058264527183730370ex_rat @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6293_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1935376822645274424al_nat @ G @ T3 )
              = ( groups1935376822645274424al_nat @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6294_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > nat,H2: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5693394587270226106ex_nat @ G @ T3 )
              = ( groups5693394587270226106ex_nat @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6295_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: real > int,H2: real > int] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1932886352136224148al_int @ G @ T3 )
              = ( groups1932886352136224148al_int @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_6296_sum_Osubset__diff,axiom,
    ! [B5: set_complex,A4: set_complex,G: complex > real] :
      ( ( ord_le211207098394363844omplex @ B5 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups5808333547571424918x_real @ G @ A4 )
          = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups5808333547571424918x_real @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6297_sum_Osubset__diff,axiom,
    ! [B5: set_nat,A4: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ B5 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups2906978787729119204at_rat @ G @ A4 )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups2906978787729119204at_rat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6298_sum_Osubset__diff,axiom,
    ! [B5: set_complex,A4: set_complex,G: complex > rat] :
      ( ( ord_le211207098394363844omplex @ B5 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups5058264527183730370ex_rat @ G @ A4 )
          = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups5058264527183730370ex_rat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6299_sum_Osubset__diff,axiom,
    ! [B5: set_complex,A4: set_complex,G: complex > nat] :
      ( ( ord_le211207098394363844omplex @ B5 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups5693394587270226106ex_nat @ G @ A4 )
          = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups5693394587270226106ex_nat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6300_sum_Osubset__diff,axiom,
    ! [B5: set_nat,A4: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ B5 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups3539618377306564664at_int @ G @ A4 )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups3539618377306564664at_int @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6301_sum_Osubset__diff,axiom,
    ! [B5: set_complex,A4: set_complex,G: complex > int] :
      ( ( ord_le211207098394363844omplex @ B5 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups5690904116761175830ex_int @ G @ A4 )
          = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups5690904116761175830ex_int @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6302_sum_Osubset__diff,axiom,
    ! [B5: set_int,A4: set_int,G: int > real] :
      ( ( ord_less_eq_set_int @ B5 @ A4 )
     => ( ( finite_finite_int @ A4 )
       => ( ( groups8778361861064173332t_real @ G @ A4 )
          = ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups8778361861064173332t_real @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6303_sum_Osubset__diff,axiom,
    ! [B5: set_int,A4: set_int,G: int > rat] :
      ( ( ord_less_eq_set_int @ B5 @ A4 )
     => ( ( finite_finite_int @ A4 )
       => ( ( groups3906332499630173760nt_rat @ G @ A4 )
          = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups3906332499630173760nt_rat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6304_sum_Osubset__diff,axiom,
    ! [B5: set_int,A4: set_int,G: int > nat] :
      ( ( ord_less_eq_set_int @ B5 @ A4 )
     => ( ( finite_finite_int @ A4 )
       => ( ( groups4541462559716669496nt_nat @ G @ A4 )
          = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups4541462559716669496nt_nat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6305_sum_Osubset__diff,axiom,
    ! [B5: set_int,A4: set_int,G: int > int] :
      ( ( ord_less_eq_set_int @ B5 @ A4 )
     => ( ( finite_finite_int @ A4 )
       => ( ( groups4538972089207619220nt_int @ G @ A4 )
          = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups4538972089207619220nt_int @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_6306_sum__diff,axiom,
    ! [A4: set_nat,B5: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_eq_set_nat @ B5 @ A4 )
       => ( ( groups2073611262835488442omplex @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
          = ( minus_minus_complex @ ( groups2073611262835488442omplex @ F @ A4 ) @ ( groups2073611262835488442omplex @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_6307_sum__diff,axiom,
    ! [A4: set_complex,B5: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ord_le211207098394363844omplex @ B5 @ A4 )
       => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
          = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_6308_sum__diff,axiom,
    ! [A4: set_nat,B5: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_eq_set_nat @ B5 @ A4 )
       => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
          = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_6309_sum__diff,axiom,
    ! [A4: set_complex,B5: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ord_le211207098394363844omplex @ B5 @ A4 )
       => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
          = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_6310_sum__diff,axiom,
    ! [A4: set_nat,B5: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_eq_set_nat @ B5 @ A4 )
       => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
          = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_6311_sum__diff,axiom,
    ! [A4: set_complex,B5: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ord_le211207098394363844omplex @ B5 @ A4 )
       => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
          = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_6312_sum__diff,axiom,
    ! [A4: set_int,B5: set_int,F: int > complex] :
      ( ( finite_finite_int @ A4 )
     => ( ( ord_less_eq_set_int @ B5 @ A4 )
       => ( ( groups3049146728041665814omplex @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
          = ( minus_minus_complex @ ( groups3049146728041665814omplex @ F @ A4 ) @ ( groups3049146728041665814omplex @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_6313_sum__diff,axiom,
    ! [A4: set_int,B5: set_int,F: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ( ord_less_eq_set_int @ B5 @ A4 )
       => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
          = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_6314_sum__diff,axiom,
    ! [A4: set_int,B5: set_int,F: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( ord_less_eq_set_int @ B5 @ A4 )
       => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
          = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_6315_sum__diff,axiom,
    ! [A4: set_int,B5: set_int,F: int > int] :
      ( ( finite_finite_int @ A4 )
     => ( ( ord_less_eq_set_int @ B5 @ A4 )
       => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
          = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ F @ B5 ) ) ) ) ) ).

% sum_diff
thf(fact_6316_ln__add__one__self__le__self,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) @ X3 ) ) ).

% ln_add_one_self_le_self
thf(fact_6317_ln__add__one__self__le__self2,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) @ X3 ) ) ).

% ln_add_one_self_le_self2
thf(fact_6318_ln__mult,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ( ln_ln_real @ ( times_times_real @ X3 @ Y3 ) )
          = ( plus_plus_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).

% ln_mult
thf(fact_6319_ln__eq__minus__one,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ( ln_ln_real @ X3 )
          = ( minus_minus_real @ X3 @ one_one_real ) )
       => ( X3 = one_one_real ) ) ) ).

% ln_eq_minus_one
thf(fact_6320_ln__div,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ( ln_ln_real @ ( divide_divide_real @ X3 @ Y3 ) )
          = ( minus_minus_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).

% ln_div
thf(fact_6321_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_6322_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_6323_odd__numeral,axiom,
    ! [N: num] :
      ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ).

% odd_numeral
thf(fact_6324_odd__numeral,axiom,
    ! [N: num] :
      ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).

% odd_numeral
thf(fact_6325_odd__numeral,axiom,
    ! [N: num] :
      ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).

% odd_numeral
thf(fact_6326_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != zero_zero_nat ) ).

% cong_exp_iff_simps(3)
thf(fact_6327_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != zero_zero_int ) ).

% cong_exp_iff_simps(3)
thf(fact_6328_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(3)
thf(fact_6329_power3__eq__cube,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_complex @ ( times_times_complex @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_6330_power3__eq__cube,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_real @ ( times_times_real @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_6331_power3__eq__cube,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_6332_power3__eq__cube,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_6333_power3__eq__cube,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_6334_numeral__3__eq__3,axiom,
    ( ( numeral_numeral_nat @ ( bit1 @ one ) )
    = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).

% numeral_3_eq_3
thf(fact_6335_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_6336_sum__mono2,axiom,
    ! [B5: set_real,A4: set_real,F: real > real] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A4 @ B5 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
         => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_6337_sum__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
         => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_6338_sum__mono2,axiom,
    ! [B5: set_real,A4: set_real,F: real > rat] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A4 @ B5 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
         => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_6339_sum__mono2,axiom,
    ! [B5: set_nat,A4: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( ! [B3: nat] :
              ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
         => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_6340_sum__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
         => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_6341_sum__mono2,axiom,
    ! [B5: set_real,A4: set_real,F: real > nat] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A4 @ B5 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
         => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_6342_sum__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
         => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_6343_sum__mono2,axiom,
    ! [B5: set_real,A4: set_real,F: real > int] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A4 @ B5 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
         => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A4 ) @ ( groups1932886352136224148al_int @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_6344_sum__mono2,axiom,
    ! [B5: set_nat,A4: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( ! [B3: nat] :
              ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
         => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_6345_sum__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
         => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ F @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_6346_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_6347_ln__le__minus__one,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( minus_minus_real @ X3 @ one_one_real ) ) ) ).

% ln_le_minus_one
thf(fact_6348_ln__one__minus__pos__upper__bound,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ one_one_real )
       => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X3 ) ) @ ( uminus_uminus_real @ X3 ) ) ) ) ).

% ln_one_minus_pos_upper_bound
thf(fact_6349_ln__diff__le,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) ) @ ( divide_divide_real @ ( minus_minus_real @ X3 @ Y3 ) @ Y3 ) ) ) ) ).

% ln_diff_le
thf(fact_6350_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
thf(fact_6351_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
thf(fact_6352_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
thf(fact_6353_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
thf(fact_6354_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
thf(fact_6355_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
thf(fact_6356_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_6357_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_6358_sum__strict__mono2,axiom,
    ! [B5: set_real,A4: set_real,B2: real,F: real > real] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A4 @ B5 )
       => ( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A4 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B2 ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ B5 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
             => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6359_sum__strict__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,B2: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B2 ) )
           => ( ! [X5: complex] :
                  ( ( member_complex @ X5 @ B5 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
             => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6360_sum__strict__mono2,axiom,
    ! [B5: set_real,A4: set_real,B2: real,F: real > rat] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A4 @ B5 )
       => ( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A4 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B2 ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ B5 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
             => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6361_sum__strict__mono2,axiom,
    ! [B5: set_nat,A4: set_nat,B2: nat,F: nat > rat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( ( member_nat @ B2 @ ( minus_minus_set_nat @ B5 @ A4 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B2 ) )
           => ( ! [X5: nat] :
                  ( ( member_nat @ X5 @ B5 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
             => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6362_sum__strict__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,B2: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B2 ) )
           => ( ! [X5: complex] :
                  ( ( member_complex @ X5 @ B5 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
             => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6363_sum__strict__mono2,axiom,
    ! [B5: set_real,A4: set_real,B2: real,F: real > nat] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A4 @ B5 )
       => ( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A4 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B2 ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ B5 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
             => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6364_sum__strict__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,B2: complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B2 ) )
           => ( ! [X5: complex] :
                  ( ( member_complex @ X5 @ B5 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
             => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6365_sum__strict__mono2,axiom,
    ! [B5: set_real,A4: set_real,B2: real,F: real > int] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A4 @ B5 )
       => ( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A4 ) )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ B2 ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ B5 )
                 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
             => ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A4 ) @ ( groups1932886352136224148al_int @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6366_sum__strict__mono2,axiom,
    ! [B5: set_nat,A4: set_nat,B2: nat,F: nat > int] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( ( member_nat @ B2 @ ( minus_minus_set_nat @ B5 @ A4 ) )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ B2 ) )
           => ( ! [X5: nat] :
                  ( ( member_nat @ X5 @ B5 )
                 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
             => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6367_sum__strict__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,B2: complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ B2 ) )
           => ( ! [X5: complex] :
                  ( ( member_complex @ X5 @ B5 )
                 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
             => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ F @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_6368_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L2: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_6369_ln__one__plus__pos__lower__bound,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ X3 @ one_one_real )
       => ( ord_less_eq_real @ ( minus_minus_real @ X3 @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) ) ) ) ).

% ln_one_plus_pos_lower_bound
thf(fact_6370_divmod__step__int__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L2: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_int_def
thf(fact_6371_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_6372_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_6373_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_6374_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) @ one_one_int ) )
        @ ( unique5052692396658037445od_int @ M @ N ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_6375_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) @ one_one_nat ) )
        @ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_6376_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( produc6916734918728496179nteger
        @ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) @ one_one_Code_integer ) )
        @ ( unique3479559517661332726nteger @ M @ N ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_6377_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_6378_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_6379_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_6380_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_6381_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_6382_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_6383_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
     => ( ( ord_less_eq_real @ X3 @ zero_zero_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) @ X3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_6384_abs__idempotent,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_idempotent
thf(fact_6385_abs__idempotent,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_idempotent
thf(fact_6386_abs__idempotent,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_idempotent
thf(fact_6387_abs__idempotent,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_idempotent
thf(fact_6388_abs__abs,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_abs
thf(fact_6389_abs__abs,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_abs
thf(fact_6390_abs__abs,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_abs
thf(fact_6391_abs__abs,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_abs
thf(fact_6392_case__prodI,axiom,
    ! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
      ( ( F @ A @ B2 )
     => ( produc127349428274296955eger_o @ F @ ( produc6137756002093451184nteger @ A @ B2 ) ) ) ).

% case_prodI
thf(fact_6393_case__prodI,axiom,
    ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
      ( ( F @ A @ B2 )
     => ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B2 ) ) ) ).

% case_prodI
thf(fact_6394_case__prodI,axiom,
    ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
      ( ( F @ A @ B2 )
     => ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B2 ) ) ) ).

% case_prodI
thf(fact_6395_case__prodI,axiom,
    ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B2: product_prod_int_int] :
      ( ( F @ A @ B2 )
     => ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B2 ) ) ) ).

% case_prodI
thf(fact_6396_case__prodI,axiom,
    ! [F: int > int > $o,A: int,B2: int] :
      ( ( F @ A @ B2 )
     => ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B2 ) ) ) ).

% case_prodI
thf(fact_6397_case__prodI2,axiom,
    ! [P4: produc8763457246119570046nteger,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
          ( ( P4
            = ( produc6137756002093451184nteger @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc127349428274296955eger_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_6398_case__prodI2,axiom,
    ! [P4: produc1908205239877642774nteger,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
          ( ( P4
            = ( produc8603105652947943368nteger @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc6253627499356882019eger_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_6399_case__prodI2,axiom,
    ! [P4: produc2285326912895808259nt_int,C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
          ( ( P4
            = ( produc5700946648718959541nt_int @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc1573362020775583542_int_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_6400_case__prodI2,axiom,
    ! [P4: produc7773217078559923341nt_int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
          ( ( P4
            = ( produc4305682042979456191nt_int @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc2558449545302689196_int_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_6401_case__prodI2,axiom,
    ! [P4: product_prod_int_int,C: int > int > $o] :
      ( ! [A3: int,B3: int] :
          ( ( P4
            = ( product_Pair_int_int @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc4947309494688390418_int_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_6402_mem__case__prodI,axiom,
    ! [Z: nat,C: int > int > set_nat,A: int,B2: int] :
      ( ( member_nat @ Z @ ( C @ A @ B2 ) )
     => ( member_nat @ Z @ ( produc4251311855443802252et_nat @ C @ ( product_Pair_int_int @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_6403_mem__case__prodI,axiom,
    ! [Z: real,C: int > int > set_real,A: int,B2: int] :
      ( ( member_real @ Z @ ( C @ A @ B2 ) )
     => ( member_real @ Z @ ( produc6452406959799940328t_real @ C @ ( product_Pair_int_int @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_6404_mem__case__prodI,axiom,
    ! [Z: int,C: int > int > set_int,A: int,B2: int] :
      ( ( member_int @ Z @ ( C @ A @ B2 ) )
     => ( member_int @ Z @ ( produc73460835934605544et_int @ C @ ( product_Pair_int_int @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_6405_mem__case__prodI,axiom,
    ! [Z: complex,C: int > int > set_complex,A: int,B2: int] :
      ( ( member_complex @ Z @ ( C @ A @ B2 ) )
     => ( member_complex @ Z @ ( produc8580519160106071146omplex @ C @ ( product_Pair_int_int @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_6406_mem__case__prodI,axiom,
    ! [Z: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,A: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
      ( ( member_nat @ Z @ ( C @ A @ B2 ) )
     => ( member_nat @ Z @ ( produc3558942015123893603et_nat @ C @ ( produc6137756002093451184nteger @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_6407_mem__case__prodI,axiom,
    ! [Z: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real,A: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
      ( ( member_real @ Z @ ( C @ A @ B2 ) )
     => ( member_real @ Z @ ( produc815715089573277247t_real @ C @ ( produc6137756002093451184nteger @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_6408_mem__case__prodI,axiom,
    ! [Z: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,A: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
      ( ( member_int @ Z @ ( C @ A @ B2 ) )
     => ( member_int @ Z @ ( produc8604463032469472703et_int @ C @ ( produc6137756002093451184nteger @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_6409_mem__case__prodI,axiom,
    ! [Z: complex,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_complex,A: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
      ( ( member_complex @ Z @ ( C @ A @ B2 ) )
     => ( member_complex @ Z @ ( produc2592262431452330817omplex @ C @ ( produc6137756002093451184nteger @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_6410_mem__case__prodI,axiom,
    ! [Z: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,A: int > option6357759511663192854e_term,B2: product_prod_int_int] :
      ( ( member_nat @ Z @ ( C @ A @ B2 ) )
     => ( member_nat @ Z @ ( produc8289552606927098482et_nat @ C @ ( produc4305682042979456191nt_int @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_6411_mem__case__prodI,axiom,
    ! [Z: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real,A: int > option6357759511663192854e_term,B2: product_prod_int_int] :
      ( ( member_real @ Z @ ( C @ A @ B2 ) )
     => ( member_real @ Z @ ( produc8709739885379107790t_real @ C @ ( produc4305682042979456191nt_int @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_6412_mem__case__prodI2,axiom,
    ! [P4: product_prod_int_int,Z: nat,C: int > int > set_nat] :
      ( ! [A3: int,B3: int] :
          ( ( P4
            = ( product_Pair_int_int @ A3 @ B3 ) )
         => ( member_nat @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_nat @ Z @ ( produc4251311855443802252et_nat @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_6413_mem__case__prodI2,axiom,
    ! [P4: product_prod_int_int,Z: real,C: int > int > set_real] :
      ( ! [A3: int,B3: int] :
          ( ( P4
            = ( product_Pair_int_int @ A3 @ B3 ) )
         => ( member_real @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_real @ Z @ ( produc6452406959799940328t_real @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_6414_mem__case__prodI2,axiom,
    ! [P4: product_prod_int_int,Z: int,C: int > int > set_int] :
      ( ! [A3: int,B3: int] :
          ( ( P4
            = ( product_Pair_int_int @ A3 @ B3 ) )
         => ( member_int @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_int @ Z @ ( produc73460835934605544et_int @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_6415_mem__case__prodI2,axiom,
    ! [P4: product_prod_int_int,Z: complex,C: int > int > set_complex] :
      ( ! [A3: int,B3: int] :
          ( ( P4
            = ( product_Pair_int_int @ A3 @ B3 ) )
         => ( member_complex @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_complex @ Z @ ( produc8580519160106071146omplex @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_6416_mem__case__prodI2,axiom,
    ! [P4: produc8763457246119570046nteger,Z: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat] :
      ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
          ( ( P4
            = ( produc6137756002093451184nteger @ A3 @ B3 ) )
         => ( member_nat @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_nat @ Z @ ( produc3558942015123893603et_nat @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_6417_mem__case__prodI2,axiom,
    ! [P4: produc8763457246119570046nteger,Z: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real] :
      ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
          ( ( P4
            = ( produc6137756002093451184nteger @ A3 @ B3 ) )
         => ( member_real @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_real @ Z @ ( produc815715089573277247t_real @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_6418_mem__case__prodI2,axiom,
    ! [P4: produc8763457246119570046nteger,Z: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int] :
      ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
          ( ( P4
            = ( produc6137756002093451184nteger @ A3 @ B3 ) )
         => ( member_int @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_int @ Z @ ( produc8604463032469472703et_int @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_6419_mem__case__prodI2,axiom,
    ! [P4: produc8763457246119570046nteger,Z: complex,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_complex] :
      ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
          ( ( P4
            = ( produc6137756002093451184nteger @ A3 @ B3 ) )
         => ( member_complex @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_complex @ Z @ ( produc2592262431452330817omplex @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_6420_mem__case__prodI2,axiom,
    ! [P4: produc7773217078559923341nt_int,Z: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat] :
      ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
          ( ( P4
            = ( produc4305682042979456191nt_int @ A3 @ B3 ) )
         => ( member_nat @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_nat @ Z @ ( produc8289552606927098482et_nat @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_6421_mem__case__prodI2,axiom,
    ! [P4: produc7773217078559923341nt_int,Z: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real] :
      ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
          ( ( P4
            = ( produc4305682042979456191nt_int @ A3 @ B3 ) )
         => ( member_real @ Z @ ( C @ A3 @ B3 ) ) )
     => ( member_real @ Z @ ( produc8709739885379107790t_real @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_6422_case__prodI2_H,axiom,
    ! [P4: product_prod_nat_nat,C: nat > nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ( product_Pair_nat_nat @ A3 @ B3 )
            = P4 )
         => ( C @ A3 @ B3 @ X3 ) )
     => ( produc8739625826339149834_nat_o @ C @ P4 @ X3 ) ) ).

% case_prodI2'
thf(fact_6423_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( abs_abs_Code_integer @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
thf(fact_6424_abs__0__eq,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( abs_abs_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% abs_0_eq
thf(fact_6425_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( abs_abs_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% abs_0_eq
thf(fact_6426_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( abs_abs_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% abs_0_eq
thf(fact_6427_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
thf(fact_6428_abs__eq__0,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0
thf(fact_6429_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0
thf(fact_6430_abs__eq__0,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0
thf(fact_6431_abs__zero,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_zero
thf(fact_6432_abs__zero,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_zero
thf(fact_6433_abs__zero,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_zero
thf(fact_6434_abs__zero,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_zero
thf(fact_6435_abs__0,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_0
thf(fact_6436_abs__0,axiom,
    ( ( abs_abs_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% abs_0
thf(fact_6437_abs__0,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_0
thf(fact_6438_abs__0,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_0
thf(fact_6439_abs__0,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_0
thf(fact_6440_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% abs_numeral
thf(fact_6441_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ N ) ) ).

% abs_numeral
thf(fact_6442_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ N ) ) ).

% abs_numeral
thf(fact_6443_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% abs_numeral
thf(fact_6444_abs__mult__self__eq,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
      = ( times_3573771949741848930nteger @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_6445_abs__mult__self__eq,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
      = ( times_times_real @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_6446_abs__mult__self__eq,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
      = ( times_times_rat @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_6447_abs__mult__self__eq,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
      = ( times_times_int @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_6448_abs__1,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_1
thf(fact_6449_abs__1,axiom,
    ( ( abs_abs_complex @ one_one_complex )
    = one_one_complex ) ).

% abs_1
thf(fact_6450_abs__1,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_1
thf(fact_6451_abs__1,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_1
thf(fact_6452_abs__1,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_1
thf(fact_6453_abs__add__abs,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) )
      = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).

% abs_add_abs
thf(fact_6454_abs__add__abs,axiom,
    ! [A: real,B2: real] :
      ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) )
      = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).

% abs_add_abs
thf(fact_6455_abs__add__abs,axiom,
    ! [A: rat,B2: rat] :
      ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) )
      = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).

% abs_add_abs
thf(fact_6456_abs__add__abs,axiom,
    ! [A: int,B2: int] :
      ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) )
      = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).

% abs_add_abs
thf(fact_6457_abs__divide,axiom,
    ! [A: complex,B2: complex] :
      ( ( abs_abs_complex @ ( divide1717551699836669952omplex @ A @ B2 ) )
      = ( divide1717551699836669952omplex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B2 ) ) ) ).

% abs_divide
thf(fact_6458_abs__divide,axiom,
    ! [A: real,B2: real] :
      ( ( abs_abs_real @ ( divide_divide_real @ A @ B2 ) )
      = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).

% abs_divide
thf(fact_6459_abs__divide,axiom,
    ! [A: rat,B2: rat] :
      ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B2 ) )
      = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).

% abs_divide
thf(fact_6460_abs__minus__cancel,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus_cancel
thf(fact_6461_abs__minus__cancel,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus_cancel
thf(fact_6462_abs__minus__cancel,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus_cancel
thf(fact_6463_abs__minus__cancel,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus_cancel
thf(fact_6464_abs__minus,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus
thf(fact_6465_abs__minus,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus
thf(fact_6466_abs__minus,axiom,
    ! [A: complex] :
      ( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A ) )
      = ( abs_abs_complex @ A ) ) ).

% abs_minus
thf(fact_6467_abs__minus,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus
thf(fact_6468_abs__minus,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus
thf(fact_6469_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_6470_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_6471_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_6472_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_6473_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_6474_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_6475_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_6476_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_6477_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% abs_bool_eq
thf(fact_6478_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
      = ( zero_n2052037380579107095ol_rat @ P ) ) ).

% abs_bool_eq
thf(fact_6479_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% abs_bool_eq
thf(fact_6480_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
      = ( zero_n356916108424825756nteger @ P ) ) ).

% abs_bool_eq
thf(fact_6481_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ one_one_complex )
    = one_one_complex ) ).

% dbl_dec_simps(3)
thf(fact_6482_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ one_one_real )
    = one_one_real ) ).

% dbl_dec_simps(3)
thf(fact_6483_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
    = one_one_rat ) ).

% dbl_dec_simps(3)
thf(fact_6484_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ one_one_int )
    = one_one_int ) ).

% dbl_dec_simps(3)
thf(fact_6485_abs__le__zero__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
thf(fact_6486_abs__le__zero__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_le_zero_iff
thf(fact_6487_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
thf(fact_6488_abs__le__zero__iff,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_le_zero_iff
thf(fact_6489_abs__le__self__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% abs_le_self_iff
thf(fact_6490_abs__le__self__iff,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% abs_le_self_iff
thf(fact_6491_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% abs_le_self_iff
thf(fact_6492_abs__le__self__iff,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% abs_le_self_iff
thf(fact_6493_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_6494_abs__of__nonneg,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_6495_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_6496_abs__of__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_6497_zero__less__abs__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
      = ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
thf(fact_6498_zero__less__abs__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
      = ( A != zero_zero_real ) ) ).

% zero_less_abs_iff
thf(fact_6499_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
      = ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
thf(fact_6500_zero__less__abs__iff,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
      = ( A != zero_zero_int ) ) ).

% zero_less_abs_iff
thf(fact_6501_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_6502_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_6503_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% abs_neg_numeral
thf(fact_6504_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_6505_abs__neg__one,axiom,
    ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
    = one_one_real ) ).

% abs_neg_one
thf(fact_6506_abs__neg__one,axiom,
    ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
    = one_one_int ) ).

% abs_neg_one
thf(fact_6507_abs__neg__one,axiom,
    ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = one_one_Code_integer ) ).

% abs_neg_one
thf(fact_6508_abs__neg__one,axiom,
    ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = one_one_rat ) ).

% abs_neg_one
thf(fact_6509_abs__power__minus,axiom,
    ! [A: real,N: nat] :
      ( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
      = ( abs_abs_real @ ( power_power_real @ A @ N ) ) ) ).

% abs_power_minus
thf(fact_6510_abs__power__minus,axiom,
    ! [A: int,N: nat] :
      ( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
      = ( abs_abs_int @ ( power_power_int @ A @ N ) ) ) ).

% abs_power_minus
thf(fact_6511_abs__power__minus,axiom,
    ! [A: code_integer,N: nat] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
      = ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% abs_power_minus
thf(fact_6512_abs__power__minus,axiom,
    ! [A: rat,N: nat] :
      ( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
      = ( abs_abs_rat @ ( power_power_rat @ A @ N ) ) ) ).

% abs_power_minus
thf(fact_6513_pred__numeral__simps_I1_J,axiom,
    ( ( pred_numeral @ one )
    = zero_zero_nat ) ).

% pred_numeral_simps(1)
thf(fact_6514_eq__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ( numeral_numeral_nat @ K )
        = ( suc @ N ) )
      = ( ( pred_numeral @ K )
        = N ) ) ).

% eq_numeral_Suc
thf(fact_6515_Suc__eq__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ( suc @ N )
        = ( numeral_numeral_nat @ K ) )
      = ( N
        = ( pred_numeral @ K ) ) ) ).

% Suc_eq_numeral
thf(fact_6516_sum__abs,axiom,
    ! [F: int > int,A4: set_int] :
      ( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A4 ) )
      @ ( groups4538972089207619220nt_int
        @ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
        @ A4 ) ) ).

% sum_abs
thf(fact_6517_sum__abs,axiom,
    ! [F: nat > real,A4: set_nat] :
      ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A4 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
        @ A4 ) ) ).

% sum_abs
thf(fact_6518_zero__le__divide__abs__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B2 ) ) )
      = ( ( ord_less_eq_real @ zero_zero_real @ A )
        | ( B2 = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
thf(fact_6519_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B2 ) ) )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
        | ( B2 = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
thf(fact_6520_divide__le__0__abs__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B2 ) ) @ zero_zero_real )
      = ( ( ord_less_eq_real @ A @ zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
thf(fact_6521_divide__le__0__abs__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B2 ) ) @ zero_zero_rat )
      = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
thf(fact_6522_abs__of__nonpos,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( abs_abs_real @ A )
        = ( uminus_uminus_real @ A ) ) ) ).

% abs_of_nonpos
thf(fact_6523_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_nonpos
thf(fact_6524_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( abs_abs_rat @ A )
        = ( uminus_uminus_rat @ A ) ) ) ).

% abs_of_nonpos
thf(fact_6525_abs__of__nonpos,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( abs_abs_int @ A )
        = ( uminus_uminus_int @ A ) ) ) ).

% abs_of_nonpos
thf(fact_6526_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_6527_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_6528_artanh__minus__real,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( ( artanh_real @ ( uminus_uminus_real @ X3 ) )
        = ( uminus_uminus_real @ ( artanh_real @ X3 ) ) ) ) ).

% artanh_minus_real
thf(fact_6529_pred__numeral__simps_I3_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit1 @ K ) )
      = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).

% pred_numeral_simps(3)
thf(fact_6530_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_6531_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_6532_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_6533_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_6534_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_6535_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_6536_sum__abs__ge__zero,axiom,
    ! [F: int > int,A4: set_int] :
      ( ord_less_eq_int @ zero_zero_int
      @ ( groups4538972089207619220nt_int
        @ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
        @ A4 ) ) ).

% sum_abs_ge_zero
thf(fact_6537_sum__abs__ge__zero,axiom,
    ! [F: nat > real,A4: set_nat] :
      ( ord_less_eq_real @ zero_zero_real
      @ ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
        @ A4 ) ) ).

% sum_abs_ge_zero
thf(fact_6538_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_dec_simps(2)
thf(fact_6539_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_dec_simps(2)
thf(fact_6540_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_dec_simps(2)
thf(fact_6541_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_dec_simps(2)
thf(fact_6542_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_dec_simps(2)
thf(fact_6543_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_6544_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_6545_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) )
      = ( ( A != zero_z3403309356797280102nteger )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_6546_zero__less__power__abs__iff,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
      = ( ( A != zero_zero_real )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_6547_zero__less__power__abs__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) )
      = ( ( A != zero_zero_rat )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_6548_zero__less__power__abs__iff,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
      = ( ( A != zero_zero_int )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_6549_abs__power2,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_6550_abs__power2,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_6551_abs__power2,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_6552_abs__power2,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_6553_power2__abs,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_6554_power2__abs,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_6555_power2__abs,axiom,
    ! [A: real] :
      ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_6556_power2__abs,axiom,
    ! [A: int] :
      ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_6557_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > complex] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_6558_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_6559_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_6560_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_6561_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_6562_dvd__numeral__simp,axiom,
    ! [M: num,N: num] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_6563_dvd__numeral__simp,axiom,
    ! [M: num,N: num] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_6564_dvd__numeral__simp,axiom,
    ! [M: num,N: num] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
      = ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_6565_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique5052692396658037445od_int @ M @ one )
      = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).

% divmod_algorithm_code(2)
thf(fact_6566_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique5055182867167087721od_nat @ M @ one )
      = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).

% divmod_algorithm_code(2)
thf(fact_6567_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique3479559517661332726nteger @ M @ one )
      = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).

% divmod_algorithm_code(2)
thf(fact_6568_sum__zero__power,axiom,
    ! [A4: set_nat,C: nat > complex] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
            @ A4 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
            @ A4 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power
thf(fact_6569_sum__zero__power,axiom,
    ! [A4: set_nat,C: nat > rat] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
            @ A4 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
            @ A4 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power
thf(fact_6570_sum__zero__power,axiom,
    ! [A4: set_nat,C: nat > real] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
            @ A4 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
            @ A4 )
          = zero_zero_real ) ) ) ).

% sum_zero_power
thf(fact_6571_power__even__abs__numeral,axiom,
    ! [W: num,A: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ W ) )
        = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_6572_power__even__abs__numeral,axiom,
    ! [W: num,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ W ) )
        = ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_6573_power__even__abs__numeral,axiom,
    ! [W: num,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ W ) )
        = ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_6574_power__even__abs__numeral,axiom,
    ! [W: num,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ W ) )
        = ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_6575_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_6576_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_6577_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_6578_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_6579_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_6580_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_6581_sum__zero__power_H,axiom,
    ! [A4: set_nat,C: nat > complex,D: nat > complex] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
            @ A4 )
          = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
            @ A4 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power'
thf(fact_6582_sum__zero__power_H,axiom,
    ! [A4: set_nat,C: nat > rat,D: nat > rat] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
            @ A4 )
          = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
            @ A4 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power'
thf(fact_6583_sum__zero__power_H,axiom,
    ! [A4: set_nat,C: nat > real,D: nat > real] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
            @ A4 )
          = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
            @ A4 )
          = zero_zero_real ) ) ) ).

% sum_zero_power'
thf(fact_6584_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_6585_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_6586_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_6587_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) )
        @ ( unique5052692396658037445od_int @ M @ N ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_6588_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) )
        @ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_6589_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( produc6916734918728496179nteger
        @ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) )
        @ ( unique3479559517661332726nteger @ M @ N ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_6590_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_6591_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_num @ M @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_6592_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_num @ M @ N )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_6593_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_num @ M @ N )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_6594_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_num @ M @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_6595_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_num @ M @ N )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_6596_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_num @ M @ N )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_6597_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_6598_abs__ge__self,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).

% abs_ge_self
thf(fact_6599_abs__ge__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_self
thf(fact_6600_abs__ge__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).

% abs_ge_self
thf(fact_6601_abs__ge__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).

% abs_ge_self
thf(fact_6602_abs__le__D1,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 )
     => ( ord_less_eq_real @ A @ B2 ) ) ).

% abs_le_D1
thf(fact_6603_abs__le__D1,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B2 )
     => ( ord_le3102999989581377725nteger @ A @ B2 ) ) ).

% abs_le_D1
thf(fact_6604_abs__le__D1,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B2 )
     => ( ord_less_eq_rat @ A @ B2 ) ) ).

% abs_le_D1
thf(fact_6605_abs__le__D1,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 )
     => ( ord_less_eq_int @ A @ B2 ) ) ).

% abs_le_D1
thf(fact_6606_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
thf(fact_6607_abs__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( ( abs_abs_complex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% abs_eq_0_iff
thf(fact_6608_abs__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0_iff
thf(fact_6609_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
thf(fact_6610_abs__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0_iff
thf(fact_6611_abs__mult,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) )
      = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).

% abs_mult
thf(fact_6612_abs__mult,axiom,
    ! [A: real,B2: real] :
      ( ( abs_abs_real @ ( times_times_real @ A @ B2 ) )
      = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).

% abs_mult
thf(fact_6613_abs__mult,axiom,
    ! [A: rat,B2: rat] :
      ( ( abs_abs_rat @ ( times_times_rat @ A @ B2 ) )
      = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).

% abs_mult
thf(fact_6614_abs__mult,axiom,
    ! [A: int,B2: int] :
      ( ( abs_abs_int @ ( times_times_int @ A @ B2 ) )
      = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).

% abs_mult
thf(fact_6615_abs__one,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_one
thf(fact_6616_abs__one,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_one
thf(fact_6617_abs__one,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_one
thf(fact_6618_abs__one,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_one
thf(fact_6619_abs__minus__commute,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B2 ) )
      = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B2 @ A ) ) ) ).

% abs_minus_commute
thf(fact_6620_abs__minus__commute,axiom,
    ! [A: real,B2: real] :
      ( ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) )
      = ( abs_abs_real @ ( minus_minus_real @ B2 @ A ) ) ) ).

% abs_minus_commute
thf(fact_6621_abs__minus__commute,axiom,
    ! [A: rat,B2: rat] :
      ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B2 ) )
      = ( abs_abs_rat @ ( minus_minus_rat @ B2 @ A ) ) ) ).

% abs_minus_commute
thf(fact_6622_abs__minus__commute,axiom,
    ! [A: int,B2: int] :
      ( ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) )
      = ( abs_abs_int @ ( minus_minus_int @ B2 @ A ) ) ) ).

% abs_minus_commute
thf(fact_6623_abs__eq__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( abs_abs_real @ X3 )
        = ( abs_abs_real @ Y3 ) )
      = ( ( X3 = Y3 )
        | ( X3
          = ( uminus_uminus_real @ Y3 ) ) ) ) ).

% abs_eq_iff
thf(fact_6624_abs__eq__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( ( abs_abs_int @ X3 )
        = ( abs_abs_int @ Y3 ) )
      = ( ( X3 = Y3 )
        | ( X3
          = ( uminus_uminus_int @ Y3 ) ) ) ) ).

% abs_eq_iff
thf(fact_6625_abs__eq__iff,axiom,
    ! [X3: code_integer,Y3: code_integer] :
      ( ( ( abs_abs_Code_integer @ X3 )
        = ( abs_abs_Code_integer @ Y3 ) )
      = ( ( X3 = Y3 )
        | ( X3
          = ( uminus1351360451143612070nteger @ Y3 ) ) ) ) ).

% abs_eq_iff
thf(fact_6626_abs__eq__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ( abs_abs_rat @ X3 )
        = ( abs_abs_rat @ Y3 ) )
      = ( ( X3 = Y3 )
        | ( X3
          = ( uminus_uminus_rat @ Y3 ) ) ) ) ).

% abs_eq_iff
thf(fact_6627_power__abs,axiom,
    ! [A: code_integer,N: nat] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) )
      = ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).

% power_abs
thf(fact_6628_power__abs,axiom,
    ! [A: rat,N: nat] :
      ( ( abs_abs_rat @ ( power_power_rat @ A @ N ) )
      = ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).

% power_abs
thf(fact_6629_power__abs,axiom,
    ! [A: real,N: nat] :
      ( ( abs_abs_real @ ( power_power_real @ A @ N ) )
      = ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).

% power_abs
thf(fact_6630_power__abs,axiom,
    ! [A: int,N: nat] :
      ( ( abs_abs_int @ ( power_power_int @ A @ N ) )
      = ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).

% power_abs
thf(fact_6631_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_6632_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_6633_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_6634_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_6635_mem__case__prodE,axiom,
    ! [Z: nat,C: int > int > set_nat,P4: product_prod_int_int] :
      ( ( member_nat @ Z @ ( produc4251311855443802252et_nat @ C @ P4 ) )
     => ~ ! [X5: int,Y4: int] :
            ( ( P4
              = ( product_Pair_int_int @ X5 @ Y4 ) )
           => ~ ( member_nat @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6636_mem__case__prodE,axiom,
    ! [Z: real,C: int > int > set_real,P4: product_prod_int_int] :
      ( ( member_real @ Z @ ( produc6452406959799940328t_real @ C @ P4 ) )
     => ~ ! [X5: int,Y4: int] :
            ( ( P4
              = ( product_Pair_int_int @ X5 @ Y4 ) )
           => ~ ( member_real @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6637_mem__case__prodE,axiom,
    ! [Z: int,C: int > int > set_int,P4: product_prod_int_int] :
      ( ( member_int @ Z @ ( produc73460835934605544et_int @ C @ P4 ) )
     => ~ ! [X5: int,Y4: int] :
            ( ( P4
              = ( product_Pair_int_int @ X5 @ Y4 ) )
           => ~ ( member_int @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6638_mem__case__prodE,axiom,
    ! [Z: complex,C: int > int > set_complex,P4: product_prod_int_int] :
      ( ( member_complex @ Z @ ( produc8580519160106071146omplex @ C @ P4 ) )
     => ~ ! [X5: int,Y4: int] :
            ( ( P4
              = ( product_Pair_int_int @ X5 @ Y4 ) )
           => ~ ( member_complex @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6639_mem__case__prodE,axiom,
    ! [Z: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,P4: produc8763457246119570046nteger] :
      ( ( member_nat @ Z @ ( produc3558942015123893603et_nat @ C @ P4 ) )
     => ~ ! [X5: code_integer > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
            ( ( P4
              = ( produc6137756002093451184nteger @ X5 @ Y4 ) )
           => ~ ( member_nat @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6640_mem__case__prodE,axiom,
    ! [Z: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real,P4: produc8763457246119570046nteger] :
      ( ( member_real @ Z @ ( produc815715089573277247t_real @ C @ P4 ) )
     => ~ ! [X5: code_integer > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
            ( ( P4
              = ( produc6137756002093451184nteger @ X5 @ Y4 ) )
           => ~ ( member_real @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6641_mem__case__prodE,axiom,
    ! [Z: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,P4: produc8763457246119570046nteger] :
      ( ( member_int @ Z @ ( produc8604463032469472703et_int @ C @ P4 ) )
     => ~ ! [X5: code_integer > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
            ( ( P4
              = ( produc6137756002093451184nteger @ X5 @ Y4 ) )
           => ~ ( member_int @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6642_mem__case__prodE,axiom,
    ! [Z: complex,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_complex,P4: produc8763457246119570046nteger] :
      ( ( member_complex @ Z @ ( produc2592262431452330817omplex @ C @ P4 ) )
     => ~ ! [X5: code_integer > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
            ( ( P4
              = ( produc6137756002093451184nteger @ X5 @ Y4 ) )
           => ~ ( member_complex @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6643_mem__case__prodE,axiom,
    ! [Z: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,P4: produc7773217078559923341nt_int] :
      ( ( member_nat @ Z @ ( produc8289552606927098482et_nat @ C @ P4 ) )
     => ~ ! [X5: int > option6357759511663192854e_term,Y4: product_prod_int_int] :
            ( ( P4
              = ( produc4305682042979456191nt_int @ X5 @ Y4 ) )
           => ~ ( member_nat @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6644_mem__case__prodE,axiom,
    ! [Z: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real,P4: produc7773217078559923341nt_int] :
      ( ( member_real @ Z @ ( produc8709739885379107790t_real @ C @ P4 ) )
     => ~ ! [X5: int > option6357759511663192854e_term,Y4: product_prod_int_int] :
            ( ( P4
              = ( produc4305682042979456191nt_int @ X5 @ Y4 ) )
           => ~ ( member_real @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6645_case__prodD,axiom,
    ! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
      ( ( produc127349428274296955eger_o @ F @ ( produc6137756002093451184nteger @ A @ B2 ) )
     => ( F @ A @ B2 ) ) ).

% case_prodD
thf(fact_6646_case__prodD,axiom,
    ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
      ( ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B2 ) )
     => ( F @ A @ B2 ) ) ).

% case_prodD
thf(fact_6647_case__prodD,axiom,
    ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
      ( ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B2 ) )
     => ( F @ A @ B2 ) ) ).

% case_prodD
thf(fact_6648_case__prodD,axiom,
    ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B2: product_prod_int_int] :
      ( ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B2 ) )
     => ( F @ A @ B2 ) ) ).

% case_prodD
thf(fact_6649_case__prodD,axiom,
    ! [F: int > int > $o,A: int,B2: int] :
      ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B2 ) )
     => ( F @ A @ B2 ) ) ).

% case_prodD
thf(fact_6650_case__prodE,axiom,
    ! [C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P4: produc8763457246119570046nteger] :
      ( ( produc127349428274296955eger_o @ C @ P4 )
     => ~ ! [X5: code_integer > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
            ( ( P4
              = ( produc6137756002093451184nteger @ X5 @ Y4 ) )
           => ~ ( C @ X5 @ Y4 ) ) ) ).

% case_prodE
thf(fact_6651_case__prodE,axiom,
    ! [C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P4: produc1908205239877642774nteger] :
      ( ( produc6253627499356882019eger_o @ C @ P4 )
     => ~ ! [X5: produc6241069584506657477e_term > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
            ( ( P4
              = ( produc8603105652947943368nteger @ X5 @ Y4 ) )
           => ~ ( C @ X5 @ Y4 ) ) ) ).

% case_prodE
thf(fact_6652_case__prodE,axiom,
    ! [C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,P4: produc2285326912895808259nt_int] :
      ( ( produc1573362020775583542_int_o @ C @ P4 )
     => ~ ! [X5: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int] :
            ( ( P4
              = ( produc5700946648718959541nt_int @ X5 @ Y4 ) )
           => ~ ( C @ X5 @ Y4 ) ) ) ).

% case_prodE
thf(fact_6653_case__prodE,axiom,
    ! [C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,P4: produc7773217078559923341nt_int] :
      ( ( produc2558449545302689196_int_o @ C @ P4 )
     => ~ ! [X5: int > option6357759511663192854e_term,Y4: product_prod_int_int] :
            ( ( P4
              = ( produc4305682042979456191nt_int @ X5 @ Y4 ) )
           => ~ ( C @ X5 @ Y4 ) ) ) ).

% case_prodE
thf(fact_6654_case__prodE,axiom,
    ! [C: int > int > $o,P4: product_prod_int_int] :
      ( ( produc4947309494688390418_int_o @ C @ P4 )
     => ~ ! [X5: int,Y4: int] :
            ( ( P4
              = ( product_Pair_int_int @ X5 @ Y4 ) )
           => ~ ( C @ X5 @ Y4 ) ) ) ).

% case_prodE
thf(fact_6655_case__prodD_H,axiom,
    ! [R2: nat > nat > product_prod_nat_nat > $o,A: nat,B2: nat,C: product_prod_nat_nat] :
      ( ( produc8739625826339149834_nat_o @ R2 @ ( product_Pair_nat_nat @ A @ B2 ) @ C )
     => ( R2 @ A @ B2 @ C ) ) ).

% case_prodD'
thf(fact_6656_case__prodE_H,axiom,
    ! [C: nat > nat > product_prod_nat_nat > $o,P4: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( produc8739625826339149834_nat_o @ C @ P4 @ Z )
     => ~ ! [X5: nat,Y4: nat] :
            ( ( P4
              = ( product_Pair_nat_nat @ X5 @ Y4 ) )
           => ~ ( C @ X5 @ Y4 @ Z ) ) ) ).

% case_prodE'
thf(fact_6657_abs__ge__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_zero
thf(fact_6658_abs__ge__zero,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).

% abs_ge_zero
thf(fact_6659_abs__ge__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).

% abs_ge_zero
thf(fact_6660_abs__ge__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).

% abs_ge_zero
thf(fact_6661_abs__not__less__zero,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).

% abs_not_less_zero
thf(fact_6662_abs__not__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).

% abs_not_less_zero
thf(fact_6663_abs__not__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).

% abs_not_less_zero
thf(fact_6664_abs__not__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_6665_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_6666_abs__of__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_6667_abs__of__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_6668_abs__of__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_6669_abs__triangle__ineq,axiom,
    ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B2 ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).

% abs_triangle_ineq
thf(fact_6670_abs__triangle__ineq,axiom,
    ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B2 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).

% abs_triangle_ineq
thf(fact_6671_abs__triangle__ineq,axiom,
    ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B2 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).

% abs_triangle_ineq
thf(fact_6672_abs__triangle__ineq,axiom,
    ! [A: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B2 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).

% abs_triangle_ineq
thf(fact_6673_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer,D: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
     => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B2 ) @ D )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_6674_abs__mult__less,axiom,
    ! [A: real,C: real,B2: real,D: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
     => ( ( ord_less_real @ ( abs_abs_real @ B2 ) @ D )
       => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) @ ( times_times_real @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_6675_abs__mult__less,axiom,
    ! [A: rat,C: rat,B2: rat,D: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
     => ( ( ord_less_rat @ ( abs_abs_rat @ B2 ) @ D )
       => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_6676_abs__mult__less,axiom,
    ! [A: int,C: int,B2: int,D: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
     => ( ( ord_less_int @ ( abs_abs_int @ B2 ) @ D )
       => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) @ ( times_times_int @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_6677_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ).

% abs_triangle_ineq2
thf(fact_6678_abs__triangle__ineq2,axiom,
    ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) ) ) ).

% abs_triangle_ineq2
thf(fact_6679_abs__triangle__ineq2,axiom,
    ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B2 ) ) ) ).

% abs_triangle_ineq2
thf(fact_6680_abs__triangle__ineq2,axiom,
    ! [A: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) ) ) ).

% abs_triangle_ineq2
thf(fact_6681_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ).

% abs_triangle_ineq3
thf(fact_6682_abs__triangle__ineq3,axiom,
    ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) ) ) ).

% abs_triangle_ineq3
thf(fact_6683_abs__triangle__ineq3,axiom,
    ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B2 ) ) ) ).

% abs_triangle_ineq3
thf(fact_6684_abs__triangle__ineq3,axiom,
    ! [A: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) ) ) ).

% abs_triangle_ineq3
thf(fact_6685_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B2 @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_6686_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) @ ( abs_abs_real @ ( minus_minus_real @ B2 @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_6687_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B2 @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_6688_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B2 @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_6689_nonzero__abs__divide,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( abs_abs_real @ ( divide_divide_real @ A @ B2 ) )
        = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ) ).

% nonzero_abs_divide
thf(fact_6690_nonzero__abs__divide,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B2 ) )
        = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ) ).

% nonzero_abs_divide
thf(fact_6691_abs__leI,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 )
       => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 ) ) ) ).

% abs_leI
thf(fact_6692_abs__leI,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B2 )
     => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
       => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B2 ) ) ) ).

% abs_leI
thf(fact_6693_abs__leI,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A @ B2 )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B2 )
       => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B2 ) ) ) ).

% abs_leI
thf(fact_6694_abs__leI,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ A @ B2 )
     => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 )
       => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 ) ) ) ).

% abs_leI
thf(fact_6695_abs__le__D2,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ).

% abs_le_D2
thf(fact_6696_abs__le__D2,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B2 )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).

% abs_le_D2
thf(fact_6697_abs__le__D2,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B2 )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ).

% abs_le_D2
thf(fact_6698_abs__le__D2,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).

% abs_le_D2
thf(fact_6699_abs__le__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 )
      = ( ( ord_less_eq_real @ A @ B2 )
        & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ) ).

% abs_le_iff
thf(fact_6700_abs__le__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B2 )
      = ( ( ord_le3102999989581377725nteger @ A @ B2 )
        & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ) ).

% abs_le_iff
thf(fact_6701_abs__le__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B2 )
      = ( ( ord_less_eq_rat @ A @ B2 )
        & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ) ).

% abs_le_iff
thf(fact_6702_abs__le__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 )
      = ( ( ord_less_eq_int @ A @ B2 )
        & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).

% abs_le_iff
thf(fact_6703_abs__ge__minus__self,axiom,
    ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).

% abs_ge_minus_self
thf(fact_6704_abs__ge__minus__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_minus_self
thf(fact_6705_abs__ge__minus__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).

% abs_ge_minus_self
thf(fact_6706_abs__ge__minus__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).

% abs_ge_minus_self
thf(fact_6707_abs__less__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A ) @ B2 )
      = ( ( ord_less_real @ A @ B2 )
        & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ) ).

% abs_less_iff
thf(fact_6708_abs__less__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A ) @ B2 )
      = ( ( ord_less_int @ A @ B2 )
        & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).

% abs_less_iff
thf(fact_6709_abs__less__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B2 )
      = ( ( ord_le6747313008572928689nteger @ A @ B2 )
        & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ) ).

% abs_less_iff
thf(fact_6710_abs__less__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B2 )
      = ( ( ord_less_rat @ A @ B2 )
        & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ) ).

% abs_less_iff
thf(fact_6711_sin__bound__lemma,axiom,
    ! [X3: real,Y3: real,U: real,V: real] :
      ( ( X3 = Y3 )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X3 @ U ) @ Y3 ) ) @ V ) ) ) ).

% sin_bound_lemma
thf(fact_6712_numeral__eq__Suc,axiom,
    ( numeral_numeral_nat
    = ( ^ [K2: num] : ( suc @ ( pred_numeral @ K2 ) ) ) ) ).

% numeral_eq_Suc
thf(fact_6713_sum__subtractf__nat,axiom,
    ! [A4: set_real,G: real > nat,F: real > nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups1935376822645274424al_nat
          @ ^ [X: real] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ G @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6714_sum__subtractf__nat,axiom,
    ! [A4: set_int,G: int > nat,F: int > nat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X: int] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6715_sum__subtractf__nat,axiom,
    ! [A4: set_complex,G: complex > nat,F: complex > nat] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A4 )
         => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups5693394587270226106ex_nat
          @ ^ [X: complex] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6716_sum__subtractf__nat,axiom,
    ! [A4: set_nat,G: nat > nat,F: nat > nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups3542108847815614940at_nat
          @ ^ [X: nat] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ G @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6717_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_6718_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > real,M: nat,K: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_6719_dense__eq0__I,axiom,
    ! [X3: real] :
      ( ! [E2: real] :
          ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ E2 ) )
     => ( X3 = zero_zero_real ) ) ).

% dense_eq0_I
thf(fact_6720_dense__eq0__I,axiom,
    ! [X3: rat] :
      ( ! [E2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E2 )
         => ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ E2 ) )
     => ( X3 = zero_zero_rat ) ) ).

% dense_eq0_I
thf(fact_6721_abs__mult__pos,axiom,
    ! [X3: code_integer,Y3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
     => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y3 ) @ X3 )
        = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y3 @ X3 ) ) ) ) ).

% abs_mult_pos
thf(fact_6722_abs__mult__pos,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( times_times_real @ ( abs_abs_real @ Y3 ) @ X3 )
        = ( abs_abs_real @ ( times_times_real @ Y3 @ X3 ) ) ) ) ).

% abs_mult_pos
thf(fact_6723_abs__mult__pos,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( times_times_rat @ ( abs_abs_rat @ Y3 ) @ X3 )
        = ( abs_abs_rat @ ( times_times_rat @ Y3 @ X3 ) ) ) ) ).

% abs_mult_pos
thf(fact_6724_abs__mult__pos,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ( times_times_int @ ( abs_abs_int @ Y3 ) @ X3 )
        = ( abs_abs_int @ ( times_times_int @ Y3 @ X3 ) ) ) ) ).

% abs_mult_pos
thf(fact_6725_abs__eq__mult,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
        & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
          | ( ord_le3102999989581377725nteger @ B2 @ zero_z3403309356797280102nteger ) ) )
     => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) )
        = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ) ).

% abs_eq_mult
thf(fact_6726_abs__eq__mult,axiom,
    ! [A: real,B2: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          | ( ord_less_eq_real @ A @ 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 @ A @ B2 ) )
        = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ) ).

% abs_eq_mult
thf(fact_6727_abs__eq__mult,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          | ( ord_less_eq_rat @ A @ 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 @ A @ B2 ) )
        = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ) ).

% abs_eq_mult
thf(fact_6728_abs__eq__mult,axiom,
    ! [A: int,B2: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          | ( ord_less_eq_int @ A @ 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 @ A @ B2 ) )
        = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ) ).

% abs_eq_mult
thf(fact_6729_abs__eq__iff_H,axiom,
    ! [A: real,B2: real] :
      ( ( ( abs_abs_real @ A )
        = B2 )
      = ( ( ord_less_eq_real @ zero_zero_real @ B2 )
        & ( ( A = B2 )
          | ( A
            = ( uminus_uminus_real @ B2 ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_6730_abs__eq__iff_H,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = B2 )
      = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
        & ( ( A = B2 )
          | ( A
            = ( uminus1351360451143612070nteger @ B2 ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_6731_abs__eq__iff_H,axiom,
    ! [A: rat,B2: rat] :
      ( ( ( abs_abs_rat @ A )
        = B2 )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
        & ( ( A = B2 )
          | ( A
            = ( uminus_uminus_rat @ B2 ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_6732_abs__eq__iff_H,axiom,
    ! [A: int,B2: int] :
      ( ( ( abs_abs_int @ A )
        = B2 )
      = ( ( ord_less_eq_int @ zero_zero_int @ B2 )
        & ( ( A = B2 )
          | ( A
            = ( uminus_uminus_int @ B2 ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_6733_eq__abs__iff_H,axiom,
    ! [A: real,B2: real] :
      ( ( A
        = ( abs_abs_real @ B2 ) )
      = ( ( ord_less_eq_real @ zero_zero_real @ A )
        & ( ( B2 = A )
          | ( B2
            = ( uminus_uminus_real @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_6734_eq__abs__iff_H,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( A
        = ( abs_abs_Code_integer @ B2 ) )
      = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
        & ( ( B2 = A )
          | ( B2
            = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_6735_eq__abs__iff_H,axiom,
    ! [A: rat,B2: rat] :
      ( ( A
        = ( abs_abs_rat @ B2 ) )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
        & ( ( B2 = A )
          | ( B2
            = ( uminus_uminus_rat @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_6736_eq__abs__iff_H,axiom,
    ! [A: int,B2: int] :
      ( ( A
        = ( abs_abs_int @ B2 ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ A )
        & ( ( B2 = A )
          | ( B2
            = ( uminus_uminus_int @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_6737_abs__minus__le__zero,axiom,
    ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).

% abs_minus_le_zero
thf(fact_6738_abs__minus__le__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).

% abs_minus_le_zero
thf(fact_6739_abs__minus__le__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).

% abs_minus_le_zero
thf(fact_6740_abs__minus__le__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).

% abs_minus_le_zero
thf(fact_6741_abs__div__pos,axiom,
    ! [Y3: real,X3: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ( ( divide_divide_real @ ( abs_abs_real @ X3 ) @ Y3 )
        = ( abs_abs_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).

% abs_div_pos
thf(fact_6742_abs__div__pos,axiom,
    ! [Y3: rat,X3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y3 )
     => ( ( divide_divide_rat @ ( abs_abs_rat @ X3 ) @ Y3 )
        = ( abs_abs_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).

% abs_div_pos
thf(fact_6743_zero__le__power__abs,axiom,
    ! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).

% zero_le_power_abs
thf(fact_6744_zero__le__power__abs,axiom,
    ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).

% zero_le_power_abs
thf(fact_6745_zero__le__power__abs,axiom,
    ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).

% zero_le_power_abs
thf(fact_6746_zero__le__power__abs,axiom,
    ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).

% zero_le_power_abs
thf(fact_6747_abs__if,axiom,
    ( abs_abs_real
    = ( ^ [A5: real] : ( if_real @ ( ord_less_real @ A5 @ zero_zero_real ) @ ( uminus_uminus_real @ A5 ) @ A5 ) ) ) ).

% abs_if
thf(fact_6748_abs__if,axiom,
    ( abs_abs_int
    = ( ^ [A5: int] : ( if_int @ ( ord_less_int @ A5 @ zero_zero_int ) @ ( uminus_uminus_int @ A5 ) @ A5 ) ) ) ).

% abs_if
thf(fact_6749_abs__if,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A5: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A5 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A5 ) @ A5 ) ) ) ).

% abs_if
thf(fact_6750_abs__if,axiom,
    ( abs_abs_rat
    = ( ^ [A5: rat] : ( if_rat @ ( ord_less_rat @ A5 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A5 ) @ A5 ) ) ) ).

% abs_if
thf(fact_6751_abs__if__raw,axiom,
    ( abs_abs_real
    = ( ^ [A5: real] : ( if_real @ ( ord_less_real @ A5 @ zero_zero_real ) @ ( uminus_uminus_real @ A5 ) @ A5 ) ) ) ).

% abs_if_raw
thf(fact_6752_abs__if__raw,axiom,
    ( abs_abs_int
    = ( ^ [A5: int] : ( if_int @ ( ord_less_int @ A5 @ zero_zero_int ) @ ( uminus_uminus_int @ A5 ) @ A5 ) ) ) ).

% abs_if_raw
thf(fact_6753_abs__if__raw,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A5: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A5 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A5 ) @ A5 ) ) ) ).

% abs_if_raw
thf(fact_6754_abs__if__raw,axiom,
    ( abs_abs_rat
    = ( ^ [A5: rat] : ( if_rat @ ( ord_less_rat @ A5 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A5 ) @ A5 ) ) ) ).

% abs_if_raw
thf(fact_6755_abs__of__neg,axiom,
    ! [A: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( abs_abs_real @ A )
        = ( uminus_uminus_real @ A ) ) ) ).

% abs_of_neg
thf(fact_6756_abs__of__neg,axiom,
    ! [A: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( abs_abs_int @ A )
        = ( uminus_uminus_int @ A ) ) ) ).

% abs_of_neg
thf(fact_6757_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_neg
thf(fact_6758_abs__of__neg,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( abs_abs_rat @ A )
        = ( uminus_uminus_rat @ A ) ) ) ).

% abs_of_neg
thf(fact_6759_abs__diff__le__iff,axiom,
    ! [X3: code_integer,A: code_integer,R: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X3 @ A ) ) @ R )
      = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R ) @ X3 )
        & ( ord_le3102999989581377725nteger @ X3 @ ( plus_p5714425477246183910nteger @ A @ R ) ) ) ) ).

% abs_diff_le_iff
thf(fact_6760_abs__diff__le__iff,axiom,
    ! [X3: real,A: real,R: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ A ) ) @ R )
      = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R ) @ X3 )
        & ( ord_less_eq_real @ X3 @ ( plus_plus_real @ A @ R ) ) ) ) ).

% abs_diff_le_iff
thf(fact_6761_abs__diff__le__iff,axiom,
    ! [X3: rat,A: rat,R: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X3 @ A ) ) @ R )
      = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R ) @ X3 )
        & ( ord_less_eq_rat @ X3 @ ( plus_plus_rat @ A @ R ) ) ) ) ).

% abs_diff_le_iff
thf(fact_6762_abs__diff__le__iff,axiom,
    ! [X3: int,A: int,R: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ A ) ) @ R )
      = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R ) @ X3 )
        & ( ord_less_eq_int @ X3 @ ( plus_plus_int @ A @ R ) ) ) ) ).

% abs_diff_le_iff
thf(fact_6763_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B2 ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).

% abs_triangle_ineq4
thf(fact_6764_abs__triangle__ineq4,axiom,
    ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).

% abs_triangle_ineq4
thf(fact_6765_abs__triangle__ineq4,axiom,
    ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B2 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).

% abs_triangle_ineq4
thf(fact_6766_abs__triangle__ineq4,axiom,
    ! [A: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).

% abs_triangle_ineq4
thf(fact_6767_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B2 @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_6768_abs__diff__triangle__ineq,axiom,
    ! [A: real,B2: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B2 @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_6769_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B2 @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_6770_abs__diff__triangle__ineq,axiom,
    ! [A: int,B2: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B2 @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_6771_abs__diff__less__iff,axiom,
    ! [X3: code_integer,A: code_integer,R: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X3 @ A ) ) @ R )
      = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R ) @ X3 )
        & ( ord_le6747313008572928689nteger @ X3 @ ( plus_p5714425477246183910nteger @ A @ R ) ) ) ) ).

% abs_diff_less_iff
thf(fact_6772_abs__diff__less__iff,axiom,
    ! [X3: real,A: real,R: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ A ) ) @ R )
      = ( ( ord_less_real @ ( minus_minus_real @ A @ R ) @ X3 )
        & ( ord_less_real @ X3 @ ( plus_plus_real @ A @ R ) ) ) ) ).

% abs_diff_less_iff
thf(fact_6773_abs__diff__less__iff,axiom,
    ! [X3: rat,A: rat,R: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X3 @ A ) ) @ R )
      = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R ) @ X3 )
        & ( ord_less_rat @ X3 @ ( plus_plus_rat @ A @ R ) ) ) ) ).

% abs_diff_less_iff
thf(fact_6774_abs__diff__less__iff,axiom,
    ! [X3: int,A: int,R: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ A ) ) @ R )
      = ( ( ord_less_int @ ( minus_minus_int @ A @ R ) @ X3 )
        & ( ord_less_int @ X3 @ ( plus_plus_int @ A @ R ) ) ) ) ).

% abs_diff_less_iff
thf(fact_6775_sum__SucD,axiom,
    ! [F: nat > nat,A4: set_nat,N: nat] :
      ( ( ( groups3542108847815614940at_nat @ F @ A4 )
        = ( suc @ N ) )
     => ? [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
          & ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) ) ) ).

% sum_SucD
thf(fact_6776_abs__real__def,axiom,
    ( abs_abs_real
    = ( ^ [A5: real] : ( if_real @ ( ord_less_real @ A5 @ zero_zero_real ) @ ( uminus_uminus_real @ A5 ) @ A5 ) ) ) ).

% abs_real_def
thf(fact_6777_lemma__interval__lt,axiom,
    ! [A: real,X3: real,B2: real] :
      ( ( ord_less_real @ A @ X3 )
     => ( ( ord_less_real @ X3 @ B2 )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [Y5: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y5 ) ) @ D3 )
               => ( ( ord_less_real @ A @ Y5 )
                  & ( ord_less_real @ Y5 @ B2 ) ) ) ) ) ) ).

% lemma_interval_lt
thf(fact_6778_sum__power__add,axiom,
    ! [X3: complex,M: nat,I5: set_nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [I4: nat] : ( power_power_complex @ X3 @ ( plus_plus_nat @ M @ I4 ) )
        @ I5 )
      = ( times_times_complex @ ( power_power_complex @ X3 @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_6779_sum__power__add,axiom,
    ! [X3: rat,M: nat,I5: set_nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I4: nat] : ( power_power_rat @ X3 @ ( plus_plus_nat @ M @ I4 ) )
        @ I5 )
      = ( times_times_rat @ ( power_power_rat @ X3 @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_6780_sum__power__add,axiom,
    ! [X3: int,M: nat,I5: set_nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I4: nat] : ( power_power_int @ X3 @ ( plus_plus_nat @ M @ I4 ) )
        @ I5 )
      = ( times_times_int @ ( power_power_int @ X3 @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_6781_sum__power__add,axiom,
    ! [X3: real,M: nat,I5: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( power_power_real @ X3 @ ( plus_plus_nat @ M @ I4 ) )
        @ I5 )
      = ( times_times_real @ ( power_power_real @ X3 @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_6782_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N: nat,M: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_6783_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > real,N: nat,M: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_6784_pred__numeral__def,axiom,
    ( pred_numeral
    = ( ^ [K2: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K2 ) @ one_one_nat ) ) ) ).

% pred_numeral_def
thf(fact_6785_sum__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( groups7754918857620584856omplex
          @ ^ [X: complex] : X
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = C ) ) )
        = zero_zero_complex ) ) ).

% sum_nth_roots
thf(fact_6786_sum__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( groups7754918857620584856omplex
          @ ^ [X: complex] : X
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = one_one_complex ) ) )
        = zero_zero_complex ) ) ).

% sum_roots_unity
thf(fact_6787_abs__add__one__gt__zero,axiom,
    ! [X3: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X3 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_6788_abs__add__one__gt__zero,axiom,
    ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X3 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_6789_abs__add__one__gt__zero,axiom,
    ! [X3: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X3 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_6790_abs__add__one__gt__zero,axiom,
    ! [X3: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X3 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_6791_sum__diff__nat,axiom,
    ! [B5: set_complex,A4: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ B5 @ A4 )
       => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
          = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_6792_sum__diff__nat,axiom,
    ! [B5: set_int,A4: set_int,F: int > nat] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ B5 @ A4 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ F @ B5 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_6793_sum__diff__nat,axiom,
    ! [B5: set_nat,A4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ B5 @ A4 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ F @ B5 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_6794_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_6795_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_6796_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_6797_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_6798_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_6799_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_6800_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_6801_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_6802_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( plus_plus_rat @ ( G @ ( suc @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_6803_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( plus_plus_int @ ( G @ ( suc @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_6804_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( plus_plus_nat @ ( G @ ( suc @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_6805_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( plus_plus_real @ ( G @ ( suc @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_6806_lemma__interval,axiom,
    ! [A: real,X3: real,B2: real] :
      ( ( ord_less_real @ A @ X3 )
     => ( ( ord_less_real @ X3 @ B2 )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [Y5: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y5 ) ) @ D3 )
               => ( ( ord_less_eq_real @ A @ Y5 )
                  & ( ord_less_eq_real @ Y5 @ B2 ) ) ) ) ) ) ).

% lemma_interval
thf(fact_6807_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( plus_plus_rat @ ( G @ M )
          @ ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_6808_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( plus_plus_int @ ( G @ M )
          @ ( groups3539618377306564664at_int
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_6809_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( plus_plus_nat @ ( G @ M )
          @ ( groups3542108847815614940at_nat
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_6810_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( plus_plus_real @ ( G @ M )
          @ ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_6811_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: nat > complex] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups2073611262835488442omplex
          @ ^ [I4: nat] : ( minus_minus_complex @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( minus_minus_complex @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_6812_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_6813_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_6814_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups6591440286371151544t_real
          @ ^ [I4: nat] : ( minus_minus_real @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_6815_abs__le__square__iff,axiom,
    ! [X3: code_integer,Y3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X3 ) @ ( abs_abs_Code_integer @ Y3 ) )
      = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6816_abs__le__square__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ ( abs_abs_real @ Y3 ) )
      = ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6817_abs__le__square__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ ( abs_abs_rat @ Y3 ) )
      = ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6818_abs__le__square__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ X3 ) @ ( abs_abs_int @ Y3 ) )
      = ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6819_abs__square__eq__1,axiom,
    ! [X3: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_Code_integer )
      = ( ( abs_abs_Code_integer @ X3 )
        = one_one_Code_integer ) ) ).

% abs_square_eq_1
thf(fact_6820_abs__square__eq__1,axiom,
    ! [X3: rat] :
      ( ( ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_rat )
      = ( ( abs_abs_rat @ X3 )
        = one_one_rat ) ) ).

% abs_square_eq_1
thf(fact_6821_abs__square__eq__1,axiom,
    ! [X3: real] :
      ( ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
      = ( ( abs_abs_real @ X3 )
        = one_one_real ) ) ).

% abs_square_eq_1
thf(fact_6822_abs__square__eq__1,axiom,
    ! [X3: int] :
      ( ( ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_int )
      = ( ( abs_abs_int @ X3 )
        = one_one_int ) ) ).

% abs_square_eq_1
thf(fact_6823_power__even__abs,axiom,
    ! [N: nat,A: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N )
        = ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% power_even_abs
thf(fact_6824_power__even__abs,axiom,
    ! [N: nat,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ N )
        = ( power_power_rat @ A @ N ) ) ) ).

% power_even_abs
thf(fact_6825_power__even__abs,axiom,
    ! [N: nat,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( abs_abs_real @ A ) @ N )
        = ( power_power_real @ A @ N ) ) ) ).

% power_even_abs
thf(fact_6826_power__even__abs,axiom,
    ! [N: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( abs_abs_int @ A ) @ N )
        = ( power_power_int @ A @ N ) ) ) ).

% power_even_abs
thf(fact_6827_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: nat > rat,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_6828_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: nat > int,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_6829_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: nat > nat,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_6830_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: nat > real,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_6831_divmod__int__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M3: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% divmod_int_def
thf(fact_6832_power2__le__iff__abs__le,axiom,
    ! [Y3: code_integer,X3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y3 )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X3 ) @ Y3 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6833_power2__le__iff__abs__le,axiom,
    ! [Y3: real,X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ Y3 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6834_power2__le__iff__abs__le,axiom,
    ! [Y3: rat,X3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ Y3 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6835_power2__le__iff__abs__le,axiom,
    ! [Y3: int,X3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_int @ ( abs_abs_int @ X3 ) @ Y3 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6836_abs__sqrt__wlog,axiom,
    ! [P: code_integer > code_integer > $o,X3: code_integer] :
      ( ! [X5: code_integer] :
          ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X5 )
         => ( P @ X5 @ ( power_8256067586552552935nteger @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_Code_integer @ X3 ) @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_6837_abs__sqrt__wlog,axiom,
    ! [P: real > real > $o,X3: real] :
      ( ! [X5: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X5 )
         => ( P @ X5 @ ( power_power_real @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_real @ X3 ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_6838_abs__sqrt__wlog,axiom,
    ! [P: rat > rat > $o,X3: rat] :
      ( ! [X5: rat] :
          ( ( ord_less_eq_rat @ zero_zero_rat @ X5 )
         => ( P @ X5 @ ( power_power_rat @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_rat @ X3 ) @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_6839_abs__sqrt__wlog,axiom,
    ! [P: int > int > $o,X3: int] :
      ( ! [X5: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X5 )
         => ( P @ X5 @ ( power_power_int @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_int @ X3 ) @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_6840_abs__square__le__1,axiom,
    ! [X3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X3 ) @ one_one_Code_integer ) ) ).

% abs_square_le_1
thf(fact_6841_abs__square__le__1,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real ) ) ).

% abs_square_le_1
thf(fact_6842_abs__square__le__1,axiom,
    ! [X3: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ one_one_rat ) ) ).

% abs_square_le_1
thf(fact_6843_abs__square__le__1,axiom,
    ! [X3: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_eq_int @ ( abs_abs_int @ X3 ) @ one_one_int ) ) ).

% abs_square_le_1
thf(fact_6844_divmod__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M3: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% divmod_def
thf(fact_6845_divmod__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M3: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).

% divmod_def
thf(fact_6846_divmod__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M3: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).

% divmod_def
thf(fact_6847_abs__square__less__1,axiom,
    ! [X3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X3 ) @ one_one_Code_integer ) ) ).

% abs_square_less_1
thf(fact_6848_abs__square__less__1,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real ) ) ).

% abs_square_less_1
thf(fact_6849_abs__square__less__1,axiom,
    ! [X3: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_rat @ ( abs_abs_rat @ X3 ) @ one_one_rat ) ) ).

% abs_square_less_1
thf(fact_6850_abs__square__less__1,axiom,
    ! [X3: int] :
      ( ( ord_less_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_int @ ( abs_abs_int @ X3 ) @ one_one_int ) ) ).

% abs_square_less_1
thf(fact_6851_power__mono__even,axiom,
    ! [N: nat,A: code_integer,B2: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B2 @ N ) ) ) ) ).

% power_mono_even
thf(fact_6852_power__mono__even,axiom,
    ! [N: nat,A: real,B2: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ).

% power_mono_even
thf(fact_6853_power__mono__even,axiom,
    ! [N: nat,A: rat,B2: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ).

% power_mono_even
thf(fact_6854_power__mono__even,axiom,
    ! [N: nat,A: int,B2: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).

% power_mono_even
thf(fact_6855_divmod_H__nat__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M3: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).

% divmod'_nat_def
thf(fact_6856_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X3: nat > code_integer,A: nat > code_integer,B2: code_integer,Delta: code_integer] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X3 @ I3 ) ) )
     => ( ( ( groups7501900531339628137nteger @ X3 @ I5 )
          = one_one_Code_integer )
       => ( ! [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7501900531339628137nteger
                  @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X3 @ I4 ) )
                  @ I5 )
                @ B2 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6857_convex__sum__bound__le,axiom,
    ! [I5: set_real,X3: real > code_integer,A: real > code_integer,B2: code_integer,Delta: code_integer] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X3 @ I3 ) ) )
     => ( ( ( groups7713935264441627589nteger @ X3 @ I5 )
          = one_one_Code_integer )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7713935264441627589nteger
                  @ ^ [I4: real] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X3 @ I4 ) )
                  @ I5 )
                @ B2 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6858_convex__sum__bound__le,axiom,
    ! [I5: set_int,X3: int > code_integer,A: int > code_integer,B2: code_integer,Delta: code_integer] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X3 @ I3 ) ) )
     => ( ( ( groups7873554091576472773nteger @ X3 @ I5 )
          = one_one_Code_integer )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7873554091576472773nteger
                  @ ^ [I4: int] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X3 @ I4 ) )
                  @ I5 )
                @ B2 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6859_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X3: complex > code_integer,A: complex > code_integer,B2: code_integer,Delta: code_integer] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X3 @ I3 ) ) )
     => ( ( ( groups6621422865394947399nteger @ X3 @ I5 )
          = one_one_Code_integer )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups6621422865394947399nteger
                  @ ^ [I4: complex] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X3 @ I4 ) )
                  @ I5 )
                @ B2 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6860_convex__sum__bound__le,axiom,
    ! [I5: set_real,X3: real > real,A: real > real,B2: real,Delta: real] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ I5 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I3 ) ) )
     => ( ( ( groups8097168146408367636l_real @ X3 @ I5 )
          = one_one_real )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ I5 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8097168146408367636l_real
                  @ ^ [I4: real] : ( times_times_real @ ( A @ I4 ) @ ( X3 @ I4 ) )
                  @ I5 )
                @ B2 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6861_convex__sum__bound__le,axiom,
    ! [I5: set_int,X3: int > real,A: int > real,B2: real,Delta: real] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ I5 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I3 ) ) )
     => ( ( ( groups8778361861064173332t_real @ X3 @ I5 )
          = one_one_real )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8778361861064173332t_real
                  @ ^ [I4: int] : ( times_times_real @ ( A @ I4 ) @ ( X3 @ I4 ) )
                  @ I5 )
                @ B2 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6862_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X3: complex > real,A: complex > real,B2: real,Delta: real] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ I5 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I3 ) ) )
     => ( ( ( groups5808333547571424918x_real @ X3 @ I5 )
          = one_one_real )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups5808333547571424918x_real
                  @ ^ [I4: complex] : ( times_times_real @ ( A @ I4 ) @ ( X3 @ I4 ) )
                  @ I5 )
                @ B2 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6863_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X3: nat > rat,A: nat > rat,B2: rat,Delta: rat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X3 @ I3 ) ) )
     => ( ( ( groups2906978787729119204at_rat @ X3 @ I5 )
          = one_one_rat )
       => ( ! [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups2906978787729119204at_rat
                  @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( X3 @ I4 ) )
                  @ I5 )
                @ B2 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6864_convex__sum__bound__le,axiom,
    ! [I5: set_real,X3: real > rat,A: real > rat,B2: rat,Delta: rat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X3 @ I3 ) ) )
     => ( ( ( groups1300246762558778688al_rat @ X3 @ I5 )
          = one_one_rat )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups1300246762558778688al_rat
                  @ ^ [I4: real] : ( times_times_rat @ ( A @ I4 ) @ ( X3 @ I4 ) )
                  @ I5 )
                @ B2 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6865_convex__sum__bound__le,axiom,
    ! [I5: set_int,X3: int > rat,A: int > rat,B2: rat,Delta: rat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X3 @ I3 ) ) )
     => ( ( ( groups3906332499630173760nt_rat @ X3 @ I5 )
          = one_one_rat )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups3906332499630173760nt_rat
                  @ ^ [I4: int] : ( times_times_rat @ ( A @ I4 ) @ ( X3 @ I4 ) )
                  @ I5 )
                @ B2 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_6866_dbl__dec__def,axiom,
    ( neg_nu6511756317524482435omplex
    = ( ^ [X: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X @ X ) @ one_one_complex ) ) ) ).

% dbl_dec_def
thf(fact_6867_dbl__dec__def,axiom,
    ( neg_nu6075765906172075777c_real
    = ( ^ [X: real] : ( minus_minus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).

% dbl_dec_def
thf(fact_6868_dbl__dec__def,axiom,
    ( neg_nu3179335615603231917ec_rat
    = ( ^ [X: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X @ X ) @ one_one_rat ) ) ) ).

% dbl_dec_def
thf(fact_6869_dbl__dec__def,axiom,
    ( neg_nu3811975205180677377ec_int
    = ( ^ [X: int] : ( minus_minus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).

% dbl_dec_def
thf(fact_6870_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: nat > complex] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( groups2073611262835488442omplex
            @ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = ( minus_minus_complex @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( groups2073611262835488442omplex
            @ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_complex ) ) ) ).

% sum_natinterval_diff
thf(fact_6871_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: nat > rat] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
thf(fact_6872_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: nat > int] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( groups3539618377306564664at_int
            @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( groups3539618377306564664at_int
            @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_int ) ) ) ).

% sum_natinterval_diff
thf(fact_6873_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: nat > real] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( groups6591440286371151544t_real
            @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( groups6591440286371151544t_real
            @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_real ) ) ) ).

% sum_natinterval_diff
thf(fact_6874_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: nat > complex] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups2073611262835488442omplex
          @ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
        = ( minus_minus_complex @ ( F @ N ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_6875_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups2906978787729119204at_rat
          @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
        = ( minus_minus_rat @ ( F @ N ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_6876_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3539618377306564664at_int
          @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
        = ( minus_minus_int @ ( F @ N ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_6877_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups6591440286371151544t_real
          @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
        = ( minus_minus_real @ ( F @ N ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_6878_mask__eq__sum__exp,axiom,
    ! [N: nat] :
      ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int )
      = ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_6879_mask__eq__sum__exp,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_6880_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X3: complex] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X3 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
        = ( minus_minus_complex @ ( power_power_complex @ X3 @ M ) @ ( power_power_complex @ X3 @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_6881_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X3: rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X3 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
        = ( minus_minus_rat @ ( power_power_rat @ X3 @ M ) @ ( power_power_rat @ X3 @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_6882_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X3: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X3 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
        = ( minus_minus_int @ ( power_power_int @ X3 @ M ) @ ( power_power_int @ X3 @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_6883_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X3: real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X3 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
        = ( minus_minus_real @ ( power_power_real @ X3 @ M ) @ ( power_power_real @ X3 @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_6884_sum_Oin__pairs,axiom,
    ! [G: nat > rat,M: nat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I4: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.in_pairs
thf(fact_6885_sum_Oin__pairs,axiom,
    ! [G: nat > int,M: nat,N: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups3539618377306564664at_int
        @ ^ [I4: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.in_pairs
thf(fact_6886_sum_Oin__pairs,axiom,
    ! [G: nat > nat,M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.in_pairs
thf(fact_6887_sum_Oin__pairs,axiom,
    ! [G: nat > real,M: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.in_pairs
thf(fact_6888_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).

% mask_eq_sum_exp_nat
thf(fact_6889_gauss__sum__nat,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( 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_6890_arith__series__nat,axiom,
    ! [A: nat,D: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( 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 ) ) @ A ) @ ( times_times_nat @ N @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series_nat
thf(fact_6891_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( 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_6892_divmod__divmod__step,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M3: num,N2: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M3 @ N2 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M3 ) ) @ ( unique5026877609467782581ep_nat @ N2 @ ( unique5055182867167087721od_nat @ M3 @ ( bit0 @ N2 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_6893_divmod__divmod__step,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M3: num,N2: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M3 @ N2 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M3 ) ) @ ( unique5024387138958732305ep_int @ N2 @ ( unique5052692396658037445od_int @ M3 @ ( bit0 @ N2 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_6894_divmod__divmod__step,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M3: num,N2: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M3 @ N2 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M3 ) ) @ ( unique4921790084139445826nteger @ N2 @ ( unique3479559517661332726nteger @ M3 @ ( bit0 @ N2 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_6895_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ X3 @ one_one_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) @ X3 ) ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_6896_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ ( 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 @ X3 ) ) @ X3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound
thf(fact_6897_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M3: nat,N2: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N2 = zero_zero_nat )
            | ( ord_less_nat @ M3 @ N2 ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M3 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M3 @ N2 ) @ N2 ) ) ) ) ) ).

% divmod_nat_if
thf(fact_6898_tanh__ln__real,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( tanh_real @ ( ln_ln_real @ X3 ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% tanh_ln_real
thf(fact_6899_arctan__double,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X3 ) )
        = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% arctan_double
thf(fact_6900_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_6901_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_6902_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_6903_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_6904_sum__gp,axiom,
    ! [N: nat,M: nat,X3: complex] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( ( X3 = one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
          & ( ( X3 != one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X3 @ M ) @ ( power_power_complex @ X3 @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X3 ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_6905_sum__gp,axiom,
    ! [N: nat,M: nat,X3: rat] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( ( X3 = one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
          & ( ( X3 != one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X3 @ M ) @ ( power_power_rat @ X3 @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X3 ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_6906_sum__gp,axiom,
    ! [N: nat,M: nat,X3: real] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( ( X3 = one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
          & ( ( X3 != one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X3 @ M ) @ ( power_power_real @ X3 @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_6907_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_6908_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( semiri5074537144036343181t_real @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_6909_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_6910_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_6911_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
      = ( semiri4939895301339042750nteger @ N ) ) ).

% abs_of_nat
thf(fact_6912_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
      = ( semiri681578069525770553at_rat @ N ) ) ).

% abs_of_nat
thf(fact_6913_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% abs_of_nat
thf(fact_6914_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
      = ( semiri5074537144036343181t_real @ N ) ) ).

% abs_of_nat
thf(fact_6915_negative__zle,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zle
thf(fact_6916_tanh__real__less__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( tanh_real @ X3 ) @ ( tanh_real @ Y3 ) )
      = ( ord_less_real @ X3 @ Y3 ) ) ).

% tanh_real_less_iff
thf(fact_6917_tanh__real__le__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X3 ) @ ( tanh_real @ Y3 ) )
      = ( ord_less_eq_real @ X3 @ Y3 ) ) ).

% tanh_real_le_iff
thf(fact_6918_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri8010041392384452111omplex @ M )
        = zero_zero_complex )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_6919_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_6920_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_6921_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_6922_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_6923_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_complex
        = ( semiri8010041392384452111omplex @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_6924_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_6925_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_6926_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_6927_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_6928_of__nat__0,axiom,
    ( ( semiri8010041392384452111omplex @ zero_zero_nat )
    = zero_zero_complex ) ).

% of_nat_0
thf(fact_6929_of__nat__0,axiom,
    ( ( semiri681578069525770553at_rat @ zero_zero_nat )
    = zero_zero_rat ) ).

% of_nat_0
thf(fact_6930_of__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% of_nat_0
thf(fact_6931_of__nat__0,axiom,
    ( ( semiri5074537144036343181t_real @ zero_zero_nat )
    = zero_zero_real ) ).

% of_nat_0
thf(fact_6932_of__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% of_nat_0
thf(fact_6933_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_6934_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_6935_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_6936_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_6937_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
      = ( numera6690914467698888265omplex @ N ) ) ).

% of_nat_numeral
thf(fact_6938_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_rat @ N ) ) ).

% of_nat_numeral
thf(fact_6939_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% of_nat_numeral
thf(fact_6940_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_real @ N ) ) ).

% of_nat_numeral
thf(fact_6941_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ N ) ) ).

% of_nat_numeral
thf(fact_6942_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_6943_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_6944_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_6945_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_6946_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_6947_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_6948_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_6949_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_6950_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_6951_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_6952_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_6953_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_6954_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri8010041392384452111omplex @ N )
        = one_one_complex )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_6955_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_6956_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_6957_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_6958_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_6959_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_complex
        = ( semiri8010041392384452111omplex @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_6960_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_6961_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_6962_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_6963_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_6964_of__nat__1,axiom,
    ( ( semiri8010041392384452111omplex @ one_one_nat )
    = one_one_complex ) ).

% of_nat_1
thf(fact_6965_of__nat__1,axiom,
    ( ( semiri681578069525770553at_rat @ one_one_nat )
    = one_one_rat ) ).

% of_nat_1
thf(fact_6966_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_6967_of__nat__1,axiom,
    ( ( semiri5074537144036343181t_real @ one_one_nat )
    = one_one_real ) ).

% of_nat_1
thf(fact_6968_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_6969_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N ) )
      = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N ) ) ).

% of_nat_power
thf(fact_6970_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
      = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).

% of_nat_power
thf(fact_6971_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
      = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).

% of_nat_power
thf(fact_6972_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
      = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).

% of_nat_power
thf(fact_6973_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X3: nat] :
      ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B2 ) @ W )
        = ( semiri8010041392384452111omplex @ X3 ) )
      = ( ( power_power_nat @ B2 @ W )
        = X3 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_6974_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X3: nat] :
      ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W )
        = ( semiri1314217659103216013at_int @ X3 ) )
      = ( ( power_power_nat @ B2 @ W )
        = X3 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_6975_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X3: nat] :
      ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W )
        = ( semiri5074537144036343181t_real @ X3 ) )
      = ( ( power_power_nat @ B2 @ W )
        = X3 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_6976_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X3: nat] :
      ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W )
        = ( semiri1316708129612266289at_nat @ X3 ) )
      = ( ( power_power_nat @ B2 @ W )
        = X3 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_6977_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: nat,B2: nat,W: nat] :
      ( ( ( semiri8010041392384452111omplex @ X3 )
        = ( power_power_complex @ ( semiri8010041392384452111omplex @ B2 ) @ W ) )
      = ( X3
        = ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_6978_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: nat,B2: nat,W: nat] :
      ( ( ( semiri1314217659103216013at_int @ X3 )
        = ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) )
      = ( X3
        = ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_6979_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: nat,B2: nat,W: nat] :
      ( ( ( semiri5074537144036343181t_real @ X3 )
        = ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) )
      = ( X3
        = ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_6980_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: nat,B2: nat,W: nat] :
      ( ( ( semiri1316708129612266289at_nat @ X3 )
        = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) )
      = ( X3
        = ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_6981_negative__zless,axiom,
    ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zless
thf(fact_6982_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_6983_zero__less__arctan__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( arctan @ X3 ) )
      = ( ord_less_real @ zero_zero_real @ X3 ) ) ).

% zero_less_arctan_iff
thf(fact_6984_arctan__less__zero__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( arctan @ X3 ) @ zero_zero_real )
      = ( ord_less_real @ X3 @ zero_zero_real ) ) ).

% arctan_less_zero_iff
thf(fact_6985_zero__le__arctan__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X3 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).

% zero_le_arctan_iff
thf(fact_6986_arctan__le__zero__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( arctan @ X3 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).

% arctan_le_zero_iff
thf(fact_6987_tanh__real__pos__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X3 ) )
      = ( ord_less_real @ zero_zero_real @ X3 ) ) ).

% tanh_real_pos_iff
thf(fact_6988_tanh__real__neg__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( tanh_real @ X3 ) @ zero_zero_real )
      = ( ord_less_real @ X3 @ zero_zero_real ) ) ).

% tanh_real_neg_iff
thf(fact_6989_tanh__real__nonpos__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X3 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).

% tanh_real_nonpos_iff
thf(fact_6990_tanh__real__nonneg__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X3 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).

% tanh_real_nonneg_iff
thf(fact_6991_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
    = one_one_complex ) ).

% dbl_inc_simps(2)
thf(fact_6992_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
    = one_one_real ) ).

% dbl_inc_simps(2)
thf(fact_6993_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
    = one_one_rat ) ).

% dbl_inc_simps(2)
thf(fact_6994_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
    = one_one_int ) ).

% dbl_inc_simps(2)
thf(fact_6995_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% of_nat_of_bool
thf(fact_6996_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2687167440665602831ol_nat @ P ) ) ).

% of_nat_of_bool
thf(fact_6997_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% of_nat_of_bool
thf(fact_6998_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n356916108424825756nteger @ P ) ) ).

% of_nat_of_bool
thf(fact_6999_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_7000_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_7001_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_inc_simps(4)
thf(fact_7002_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_inc_simps(4)
thf(fact_7003_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_7004_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
      = ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_7005_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_7006_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_7007_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_7008_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_7009_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_7010_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_7011_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_7012_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
      = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).

% of_nat_Suc
thf(fact_7013_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).

% of_nat_Suc
thf(fact_7014_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% of_nat_Suc
thf(fact_7015_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
      = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).

% of_nat_Suc
thf(fact_7016_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).

% of_nat_Suc
thf(fact_7017_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_7018_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_7019_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_7020_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_7021_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_7022_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_7023_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_7024_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_7025_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_7026_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_7027_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_7028_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_7029_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_7030_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_7031_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_7032_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_7033_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_7034_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X3: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) @ ( semiri681578069525770553at_rat @ X3 ) )
      = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X3 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_7035_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X3: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) @ ( semiri1314217659103216013at_int @ X3 ) )
      = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X3 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_7036_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X3: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) @ ( semiri5074537144036343181t_real @ X3 ) )
      = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X3 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_7037_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X3: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) @ ( semiri1316708129612266289at_nat @ X3 ) )
      = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X3 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_7038_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X3: nat,B2: nat,W: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) )
      = ( ord_less_nat @ X3 @ ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_7039_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X3: nat,B2: nat,W: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) )
      = ( ord_less_nat @ X3 @ ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_7040_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X3: nat,B2: nat,W: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) )
      = ( ord_less_nat @ X3 @ ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_7041_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X3: nat,B2: nat,W: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) )
      = ( ord_less_nat @ X3 @ ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_7042_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,X3: num,N: nat] :
      ( ( ( semiri8010041392384452111omplex @ Y3 )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X3 ) @ N ) )
      = ( Y3
        = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_7043_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,X3: num,N: nat] :
      ( ( ( semiri681578069525770553at_rat @ Y3 )
        = ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N ) )
      = ( Y3
        = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_7044_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,X3: num,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ Y3 )
        = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) )
      = ( Y3
        = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_7045_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,X3: num,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ Y3 )
        = ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N ) )
      = ( Y3
        = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_7046_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: nat,X3: num,N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ Y3 )
        = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) )
      = ( Y3
        = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_7047_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: num,N: nat,Y3: nat] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X3 ) @ N )
        = ( semiri8010041392384452111omplex @ Y3 ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
        = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_7048_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: num,N: nat,Y3: nat] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N )
        = ( semiri681578069525770553at_rat @ Y3 ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
        = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_7049_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: num,N: nat,Y3: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N )
        = ( semiri1314217659103216013at_int @ Y3 ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
        = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_7050_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: num,N: nat,Y3: nat] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N )
        = ( semiri5074537144036343181t_real @ Y3 ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
        = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_7051_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: num,N: nat,Y3: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
        = ( semiri1316708129612266289at_nat @ Y3 ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
        = Y3 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_7052_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X3: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) @ ( semiri5074537144036343181t_real @ X3 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X3 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_7053_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X3: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) @ ( semiri681578069525770553at_rat @ X3 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X3 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_7054_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X3: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) @ ( semiri1316708129612266289at_nat @ X3 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X3 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_7055_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X3: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) @ ( semiri1314217659103216013at_int @ X3 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X3 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_7056_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X3: nat,B2: nat,W: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) )
      = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_7057_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X3: nat,B2: nat,W: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) )
      = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_7058_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X3: nat,B2: nat,W: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) )
      = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_7059_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X3: nat,B2: nat,W: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) )
      = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_7060_of__nat__zero__less__power__iff,axiom,
    ! [X3: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X3 ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X3 )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_7061_of__nat__zero__less__power__iff,axiom,
    ! [X3: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X3 ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X3 )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_7062_of__nat__zero__less__power__iff,axiom,
    ! [X3: nat,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X3 ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X3 )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_7063_of__nat__zero__less__power__iff,axiom,
    ! [X3: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X3 )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_7064_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_7065_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_7066_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_7067_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X3: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X3 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_7068_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X3: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X3 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_7069_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X3: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X3 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_7070_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X3: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X3 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_7071_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X3: nat,I: num,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
      = ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_7072_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X3: nat,I: num,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
      = ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_7073_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X3: nat,I: num,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
      = ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_7074_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X3: nat,I: num,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
      = ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_7075_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X3: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X3 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_7076_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X3: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X3 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_7077_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X3: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X3 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_7078_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X3: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X3 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_7079_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X3: nat,I: num,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
      = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_7080_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X3: nat,I: num,N: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
      = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_7081_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X3: nat,I: num,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
      = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_7082_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X3: nat,I: num,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
      = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_7083_Collect__case__prod__mono,axiom,
    ! [A4: int > int > $o,B5: int > int > $o] :
      ( ( ord_le6741204236512500942_int_o @ A4 @ B5 )
     => ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ A4 ) ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ B5 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_7084_mult__of__nat__commute,axiom,
    ! [X3: nat,Y3: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X3 ) @ Y3 )
      = ( times_times_rat @ Y3 @ ( semiri681578069525770553at_rat @ X3 ) ) ) ).

% mult_of_nat_commute
thf(fact_7085_mult__of__nat__commute,axiom,
    ! [X3: nat,Y3: int] :
      ( ( times_times_int @ ( semiri1314217659103216013at_int @ X3 ) @ Y3 )
      = ( times_times_int @ Y3 @ ( semiri1314217659103216013at_int @ X3 ) ) ) ).

% mult_of_nat_commute
thf(fact_7086_mult__of__nat__commute,axiom,
    ! [X3: nat,Y3: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ X3 ) @ Y3 )
      = ( times_times_real @ Y3 @ ( semiri5074537144036343181t_real @ X3 ) ) ) ).

% mult_of_nat_commute
thf(fact_7087_mult__of__nat__commute,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ Y3 )
      = ( times_times_nat @ Y3 @ ( semiri1316708129612266289at_nat @ X3 ) ) ) ).

% mult_of_nat_commute
thf(fact_7088_arctan__less__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( arctan @ X3 ) @ ( arctan @ Y3 ) )
      = ( ord_less_real @ X3 @ Y3 ) ) ).

% arctan_less_iff
thf(fact_7089_arctan__monotone,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ( ord_less_real @ ( arctan @ X3 ) @ ( arctan @ Y3 ) ) ) ).

% arctan_monotone
thf(fact_7090_arctan__monotone_H,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ X3 @ Y3 )
     => ( ord_less_eq_real @ ( arctan @ X3 ) @ ( arctan @ Y3 ) ) ) ).

% arctan_monotone'
thf(fact_7091_arctan__le__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( arctan @ X3 ) @ ( arctan @ Y3 ) )
      = ( ord_less_eq_real @ X3 @ Y3 ) ) ).

% arctan_le_iff
thf(fact_7092_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_7093_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_7094_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_7095_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_7096_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).

% of_nat_less_0_iff
thf(fact_7097_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_7098_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).

% of_nat_less_0_iff
thf(fact_7099_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_7100_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ N ) )
     != zero_zero_complex ) ).

% of_nat_neq_0
thf(fact_7101_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
     != zero_zero_rat ) ).

% of_nat_neq_0
thf(fact_7102_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
     != zero_zero_int ) ).

% of_nat_neq_0
thf(fact_7103_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
     != zero_zero_real ) ).

% of_nat_neq_0
thf(fact_7104_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
     != zero_zero_nat ) ).

% of_nat_neq_0
thf(fact_7105_div__mult2__eq_H,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
      = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% div_mult2_eq'
thf(fact_7106_div__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% div_mult2_eq'
thf(fact_7107_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_7108_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_7109_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_7110_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_7111_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_7112_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_7113_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_7114_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_7115_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_7116_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_7117_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_7118_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_7119_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_7120_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_7121_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_7122_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_7123_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_7124_int__ops_I1_J,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% int_ops(1)
thf(fact_7125_int__ops_I3_J,axiom,
    ! [N: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% int_ops(3)
thf(fact_7126_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_7127_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_7128_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_7129_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_7130_nonneg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ~ ! [N3: nat] :
            ( K
           != ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% nonneg_int_cases
thf(fact_7131_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ? [N3: nat] :
          ( K
          = ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% zero_le_imp_eq_int
thf(fact_7132_of__nat__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N ) )
      = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% of_nat_mod
thf(fact_7133_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_7134_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_7135_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_7136_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_7137_int__ops_I5_J,axiom,
    ! [A: nat,B2: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).

% int_ops(5)
thf(fact_7138_int__ops_I2_J,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% int_ops(2)
thf(fact_7139_int__ops_I7_J,axiom,
    ! [A: nat,B2: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B2 ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).

% int_ops(7)
thf(fact_7140_zle__iff__zadd,axiom,
    ( ord_less_eq_int
    = ( ^ [W3: int,Z2: int] :
        ? [N2: nat] :
          ( Z2
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% zle_iff_zadd
thf(fact_7141_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_7142_zdiv__int,axiom,
    ! [A: nat,B2: nat] :
      ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B2 ) )
      = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).

% zdiv_int
thf(fact_7143_of__nat__max,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( semiri4216267220026989637d_enat @ ( ord_max_nat @ X3 @ Y3 ) )
      = ( ord_ma741700101516333627d_enat @ ( semiri4216267220026989637d_enat @ X3 ) @ ( semiri4216267220026989637d_enat @ Y3 ) ) ) ).

% of_nat_max
thf(fact_7144_of__nat__max,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X3 @ Y3 ) )
      = ( ord_max_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) ).

% of_nat_max
thf(fact_7145_of__nat__max,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X3 @ Y3 ) )
      = ( ord_max_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( semiri5074537144036343181t_real @ Y3 ) ) ) ).

% of_nat_max
thf(fact_7146_of__nat__max,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X3 @ Y3 ) )
      = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( semiri1316708129612266289at_nat @ Y3 ) ) ) ).

% of_nat_max
thf(fact_7147_tanh__real__lt__1,axiom,
    ! [X3: real] : ( ord_less_real @ ( tanh_real @ X3 ) @ one_one_real ) ).

% tanh_real_lt_1
thf(fact_7148_nat__less__as__int,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_less_as_int
thf(fact_7149_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_leq_as_int
thf(fact_7150_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_7151_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_7152_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_7153_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_7154_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_7155_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_7156_reals__Archimedean3,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ! [Y5: real] :
        ? [N3: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X3 ) ) ) ).

% reals_Archimedean3
thf(fact_7157_int__cases4,axiom,
    ! [M: int] :
      ( ! [N3: nat] :
          ( M
         != ( semiri1314217659103216013at_int @ N3 ) )
     => ~ ! [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( M
             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% int_cases4
thf(fact_7158_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_7159_real__of__nat__div4,axiom,
    ! [N: nat,X3: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X3 ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X3 ) ) ) ).

% real_of_nat_div4
thf(fact_7160_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_7161_int__ops_I4_J,axiom,
    ! [A: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).

% int_ops(4)
thf(fact_7162_int__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).

% int_Suc
thf(fact_7163_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W3: int,Z2: int] :
        ? [N2: nat] :
          ( Z2
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_7164_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_7165_negative__zle__0,axiom,
    ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).

% negative_zle_0
thf(fact_7166_nonpos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ~ ! [N3: nat] :
            ( K
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).

% nonpos_int_cases
thf(fact_7167_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_7168_tanh__real__gt__neg1,axiom,
    ! [X3: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X3 ) ) ).

% tanh_real_gt_neg1
thf(fact_7169_mod__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_7170_mod__mult2__eq_H,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_7171_mod__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_7172_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_7173_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_7174_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_7175_pos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ~ ! [N3: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N3 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% pos_int_cases
thf(fact_7176_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ? [N3: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N3 )
          & ( K
            = ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_7177_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N3: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N3 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
       => ~ ! [N3: nat] :
              ( ( K
                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
             => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).

% int_cases3
thf(fact_7178_nat__less__real__le,axiom,
    ( ord_less_nat
    = ( ^ [N2: nat,M3: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M3 ) ) ) ) ).

% nat_less_real_le
thf(fact_7179_nat__le__real__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [N2: nat,M3: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M3 ) @ one_one_real ) ) ) ) ).

% nat_le_real_less
thf(fact_7180_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_7181_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_7182_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_7183_negD,axiom,
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ zero_zero_int )
     => ? [N3: nat] :
          ( X3
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).

% negD
thf(fact_7184_negative__zless__0,axiom,
    ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_7185_dbl__inc__def,axiom,
    ( neg_nu8557863876264182079omplex
    = ( ^ [X: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X @ X ) @ one_one_complex ) ) ) ).

% dbl_inc_def
thf(fact_7186_dbl__inc__def,axiom,
    ( neg_nu8295874005876285629c_real
    = ( ^ [X: real] : ( plus_plus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).

% dbl_inc_def
thf(fact_7187_dbl__inc__def,axiom,
    ( neg_nu5219082963157363817nc_rat
    = ( ^ [X: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X @ X ) @ one_one_rat ) ) ) ).

% dbl_inc_def
thf(fact_7188_dbl__inc__def,axiom,
    ( neg_nu5851722552734809277nc_int
    = ( ^ [X: int] : ( plus_plus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).

% dbl_inc_def
thf(fact_7189_int__ops_I6_J,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B2 ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B2 ) )
          = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% int_ops(6)
thf(fact_7190_real__of__nat__div__aux,axiom,
    ! [X3: nat,D: nat] :
      ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( semiri5074537144036343181t_real @ D ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X3 @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X3 @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).

% real_of_nat_div_aux
thf(fact_7191_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_7192_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_7193_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_7194_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_7195_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_7196_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_7197_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_7198_real__archimedian__rdiv__eq__0,axiom,
    ! [X3: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ! [M4: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ M4 )
             => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X3 ) @ C ) )
         => ( X3 = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
thf(fact_7199_neg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ~ ! [N3: nat] :
            ( ( K
              = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% neg_int_cases
thf(fact_7200_zdiff__int__split,axiom,
    ! [P: int > $o,X3: nat,Y3: nat] :
      ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X3 @ Y3 ) ) )
      = ( ( ( ord_less_eq_nat @ Y3 @ X3 )
         => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) )
        & ( ( ord_less_nat @ X3 @ Y3 )
         => ( P @ zero_zero_int ) ) ) ) ).

% zdiff_int_split
thf(fact_7201_real__of__nat__div2,axiom,
    ! [N: nat,X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X3 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X3 ) ) ) ) ).

% real_of_nat_div2
thf(fact_7202_real__of__nat__div3,axiom,
    ! [N: nat,X3: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X3 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X3 ) ) ) @ one_one_real ) ).

% real_of_nat_div3
thf(fact_7203_ln__realpow,axiom,
    ! [X3: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ln_ln_real @ ( power_power_real @ X3 @ N ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X3 ) ) ) ) ).

% ln_realpow
thf(fact_7204_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_7205_incr__lemma,axiom,
    ! [D: int,Z: int,X3: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ Z @ ( plus_plus_int @ X3 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).

% incr_lemma
thf(fact_7206_decr__lemma,axiom,
    ! [D: int,X3: int,Z: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ ( minus_minus_int @ X3 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).

% decr_lemma
thf(fact_7207_linear__plus__1__le__power,axiom,
    ! [X3: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X3 ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X3 @ one_one_real ) @ N ) ) ) ).

% linear_plus_1_le_power
thf(fact_7208_Bernoulli__inequality,axiom,
    ! [X3: real,N: nat] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X3 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X3 ) @ N ) ) ) ).

% Bernoulli_inequality
thf(fact_7209_divmod__nat__def,axiom,
    ( divmod_nat
    = ( ^ [M3: nat,N2: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M3 @ N2 ) @ ( modulo_modulo_nat @ M3 @ N2 ) ) ) ) ).

% divmod_nat_def
thf(fact_7210_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_7211_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_7212_double__arith__series,axiom,
    ! [A: complex,D: complex,N: nat] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
        @ ( groups2073611262835488442omplex
          @ ^ [I4: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I4 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_7213_double__arith__series,axiom,
    ! [A: rat,D: rat,N: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
        @ ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I4 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_7214_double__arith__series,axiom,
    ! [A: int,D: int,N: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
        @ ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_7215_double__arith__series,axiom,
    ! [A: nat,D: nat,N: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
        @ ( groups3542108847815614940at_nat
          @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_7216_double__arith__series,axiom,
    ! [A: real,D: real,N: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I4: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I4 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_7217_double__gauss__sum,axiom,
    ! [N: nat] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).

% double_gauss_sum
thf(fact_7218_double__gauss__sum,axiom,
    ! [N: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).

% double_gauss_sum
thf(fact_7219_double__gauss__sum,axiom,
    ! [N: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).

% double_gauss_sum
thf(fact_7220_double__gauss__sum,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).

% double_gauss_sum
thf(fact_7221_double__gauss__sum,axiom,
    ! [N: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).

% double_gauss_sum
thf(fact_7222_arctan__add,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
       => ( ( plus_plus_real @ ( arctan @ X3 ) @ ( arctan @ Y3 ) )
          = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X3 @ Y3 ) ) ) ) ) ) ) ).

% arctan_add
thf(fact_7223_arith__series,axiom,
    ! [A: int,D: int,N: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I4: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_7224_arith__series,axiom,
    ! [A: nat,D: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_7225_gauss__sum,axiom,
    ! [N: nat] :
      ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_7226_gauss__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_7227_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_7228_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_7229_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_7230_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_7231_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_7232_Bernoulli__inequality__even,axiom,
    ! [N: nat,X3: 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 ) @ X3 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X3 ) @ N ) ) ) ).

% Bernoulli_inequality_even
thf(fact_7233_sum__gp__offset,axiom,
    ! [X3: complex,M: nat,N: nat] :
      ( ( ( X3 = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
          = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) )
      & ( ( X3 != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
          = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X3 @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ ( suc @ N ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X3 ) ) ) ) ) ).

% sum_gp_offset
thf(fact_7234_sum__gp__offset,axiom,
    ! [X3: rat,M: nat,N: nat] :
      ( ( ( X3 = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
          = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) )
      & ( ( X3 != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
          = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X3 @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ ( suc @ N ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X3 ) ) ) ) ) ).

% sum_gp_offset
thf(fact_7235_sum__gp__offset,axiom,
    ! [X3: real,M: nat,N: nat] :
      ( ( ( X3 = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
          = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) )
      & ( ( X3 != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
          = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X3 @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( suc @ N ) ) ) ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ).

% sum_gp_offset
thf(fact_7236_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_7237_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_7238_of__nat__code__if,axiom,
    ( semiri8010041392384452111omplex
    = ( ^ [N2: nat] :
          ( if_complex @ ( N2 = zero_zero_nat ) @ zero_zero_complex
          @ ( produc1917071388513777916omplex
            @ ^ [M3: nat,Q4: nat] : ( if_complex @ ( Q4 = zero_zero_nat ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M3 ) ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M3 ) ) @ one_one_complex ) )
            @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_7239_of__nat__code__if,axiom,
    ( semiri681578069525770553at_rat
    = ( ^ [N2: nat] :
          ( if_rat @ ( N2 = zero_zero_nat ) @ zero_zero_rat
          @ ( produc6207742614233964070at_rat
            @ ^ [M3: nat,Q4: nat] : ( if_rat @ ( Q4 = zero_zero_nat ) @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M3 ) ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M3 ) ) @ one_one_rat ) )
            @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_7240_of__nat__code__if,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N2: nat] :
          ( if_int @ ( N2 = zero_zero_nat ) @ zero_zero_int
          @ ( produc6840382203811409530at_int
            @ ^ [M3: nat,Q4: nat] : ( if_int @ ( Q4 = zero_zero_nat ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) @ one_one_int ) )
            @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_7241_of__nat__code__if,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N2: nat] :
          ( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
          @ ( produc1703576794950452218t_real
            @ ^ [M3: nat,Q4: nat] : ( if_real @ ( Q4 = zero_zero_nat ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M3 ) ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M3 ) ) @ one_one_real ) )
            @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_7242_of__nat__code__if,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N2: nat] :
          ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
          @ ( produc6842872674320459806at_nat
            @ ^ [M3: nat,Q4: nat] : ( if_nat @ ( Q4 = zero_zero_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M3 ) ) @ one_one_nat ) )
            @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_7243_nat__approx__posE,axiom,
    ! [E: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ E )
     => ~ ! [N3: nat] :
            ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) ) @ E ) ) ).

% nat_approx_posE
thf(fact_7244_nat__approx__posE,axiom,
    ! [E: real] :
      ( ( ord_less_real @ zero_zero_real @ E )
     => ~ ! [N3: nat] :
            ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ E ) ) ).

% nat_approx_posE
thf(fact_7245_monoseq__arctan__series,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( topolo6980174941875973593q_real
        @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% monoseq_arctan_series
thf(fact_7246_lemma__termdiff3,axiom,
    ! [H2: real,Z: real,K5: real,N: nat] :
      ( ( H2 != zero_zero_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K5 )
       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H2 ) ) @ K5 )
         => ( 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 @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_7247_lemma__termdiff3,axiom,
    ! [H2: complex,Z: complex,K5: real,N: nat] :
      ( ( H2 != zero_zero_complex )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K5 )
       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H2 ) ) @ K5 )
         => ( 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 @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_7248_ex__less__of__nat__mult,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X3 )
     => ? [N3: nat] : ( ord_less_rat @ Y3 @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ X3 ) ) ) ).

% ex_less_of_nat_mult
thf(fact_7249_ex__less__of__nat__mult,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ? [N3: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X3 ) ) ) ).

% ex_less_of_nat_mult
thf(fact_7250_ln__series,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ( ln_ln_real @ X3 )
          = ( suminf_real
            @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X3 @ one_one_real ) @ ( suc @ N2 ) ) ) ) ) ) ) ).

% ln_series
thf(fact_7251_powser__zero,axiom,
    ! [F: nat > complex] :
      ( ( suminf_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_7252_powser__zero,axiom,
    ! [F: nat > real] :
      ( ( suminf_real
        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_7253_int__if,axiom,
    ! [P: $o,A: nat,B2: nat] :
      ( ( P
       => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B2 ) )
          = ( semiri1314217659103216013at_int @ A ) ) )
      & ( ~ P
       => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B2 ) )
          = ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% int_if
thf(fact_7254_nat__int__comparison_I1_J,axiom,
    ( ( ^ [Y6: nat,Z5: nat] : Y6 = Z5 )
    = ( ^ [A5: nat,B4: nat] :
          ( ( semiri1314217659103216013at_int @ A5 )
          = ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_int_comparison(1)
thf(fact_7255_complex__mod__minus__le__complex__mod,axiom,
    ! [X3: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X3 ) ) @ ( real_V1022390504157884413omplex @ X3 ) ) ).

% complex_mod_minus_le_complex_mod
thf(fact_7256_complex__mod__triangle__ineq2,axiom,
    ! [B2: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B2 @ A ) ) @ ( real_V1022390504157884413omplex @ B2 ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).

% complex_mod_triangle_ineq2
thf(fact_7257_monoseq__realpow,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ X3 @ one_one_real )
       => ( topolo6980174941875973593q_real @ ( power_power_real @ X3 ) ) ) ) ).

% monoseq_realpow
thf(fact_7258_real__arch__simple,axiom,
    ! [X3: real] :
    ? [N3: nat] : ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ N3 ) ) ).

% real_arch_simple
thf(fact_7259_real__arch__simple,axiom,
    ! [X3: rat] :
    ? [N3: nat] : ( ord_less_eq_rat @ X3 @ ( semiri681578069525770553at_rat @ N3 ) ) ).

% real_arch_simple
thf(fact_7260_reals__Archimedean2,axiom,
    ! [X3: rat] :
    ? [N3: nat] : ( ord_less_rat @ X3 @ ( semiri681578069525770553at_rat @ N3 ) ) ).

% reals_Archimedean2
thf(fact_7261_reals__Archimedean2,axiom,
    ! [X3: real] :
    ? [N3: nat] : ( ord_less_real @ X3 @ ( semiri5074537144036343181t_real @ N3 ) ) ).

% reals_Archimedean2
thf(fact_7262_arctan__series,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( ( arctan @ X3 )
        = ( suminf_real
          @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).

% arctan_series
thf(fact_7263_norm__divide__numeral,axiom,
    ! [A: real,W: num] :
      ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W ) ) )
      = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_divide_numeral
thf(fact_7264_norm__divide__numeral,axiom,
    ! [A: complex,W: num] :
      ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W ) ) )
      = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_divide_numeral
thf(fact_7265_norm__mult__numeral1,axiom,
    ! [W: num,A: real] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
      = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).

% norm_mult_numeral1
thf(fact_7266_norm__mult__numeral1,axiom,
    ! [W: num,A: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
      = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).

% norm_mult_numeral1
thf(fact_7267_norm__mult__numeral2,axiom,
    ! [A: real,W: num] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) )
      = ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_mult_numeral2
thf(fact_7268_norm__mult__numeral2,axiom,
    ! [A: complex,W: num] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) )
      = ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_mult_numeral2
thf(fact_7269_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_7270_norm__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_neg_numeral
thf(fact_7271_norm__le__zero__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X3 ) @ zero_zero_real )
      = ( X3 = zero_zero_real ) ) ).

% norm_le_zero_iff
thf(fact_7272_norm__le__zero__iff,axiom,
    ! [X3: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X3 ) @ zero_zero_real )
      = ( X3 = zero_zero_complex ) ) ).

% norm_le_zero_iff
thf(fact_7273_zero__less__norm__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X3 ) )
      = ( X3 != zero_zero_real ) ) ).

% zero_less_norm_iff
thf(fact_7274_zero__less__norm__iff,axiom,
    ! [X3: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X3 ) )
      = ( X3 != zero_zero_complex ) ) ).

% zero_less_norm_iff
thf(fact_7275_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_7276_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_7277_norm__minus__commute,axiom,
    ! [A: real,B2: real] :
      ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B2 ) )
      = ( real_V7735802525324610683m_real @ ( minus_minus_real @ B2 @ A ) ) ) ).

% norm_minus_commute
thf(fact_7278_norm__minus__commute,axiom,
    ! [A: complex,B2: complex] :
      ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B2 ) )
      = ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B2 @ A ) ) ) ).

% norm_minus_commute
thf(fact_7279_norm__not__less__zero,axiom,
    ! [X3: complex] :
      ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ zero_zero_real ) ).

% norm_not_less_zero
thf(fact_7280_norm__ge__zero,axiom,
    ! [X3: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X3 ) ) ).

% norm_ge_zero
thf(fact_7281_norm__mult,axiom,
    ! [X3: real,Y3: real] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ X3 @ Y3 ) )
      = ( times_times_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).

% norm_mult
thf(fact_7282_norm__mult,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ X3 @ Y3 ) )
      = ( times_times_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).

% norm_mult
thf(fact_7283_sum__norm__le,axiom,
    ! [S2: set_real,F: real > complex,G: real > real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ S2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ S2 ) ) @ ( groups8097168146408367636l_real @ G @ S2 ) ) ) ).

% sum_norm_le
thf(fact_7284_sum__norm__le,axiom,
    ! [S2: set_int,F: int > complex,G: int > real] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ S2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups3049146728041665814omplex @ F @ S2 ) ) @ ( groups8778361861064173332t_real @ G @ S2 ) ) ) ).

% sum_norm_le
thf(fact_7285_sum__norm__le,axiom,
    ! [S2: set_nat,F: nat > complex,G: nat > real] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ S2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S2 ) ) @ ( groups6591440286371151544t_real @ G @ S2 ) ) ) ).

% sum_norm_le
thf(fact_7286_sum__norm__le,axiom,
    ! [S2: set_complex,F: complex > complex,G: complex > real] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ S2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ S2 ) ) @ ( groups5808333547571424918x_real @ G @ S2 ) ) ) ).

% sum_norm_le
thf(fact_7287_sum__norm__le,axiom,
    ! [S2: set_nat,F: nat > real,G: nat > real] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ S2 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S2 ) ) @ ( groups6591440286371151544t_real @ G @ S2 ) ) ) ).

% sum_norm_le
thf(fact_7288_norm__divide,axiom,
    ! [A: real,B2: real] :
      ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B2 ) )
      = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) ).

% norm_divide
thf(fact_7289_norm__divide,axiom,
    ! [A: complex,B2: complex] :
      ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
      = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) ).

% norm_divide
thf(fact_7290_norm__power,axiom,
    ! [X3: real,N: nat] :
      ( ( real_V7735802525324610683m_real @ ( power_power_real @ X3 @ N ) )
      = ( power_power_real @ ( real_V7735802525324610683m_real @ X3 ) @ N ) ) ).

% norm_power
thf(fact_7291_norm__power,axiom,
    ! [X3: complex,N: nat] :
      ( ( real_V1022390504157884413omplex @ ( power_power_complex @ X3 @ N ) )
      = ( power_power_real @ ( real_V1022390504157884413omplex @ X3 ) @ N ) ) ).

% norm_power
thf(fact_7292_norm__sum,axiom,
    ! [F: nat > complex,A4: set_nat] :
      ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ A4 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( real_V1022390504157884413omplex @ ( F @ I4 ) )
        @ A4 ) ) ).

% norm_sum
thf(fact_7293_norm__sum,axiom,
    ! [F: complex > complex,A4: set_complex] :
      ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ A4 ) )
      @ ( groups5808333547571424918x_real
        @ ^ [I4: complex] : ( real_V1022390504157884413omplex @ ( F @ I4 ) )
        @ A4 ) ) ).

% norm_sum
thf(fact_7294_norm__sum,axiom,
    ! [F: nat > real,A4: set_nat] :
      ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ A4 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( real_V7735802525324610683m_real @ ( F @ I4 ) )
        @ A4 ) ) ).

% norm_sum
thf(fact_7295_norm__uminus__minus,axiom,
    ! [X3: real,Y3: real] :
      ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X3 ) @ Y3 ) )
      = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y3 ) ) ) ).

% norm_uminus_minus
thf(fact_7296_norm__uminus__minus,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X3 ) @ Y3 ) )
      = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) ) ).

% norm_uminus_minus
thf(fact_7297_nonzero__norm__divide,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B2 ) )
        = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) ) ).

% nonzero_norm_divide
thf(fact_7298_nonzero__norm__divide,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
        = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) ) ).

% nonzero_norm_divide
thf(fact_7299_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_7300_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_7301_norm__mult__less,axiom,
    ! [X3: real,R: real,Y3: real,S: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ R )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X3 @ Y3 ) ) @ ( times_times_real @ R @ S ) ) ) ) ).

% norm_mult_less
thf(fact_7302_norm__mult__less,axiom,
    ! [X3: complex,R: real,Y3: complex,S: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ R )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X3 @ Y3 ) ) @ ( times_times_real @ R @ S ) ) ) ) ).

% norm_mult_less
thf(fact_7303_norm__mult__ineq,axiom,
    ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X3 @ Y3 ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).

% norm_mult_ineq
thf(fact_7304_norm__mult__ineq,axiom,
    ! [X3: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X3 @ Y3 ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).

% norm_mult_ineq
thf(fact_7305_norm__triangle__lt,axiom,
    ! [X3: real,Y3: real,E: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
     => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ E ) ) ).

% norm_triangle_lt
thf(fact_7306_norm__triangle__lt,axiom,
    ! [X3: complex,Y3: complex,E: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
     => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ E ) ) ).

% norm_triangle_lt
thf(fact_7307_norm__add__less,axiom,
    ! [X3: real,R: real,Y3: real,S: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ R )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).

% norm_add_less
thf(fact_7308_norm__add__less,axiom,
    ! [X3: complex,R: real,Y3: complex,S: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ R )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).

% norm_add_less
thf(fact_7309_norm__power__ineq,axiom,
    ! [X3: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X3 @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X3 ) @ N ) ) ).

% norm_power_ineq
thf(fact_7310_norm__power__ineq,axiom,
    ! [X3: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X3 @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X3 ) @ N ) ) ).

% norm_power_ineq
thf(fact_7311_norm__triangle__mono,axiom,
    ! [A: real,R: real,B2: real,S: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B2 ) @ S )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B2 ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).

% norm_triangle_mono
thf(fact_7312_norm__triangle__mono,axiom,
    ! [A: complex,R: real,B2: complex,S: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B2 ) @ S )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B2 ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).

% norm_triangle_mono
thf(fact_7313_norm__triangle__ineq,axiom,
    ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).

% norm_triangle_ineq
thf(fact_7314_norm__triangle__ineq,axiom,
    ! [X3: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).

% norm_triangle_ineq
thf(fact_7315_norm__triangle__le,axiom,
    ! [X3: real,Y3: real,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ E ) ) ).

% norm_triangle_le
thf(fact_7316_norm__triangle__le,axiom,
    ! [X3: complex,Y3: complex,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ E ) ) ).

% norm_triangle_le
thf(fact_7317_norm__add__leD,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B2 ) ) @ C )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B2 ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).

% norm_add_leD
thf(fact_7318_norm__add__leD,axiom,
    ! [A: complex,B2: complex,C: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B2 ) ) @ C )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B2 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).

% norm_add_leD
thf(fact_7319_norm__diff__triangle__less,axiom,
    ! [X3: real,Y3: real,E1: real,Z: real,E22: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Y3 ) ) @ E1 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z ) ) @ E22 )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_7320_norm__diff__triangle__less,axiom,
    ! [X3: complex,Y3: complex,E1: real,Z: complex,E22: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Y3 ) ) @ E1 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z ) ) @ E22 )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_7321_norm__triangle__sub,axiom,
    ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y3 ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Y3 ) ) ) ) ).

% norm_triangle_sub
thf(fact_7322_norm__triangle__sub,axiom,
    ! [X3: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y3 ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Y3 ) ) ) ) ).

% norm_triangle_sub
thf(fact_7323_norm__triangle__ineq4,axiom,
    ! [A: real,B2: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B2 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) ).

% norm_triangle_ineq4
thf(fact_7324_norm__triangle__ineq4,axiom,
    ! [A: complex,B2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B2 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) ).

% norm_triangle_ineq4
thf(fact_7325_norm__diff__triangle__le,axiom,
    ! [X3: real,Y3: real,E1: real,Z: real,E22: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Y3 ) ) @ E1 )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z ) ) @ E22 )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_le
thf(fact_7326_norm__diff__triangle__le,axiom,
    ! [X3: complex,Y3: complex,E1: real,Z: complex,E22: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Y3 ) ) @ E1 )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z ) ) @ E22 )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_le
thf(fact_7327_norm__triangle__le__diff,axiom,
    ! [X3: real,Y3: real,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Y3 ) ) @ E ) ) ).

% norm_triangle_le_diff
thf(fact_7328_norm__triangle__le__diff,axiom,
    ! [X3: complex,Y3: complex,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Y3 ) ) @ E ) ) ).

% norm_triangle_le_diff
thf(fact_7329_norm__diff__ineq,axiom,
    ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B2 ) ) ) ).

% norm_diff_ineq
thf(fact_7330_norm__diff__ineq,axiom,
    ! [A: complex,B2: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B2 ) ) ) ).

% norm_diff_ineq
thf(fact_7331_norm__triangle__ineq2,axiom,
    ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B2 ) ) ) ).

% norm_triangle_ineq2
thf(fact_7332_norm__triangle__ineq2,axiom,
    ! [A: complex,B2: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B2 ) ) ) ).

% norm_triangle_ineq2
thf(fact_7333_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_7334_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_7335_norm__diff__triangle__ineq,axiom,
    ! [A: real,B2: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B2 @ D ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_7336_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B2: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B2 ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B2 @ D ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_7337_norm__triangle__ineq3,axiom,
    ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B2 ) ) ) ).

% norm_triangle_ineq3
thf(fact_7338_norm__triangle__ineq3,axiom,
    ! [A: complex,B2: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B2 ) ) ) ).

% norm_triangle_ineq3
thf(fact_7339_square__norm__one,axiom,
    ! [X3: real] :
      ( ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
     => ( ( real_V7735802525324610683m_real @ X3 )
        = one_one_real ) ) ).

% square_norm_one
thf(fact_7340_square__norm__one,axiom,
    ! [X3: complex] :
      ( ( ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_complex )
     => ( ( real_V1022390504157884413omplex @ X3 )
        = one_one_real ) ) ).

% square_norm_one
thf(fact_7341_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_7342_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_7343_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_7344_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_7345_pi__series,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( suminf_real
      @ ^ [K2: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% pi_series
thf(fact_7346_lemma__termdiff2,axiom,
    ! [H2: complex,Z: complex,N: nat] :
      ( ( H2 != zero_zero_complex )
     => ( ( 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_complex @ H2
          @ ( groups2073611262835488442omplex
            @ ^ [P5: nat] :
                ( groups2073611262835488442omplex
                @ ^ [Q4: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ Q4 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_7347_lemma__termdiff2,axiom,
    ! [H2: rat,Z: rat,N: nat] :
      ( ( H2 != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ N ) @ ( power_power_rat @ Z @ N ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_rat @ H2
          @ ( groups2906978787729119204at_rat
            @ ^ [P5: nat] :
                ( groups2906978787729119204at_rat
                @ ^ [Q4: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ Q4 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_7348_lemma__termdiff2,axiom,
    ! [H2: real,Z: real,N: nat] :
      ( ( H2 != zero_zero_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 @ H2
          @ ( groups6591440286371151544t_real
            @ ^ [P5: nat] :
                ( groups6591440286371151544t_real
                @ ^ [Q4: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ Q4 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_7349_summable__arctan__series,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( summable_real
        @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).

% summable_arctan_series
thf(fact_7350_pred__subset__eq2,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ( ord_le6741204236512500942_int_o
        @ ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
        @ ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S2 ) )
      = ( ord_le2843351958646193337nt_int @ R2 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_7351_pred__subset__eq2,axiom,
    ! [R2: set_Pr8056137968301705908nteger,S2: set_Pr8056137968301705908nteger] :
      ( ( ord_le3636971675376928563eger_o
        @ ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ R2 )
        @ ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ S2 ) )
      = ( ord_le3216752416896350996nteger @ R2 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_7352_pred__subset__eq2,axiom,
    ! [R2: set_Pr1281608226676607948nteger,S2: set_Pr1281608226676607948nteger] :
      ( ( ord_le4340812435750786203eger_o
        @ ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ R2 )
        @ ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ S2 ) )
      = ( ord_le653643898420964396nteger @ R2 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_7353_pred__subset__eq2,axiom,
    ! [R2: set_Pr9222295170931077689nt_int,S2: set_Pr9222295170931077689nt_int] :
      ( ( ord_le5643404153117327598_int_o
        @ ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ R2 )
        @ ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ S2 ) )
      = ( ord_le8725513860283290265nt_int @ R2 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_7354_pred__subset__eq2,axiom,
    ! [R2: set_Pr1872883991513573699nt_int,S2: set_Pr1872883991513573699nt_int] :
      ( ( ord_le2124322318746777828_int_o
        @ ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ R2 )
        @ ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ S2 ) )
      = ( ord_le135402666524580259nt_int @ R2 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_7355_infinite__int__iff__unbounded,axiom,
    ! [S2: set_int] :
      ( ( ~ ( finite_finite_int @ S2 ) )
      = ( ! [M3: int] :
          ? [N2: int] :
            ( ( ord_less_int @ M3 @ ( abs_abs_int @ N2 ) )
            & ( member_int @ N2 @ S2 ) ) ) ) ).

% infinite_int_iff_unbounded
thf(fact_7356_lessThan__iff,axiom,
    ! [I: rat,K: rat] :
      ( ( member_rat @ I @ ( set_ord_lessThan_rat @ K ) )
      = ( ord_less_rat @ I @ K ) ) ).

% lessThan_iff
thf(fact_7357_lessThan__iff,axiom,
    ! [I: num,K: num] :
      ( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
      = ( ord_less_num @ I @ K ) ) ).

% lessThan_iff
thf(fact_7358_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
      = ( ord_less_nat @ I @ K ) ) ).

% lessThan_iff
thf(fact_7359_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
      = ( ord_less_int @ I @ K ) ) ).

% lessThan_iff
thf(fact_7360_lessThan__iff,axiom,
    ! [I: real,K: real] :
      ( ( member_real @ I @ ( set_or5984915006950818249n_real @ K ) )
      = ( ord_less_real @ I @ K ) ) ).

% lessThan_iff
thf(fact_7361_summable__iff__shift,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real
        @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( summable_real @ F ) ) ).

% summable_iff_shift
thf(fact_7362_lessThan__subset__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X3 ) @ ( set_ord_lessThan_rat @ Y3 ) )
      = ( ord_less_eq_rat @ X3 @ Y3 ) ) ).

% lessThan_subset_iff
thf(fact_7363_lessThan__subset__iff,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X3 ) @ ( set_ord_lessThan_num @ Y3 ) )
      = ( ord_less_eq_num @ X3 @ Y3 ) ) ).

% lessThan_subset_iff
thf(fact_7364_lessThan__subset__iff,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X3 ) @ ( set_ord_lessThan_nat @ Y3 ) )
      = ( ord_less_eq_nat @ X3 @ Y3 ) ) ).

% lessThan_subset_iff
thf(fact_7365_lessThan__subset__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X3 ) @ ( set_ord_lessThan_int @ Y3 ) )
      = ( ord_less_eq_int @ X3 @ Y3 ) ) ).

% lessThan_subset_iff
thf(fact_7366_lessThan__subset__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X3 ) @ ( set_or5984915006950818249n_real @ Y3 ) )
      = ( ord_less_eq_real @ X3 @ Y3 ) ) ).

% lessThan_subset_iff
thf(fact_7367_summable__cmult__iff,axiom,
    ! [C: complex,F: nat > complex] :
      ( ( summable_complex
        @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) ) )
      = ( ( C = zero_zero_complex )
        | ( summable_complex @ F ) ) ) ).

% summable_cmult_iff
thf(fact_7368_summable__cmult__iff,axiom,
    ! [C: real,F: nat > real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
      = ( ( C = zero_zero_real )
        | ( summable_real @ F ) ) ) ).

% summable_cmult_iff
thf(fact_7369_summable__divide__iff,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex
        @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) )
      = ( ( C = zero_zero_complex )
        | ( summable_complex @ F ) ) ) ).

% summable_divide_iff
thf(fact_7370_summable__divide__iff,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) )
      = ( ( C = zero_zero_real )
        | ( summable_real @ F ) ) ) ).

% summable_divide_iff
thf(fact_7371_sum_OlessThan__Suc,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).

% sum.lessThan_Suc
thf(fact_7372_sum_OlessThan__Suc,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).

% sum.lessThan_Suc
thf(fact_7373_sum_OlessThan__Suc,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).

% sum.lessThan_Suc
thf(fact_7374_sum_OlessThan__Suc,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).

% sum.lessThan_Suc
thf(fact_7375_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_7376_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_7377_summable__comparison__test_H,axiom,
    ! [G: nat > real,N4: nat,F: nat > real] :
      ( ( summable_real @ G )
     => ( ! [N3: nat] :
            ( ( ord_less_eq_nat @ N4 @ N3 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
       => ( summable_real @ F ) ) ) ).

% summable_comparison_test'
thf(fact_7378_summable__comparison__test_H,axiom,
    ! [G: nat > real,N4: nat,F: nat > complex] :
      ( ( summable_real @ G )
     => ( ! [N3: nat] :
            ( ( ord_less_eq_nat @ N4 @ N3 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
       => ( summable_complex @ F ) ) ) ).

% summable_comparison_test'
thf(fact_7379_summable__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N7: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N7 @ N3 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real @ F ) ) ) ).

% summable_comparison_test
thf(fact_7380_summable__comparison__test,axiom,
    ! [F: nat > complex,G: nat > real] :
      ( ? [N7: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N7 @ N3 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
     => ( ( summable_real @ G )
       => ( summable_complex @ F ) ) ) ).

% summable_comparison_test
thf(fact_7381_summable__ignore__initial__segment,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) ) ) ).

% summable_ignore_initial_segment
thf(fact_7382_summable__add,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N2: nat] : ( plus_plus_real @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).

% summable_add
thf(fact_7383_summable__add,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ( summable_nat @ F )
     => ( ( summable_nat @ G )
       => ( summable_nat
          @ ^ [N2: nat] : ( plus_plus_nat @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).

% summable_add
thf(fact_7384_summable__add,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ( summable_int @ F )
     => ( ( summable_int @ G )
       => ( summable_int
          @ ^ [N2: nat] : ( plus_plus_int @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).

% summable_add
thf(fact_7385_summable__diff,axiom,
    ! [F: nat > complex,G: nat > complex] :
      ( ( summable_complex @ F )
     => ( ( summable_complex @ G )
       => ( summable_complex
          @ ^ [N2: nat] : ( minus_minus_complex @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).

% summable_diff
thf(fact_7386_summable__diff,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).

% summable_diff
thf(fact_7387_summable__divide,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( summable_complex
        @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) ) ) ).

% summable_divide
thf(fact_7388_summable__divide,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) ) ) ).

% summable_divide
thf(fact_7389_summable__mult2,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C ) ) ) ).

% summable_mult2
thf(fact_7390_summable__mult,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) ) ) ).

% summable_mult
thf(fact_7391_suminf__le__const,axiom,
    ! [F: nat > int,X3: int] :
      ( ( summable_int @ F )
     => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
       => ( ord_less_eq_int @ ( suminf_int @ F ) @ X3 ) ) ) ).

% suminf_le_const
thf(fact_7392_suminf__le__const,axiom,
    ! [F: nat > nat,X3: nat] :
      ( ( summable_nat @ F )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
       => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X3 ) ) ) ).

% suminf_le_const
thf(fact_7393_suminf__le__const,axiom,
    ! [F: nat > real,X3: real] :
      ( ( summable_real @ F )
     => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
       => ( ord_less_eq_real @ ( suminf_real @ F ) @ X3 ) ) ) ).

% suminf_le_const
thf(fact_7394_summableI__nonneg__bounded,axiom,
    ! [F: nat > int,X3: int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
       => ( summable_int @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_7395_summableI__nonneg__bounded,axiom,
    ! [F: nat > nat,X3: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
       => ( summable_nat @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_7396_summableI__nonneg__bounded,axiom,
    ! [F: nat > real,X3: real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
       => ( summable_real @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_7397_suminf__le,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
     => ( ( summable_real @ F )
       => ( ( summable_real @ G )
         => ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).

% suminf_le
thf(fact_7398_suminf__le,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ N3 ) )
     => ( ( summable_nat @ F )
       => ( ( summable_nat @ G )
         => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) ) ) ) ) ).

% suminf_le
thf(fact_7399_suminf__le,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G @ N3 ) )
     => ( ( summable_int @ F )
       => ( ( summable_int @ G )
         => ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) ) ) ) ) ).

% suminf_le
thf(fact_7400_summable__mult__D,axiom,
    ! [C: complex,F: nat > complex] :
      ( ( summable_complex
        @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) ) )
     => ( ( C != zero_zero_complex )
       => ( summable_complex @ F ) ) ) ).

% summable_mult_D
thf(fact_7401_summable__mult__D,axiom,
    ! [C: real,F: nat > real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
     => ( ( C != zero_zero_real )
       => ( summable_real @ F ) ) ) ).

% summable_mult_D
thf(fact_7402_summable__zero__power,axiom,
    summable_real @ ( power_power_real @ zero_zero_real ) ).

% summable_zero_power
thf(fact_7403_summable__zero__power,axiom,
    summable_int @ ( power_power_int @ zero_zero_int ) ).

% summable_zero_power
thf(fact_7404_summable__zero__power,axiom,
    summable_complex @ ( power_power_complex @ zero_zero_complex ) ).

% summable_zero_power
thf(fact_7405_lessThan__def,axiom,
    ( set_ord_lessThan_rat
    = ( ^ [U2: rat] :
          ( collect_rat
          @ ^ [X: rat] : ( ord_less_rat @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7406_lessThan__def,axiom,
    ( set_ord_lessThan_num
    = ( ^ [U2: num] :
          ( collect_num
          @ ^ [X: num] : ( ord_less_num @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7407_lessThan__def,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X: nat] : ( ord_less_nat @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7408_lessThan__def,axiom,
    ( set_ord_lessThan_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X: int] : ( ord_less_int @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7409_lessThan__def,axiom,
    ( set_or5984915006950818249n_real
    = ( ^ [U2: real] :
          ( collect_real
          @ ^ [X: real] : ( ord_less_real @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7410_suminf__mult,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
        = ( times_times_real @ C @ ( suminf_real @ F ) ) ) ) ).

% suminf_mult
thf(fact_7411_suminf__mult2,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( ( times_times_real @ ( suminf_real @ F ) @ C )
        = ( suminf_real
          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C ) ) ) ) ).

% suminf_mult2
thf(fact_7412_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
            @ ^ [N2: nat] : ( plus_plus_real @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).

% suminf_add
thf(fact_7413_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
            @ ^ [N2: nat] : ( plus_plus_nat @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).

% suminf_add
thf(fact_7414_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
            @ ^ [N2: nat] : ( plus_plus_int @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).

% suminf_add
thf(fact_7415_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
            @ ^ [N2: nat] : ( minus_minus_complex @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).

% suminf_diff
thf(fact_7416_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
            @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).

% suminf_diff
thf(fact_7417_suminf__divide,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( ( suminf_complex
          @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) )
        = ( divide1717551699836669952omplex @ ( suminf_complex @ F ) @ C ) ) ) ).

% suminf_divide
thf(fact_7418_suminf__divide,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) )
        = ( divide_divide_real @ ( suminf_real @ F ) @ C ) ) ) ).

% suminf_divide
thf(fact_7419_suminf__split__initial__segment,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ( suminf_real @ F )
        = ( plus_plus_real
          @ ( suminf_real
            @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) )
          @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).

% suminf_split_initial_segment
thf(fact_7420_suminf__minus__initial__segment,axiom,
    ! [F: nat > complex,K: nat] :
      ( ( summable_complex @ F )
     => ( ( suminf_complex
          @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) )
        = ( minus_minus_complex @ ( suminf_complex @ F ) @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).

% suminf_minus_initial_segment
thf(fact_7421_suminf__minus__initial__segment,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) )
        = ( minus_minus_real @ ( suminf_real @ F ) @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).

% suminf_minus_initial_segment
thf(fact_7422_finite__nat__bounded,axiom,
    ! [S2: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ? [K3: nat] : ( ord_less_eq_set_nat @ S2 @ ( set_ord_lessThan_nat @ K3 ) ) ) ).

% finite_nat_bounded
thf(fact_7423_finite__nat__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [S4: set_nat] :
        ? [K2: nat] : ( ord_less_eq_set_nat @ S4 @ ( set_ord_lessThan_nat @ K2 ) ) ) ) ).

% finite_nat_iff_bounded
thf(fact_7424_sum__less__suminf,axiom,
    ! [F: nat > int,N: nat] :
      ( ( summable_int @ F )
     => ( ! [M4: nat] :
            ( ( ord_less_eq_nat @ N @ M4 )
           => ( ord_less_int @ zero_zero_int @ ( F @ M4 ) ) )
       => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_7425_sum__less__suminf,axiom,
    ! [F: nat > nat,N: nat] :
      ( ( summable_nat @ F )
     => ( ! [M4: nat] :
            ( ( ord_less_eq_nat @ N @ M4 )
           => ( ord_less_nat @ zero_zero_nat @ ( F @ M4 ) ) )
       => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_7426_sum__less__suminf,axiom,
    ! [F: nat > real,N: nat] :
      ( ( summable_real @ F )
     => ( ! [M4: nat] :
            ( ( ord_less_eq_nat @ N @ M4 )
           => ( ord_less_real @ zero_zero_real @ ( F @ M4 ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_7427_powser__insidea,axiom,
    ! [F: nat > real,X3: real,Z: real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X3 @ N2 ) ) )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X3 ) )
       => ( summable_real
          @ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ) ) ).

% powser_insidea
thf(fact_7428_powser__insidea,axiom,
    ! [F: nat > complex,X3: complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ X3 @ N2 ) ) )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X3 ) )
       => ( summable_real
          @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ) ) ).

% powser_insidea
thf(fact_7429_suminf__eq__zero__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
       => ( ( ( suminf_real @ F )
            = zero_zero_real )
          = ( ! [N2: nat] :
                ( ( F @ N2 )
                = zero_zero_real ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7430_suminf__eq__zero__iff,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
       => ( ( ( suminf_nat @ F )
            = zero_zero_nat )
          = ( ! [N2: nat] :
                ( ( F @ N2 )
                = zero_zero_nat ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7431_suminf__eq__zero__iff,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
       => ( ( ( suminf_int @ F )
            = zero_zero_int )
          = ( ! [N2: nat] :
                ( ( F @ N2 )
                = zero_zero_int ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7432_suminf__nonneg,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7433_suminf__nonneg,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7434_suminf__nonneg,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
       => ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7435_suminf__pos,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N3: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N3 ) )
       => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).

% suminf_pos
thf(fact_7436_suminf__pos,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N3: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N3 ) )
       => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).

% suminf_pos
thf(fact_7437_suminf__pos,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N3: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N3 ) )
       => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).

% suminf_pos
thf(fact_7438_lessThan__strict__subset__iff,axiom,
    ! [M: rat,N: rat] :
      ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N ) )
      = ( ord_less_rat @ M @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_7439_lessThan__strict__subset__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_7440_lessThan__strict__subset__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_7441_lessThan__strict__subset__iff,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
      = ( ord_less_int @ M @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_7442_lessThan__strict__subset__iff,axiom,
    ! [M: real,N: real] :
      ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N ) )
      = ( ord_less_real @ M @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_7443_summable__0__powser,axiom,
    ! [F: nat > complex] :
      ( summable_complex
      @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) ) ).

% summable_0_powser
thf(fact_7444_summable__0__powser,axiom,
    ! [F: nat > real] :
      ( summable_real
      @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) ) ).

% summable_0_powser
thf(fact_7445_summable__zero__power_H,axiom,
    ! [F: nat > complex] :
      ( summable_complex
      @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) ) ).

% summable_zero_power'
thf(fact_7446_summable__zero__power_H,axiom,
    ! [F: nat > real] :
      ( summable_real
      @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) ) ).

% summable_zero_power'
thf(fact_7447_summable__zero__power_H,axiom,
    ! [F: nat > int] :
      ( summable_int
      @ ^ [N2: nat] : ( times_times_int @ ( F @ N2 ) @ ( power_power_int @ zero_zero_int @ N2 ) ) ) ).

% summable_zero_power'
thf(fact_7448_powser__split__head_I3_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
     => ( summable_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ).

% powser_split_head(3)
thf(fact_7449_powser__split__head_I3_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
     => ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) ) ) ).

% powser_split_head(3)
thf(fact_7450_summable__powser__split__head,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) )
      = ( summable_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ).

% summable_powser_split_head
thf(fact_7451_summable__powser__split__head,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) )
      = ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ).

% summable_powser_split_head
thf(fact_7452_summable__powser__ignore__initial__segment,axiom,
    ! [F: nat > complex,M: nat,Z: complex] :
      ( ( summable_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N2 @ M ) ) @ ( power_power_complex @ Z @ N2 ) ) )
      = ( summable_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_7453_summable__powser__ignore__initial__segment,axiom,
    ! [F: nat > real,M: nat,Z: real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N2 @ M ) ) @ ( power_power_real @ Z @ N2 ) ) )
      = ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_7454_pi__not__less__zero,axiom,
    ~ ( ord_less_real @ pi @ zero_zero_real ) ).

% pi_not_less_zero
thf(fact_7455_pi__gt__zero,axiom,
    ord_less_real @ zero_zero_real @ pi ).

% pi_gt_zero
thf(fact_7456_pi__ge__zero,axiom,
    ord_less_eq_real @ zero_zero_real @ pi ).

% pi_ge_zero
thf(fact_7457_sum__less__suminf2,axiom,
    ! [F: nat > int,N: nat,I: nat] :
      ( ( summable_int @ F )
     => ( ! [M4: nat] :
            ( ( ord_less_eq_nat @ N @ M4 )
           => ( ord_less_eq_int @ zero_zero_int @ ( F @ M4 ) ) )
       => ( ( ord_less_eq_nat @ N @ I )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
           => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).

% sum_less_suminf2
thf(fact_7458_sum__less__suminf2,axiom,
    ! [F: nat > nat,N: nat,I: nat] :
      ( ( summable_nat @ F )
     => ( ! [M4: nat] :
            ( ( ord_less_eq_nat @ N @ M4 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M4 ) ) )
       => ( ( ord_less_eq_nat @ N @ I )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
           => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).

% sum_less_suminf2
thf(fact_7459_sum__less__suminf2,axiom,
    ! [F: nat > real,N: nat,I: nat] :
      ( ( summable_real @ F )
     => ( ! [M4: nat] :
            ( ( ord_less_eq_nat @ N @ M4 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ M4 ) ) )
       => ( ( ord_less_eq_nat @ N @ I )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
           => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).

% sum_less_suminf2
thf(fact_7460_summable__norm__comparison__test,axiom,
    ! [F: nat > complex,G: nat > real] :
      ( ? [N7: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N7 @ N3 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) ) ).

% summable_norm_comparison_test
thf(fact_7461_summable__rabs__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N7: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N7 @ N3 )
         => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).

% summable_rabs_comparison_test
thf(fact_7462_summable__rabs,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
        @ ( suminf_real
          @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).

% summable_rabs
thf(fact_7463_suminf__pos2,axiom,
    ! [F: nat > real,I: nat] :
      ( ( summable_real @ F )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_7464_suminf__pos2,axiom,
    ! [F: nat > nat,I: nat] :
      ( ( summable_nat @ F )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_7465_suminf__pos2,axiom,
    ! [F: nat > int,I: nat] :
      ( ( summable_int @ F )
     => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
       => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
         => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_7466_suminf__pos__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
       => ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
          = ( ? [I4: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7467_suminf__pos__iff,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
          = ( ? [I4: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7468_suminf__pos__iff,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
       => ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
          = ( ? [I4: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I4 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7469_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_7470_summable__geometric,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( summable_complex @ ( power_power_complex @ C ) ) ) ).

% summable_geometric
thf(fact_7471_complete__algebra__summable__geometric,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ one_one_real )
     => ( summable_real @ ( power_power_real @ X3 ) ) ) ).

% complete_algebra_summable_geometric
thf(fact_7472_complete__algebra__summable__geometric,axiom,
    ! [X3: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ one_one_real )
     => ( summable_complex @ ( power_power_complex @ X3 ) ) ) ).

% complete_algebra_summable_geometric
thf(fact_7473_suminf__split__head,axiom,
    ! [F: nat > complex] :
      ( ( summable_complex @ F )
     => ( ( suminf_complex
          @ ^ [N2: nat] : ( F @ ( suc @ N2 ) ) )
        = ( minus_minus_complex @ ( suminf_complex @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).

% suminf_split_head
thf(fact_7474_suminf__split__head,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N2: nat] : ( F @ ( suc @ N2 ) ) )
        = ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).

% suminf_split_head
thf(fact_7475_summable__norm,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( F @ N2 ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( suminf_real @ F ) )
        @ ( suminf_real
          @ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( F @ N2 ) ) ) ) ) ).

% summable_norm
thf(fact_7476_summable__norm,axiom,
    ! [F: nat > complex] :
      ( ( summable_real
        @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( suminf_complex @ F ) )
        @ ( suminf_real
          @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) ) ).

% summable_norm
thf(fact_7477_sum_Onat__diff__reindex,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
        @ ( set_ord_lessThan_nat @ N ) )
      = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.nat_diff_reindex
thf(fact_7478_sum_Onat__diff__reindex,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
        @ ( set_ord_lessThan_nat @ N ) )
      = ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.nat_diff_reindex
thf(fact_7479_sum__le__suminf,axiom,
    ! [F: nat > int,I5: set_nat] :
      ( ( summable_int @ F )
     => ( ( finite_finite_nat @ I5 )
       => ( ! [N3: nat] :
              ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) ) )
         => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I5 ) @ ( suminf_int @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_7480_sum__le__suminf,axiom,
    ! [F: nat > nat,I5: set_nat] :
      ( ( summable_nat @ F )
     => ( ( finite_finite_nat @ I5 )
       => ( ! [N3: nat] :
              ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) ) )
         => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I5 ) @ ( suminf_nat @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_7481_sum__le__suminf,axiom,
    ! [F: nat > real,I5: set_nat] :
      ( ( summable_real @ F )
     => ( ( finite_finite_nat @ I5 )
       => ( ! [N3: nat] :
              ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) ) )
         => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I5 ) @ ( suminf_real @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_7482_sum__diff__distrib,axiom,
    ! [Q: int > nat,P: int > nat,N: int] :
      ( ! [X5: int] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
     => ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P @ ( set_ord_lessThan_int @ N ) ) @ ( groups4541462559716669496nt_nat @ Q @ ( set_ord_lessThan_int @ N ) ) )
        = ( groups4541462559716669496nt_nat
          @ ^ [X: int] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
          @ ( set_ord_lessThan_int @ N ) ) ) ) ).

% sum_diff_distrib
thf(fact_7483_sum__diff__distrib,axiom,
    ! [Q: real > nat,P: real > nat,N: real] :
      ( ! [X5: real] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
     => ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P @ ( set_or5984915006950818249n_real @ N ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N ) ) )
        = ( groups1935376822645274424al_nat
          @ ^ [X: real] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
          @ ( set_or5984915006950818249n_real @ N ) ) ) ) ).

% sum_diff_distrib
thf(fact_7484_sum__diff__distrib,axiom,
    ! [Q: nat > nat,P: nat > nat,N: nat] :
      ( ! [X5: nat] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
     => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N ) ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X: nat] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum_diff_distrib
thf(fact_7485_powser__inside,axiom,
    ! [F: nat > real,X3: real,Z: real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X3 @ N2 ) ) )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X3 ) )
       => ( summable_real
          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ) ).

% powser_inside
thf(fact_7486_powser__inside,axiom,
    ! [F: nat > complex,X3: complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ X3 @ N2 ) ) )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X3 ) )
       => ( summable_complex
          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ) ).

% powser_inside
thf(fact_7487_pi__less__4,axiom,
    ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).

% pi_less_4
thf(fact_7488_pi__ge__two,axiom,
    ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).

% pi_ge_two
thf(fact_7489_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_7490_sum__pos__lt__pair,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ! [D3: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) @ one_one_nat ) ) ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_pos_lt_pair
thf(fact_7491_powser__split__head_I1_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
     => ( ( suminf_complex
          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
        = ( plus_plus_complex @ ( F @ zero_zero_nat )
          @ ( times_times_complex
            @ ( suminf_complex
              @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) )
            @ Z ) ) ) ) ).

% powser_split_head(1)
thf(fact_7492_powser__split__head_I1_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
     => ( ( suminf_real
          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
        = ( plus_plus_real @ ( F @ zero_zero_nat )
          @ ( times_times_real
            @ ( suminf_real
              @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) )
            @ Z ) ) ) ) ).

% powser_split_head(1)
thf(fact_7493_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_7494_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_7495_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_7496_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_7497_powser__split__head_I2_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
     => ( ( times_times_complex
          @ ( suminf_complex
            @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) )
          @ Z )
        = ( minus_minus_complex
          @ ( suminf_complex
            @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_7498_powser__split__head_I2_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
     => ( ( times_times_real
          @ ( suminf_real
            @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) )
          @ Z )
        = ( minus_minus_real
          @ ( suminf_real
            @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_7499_sumr__diff__mult__const2,axiom,
    ! [F: nat > complex,N: nat,R: complex] :
      ( ( minus_minus_complex @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ R ) )
      = ( groups2073611262835488442omplex
        @ ^ [I4: nat] : ( minus_minus_complex @ ( F @ I4 ) @ R )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sumr_diff_mult_const2
thf(fact_7500_sumr__diff__mult__const2,axiom,
    ! [F: nat > rat,N: nat,R: rat] :
      ( ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ R ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ I4 ) @ R )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sumr_diff_mult_const2
thf(fact_7501_sumr__diff__mult__const2,axiom,
    ! [F: nat > int,N: nat,R: int] :
      ( ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ R ) )
      = ( groups3539618377306564664at_int
        @ ^ [I4: nat] : ( minus_minus_int @ ( F @ I4 ) @ R )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sumr_diff_mult_const2
thf(fact_7502_sumr__diff__mult__const2,axiom,
    ! [F: nat > real,N: nat,R: real] :
      ( ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ R ) )
      = ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( minus_minus_real @ ( F @ I4 ) @ R )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sumr_diff_mult_const2
thf(fact_7503_sum__lessThan__telescope,axiom,
    ! [F: nat > complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [N2: nat] : ( minus_minus_complex @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_complex @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_7504_sum__lessThan__telescope,axiom,
    ! [F: nat > rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N2: nat] : ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_7505_sum__lessThan__telescope,axiom,
    ! [F: nat > int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N2: nat] : ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_7506_sum__lessThan__telescope,axiom,
    ! [F: nat > real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N2: nat] : ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_7507_sum__lessThan__telescope_H,axiom,
    ! [F: nat > complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [N2: nat] : ( minus_minus_complex @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_complex @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_7508_sum__lessThan__telescope_H,axiom,
    ! [F: nat > rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N2: nat] : ( minus_minus_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_7509_sum__lessThan__telescope_H,axiom,
    ! [F: nat > int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N2: nat] : ( minus_minus_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_7510_sum__lessThan__telescope_H,axiom,
    ! [F: nat > real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_7511_summable__partial__sum__bound,axiom,
    ! [F: nat > complex,E: real] :
      ( ( summable_complex @ F )
     => ( ( ord_less_real @ zero_zero_real @ E )
       => ~ ! [N8: nat] :
              ~ ! [M2: nat] :
                  ( ( ord_less_eq_nat @ N8 @ M2 )
                 => ! [N9: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M2 @ N9 ) ) ) @ E ) ) ) ) ).

% summable_partial_sum_bound
thf(fact_7512_summable__partial__sum__bound,axiom,
    ! [F: nat > real,E: real] :
      ( ( summable_real @ F )
     => ( ( ord_less_real @ zero_zero_real @ E )
       => ~ ! [N8: nat] :
              ~ ! [M2: nat] :
                  ( ( ord_less_eq_nat @ N8 @ M2 )
                 => ! [N9: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M2 @ N9 ) ) ) @ E ) ) ) ) ).

% summable_partial_sum_bound
thf(fact_7513_suminf__exist__split,axiom,
    ! [R: real,F: nat > real] :
      ( ( ord_less_real @ zero_zero_real @ R )
     => ( ( summable_real @ F )
       => ? [N8: nat] :
          ! [N9: nat] :
            ( ( ord_less_eq_nat @ N8 @ N9 )
           => ( ord_less_real
              @ ( real_V7735802525324610683m_real
                @ ( suminf_real
                  @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N9 ) ) ) )
              @ R ) ) ) ) ).

% suminf_exist_split
thf(fact_7514_suminf__exist__split,axiom,
    ! [R: real,F: nat > complex] :
      ( ( ord_less_real @ zero_zero_real @ R )
     => ( ( summable_complex @ F )
       => ? [N8: nat] :
          ! [N9: nat] :
            ( ( ord_less_eq_nat @ N8 @ N9 )
           => ( ord_less_real
              @ ( real_V1022390504157884413omplex
                @ ( suminf_complex
                  @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N9 ) ) ) )
              @ R ) ) ) ) ).

% suminf_exist_split
thf(fact_7515_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_7516_Abel__lemma,axiom,
    ! [R: real,R0: real,A: nat > complex,M7: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R )
     => ( ( ord_less_real @ R @ R0 )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N3 ) ) @ ( power_power_real @ R0 @ N3 ) ) @ M7 )
         => ( summable_real
            @ ^ [N2: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N2 ) ) @ ( power_power_real @ R @ N2 ) ) ) ) ) ) ).

% Abel_lemma
thf(fact_7517_bot__empty__eq2,axiom,
    ( bot_bot_int_int_o
    = ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ bot_bo1796632182523588997nt_int ) ) ) ).

% bot_empty_eq2
thf(fact_7518_bot__empty__eq2,axiom,
    ( bot_bo5358457235160185703eger_o
    = ( ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ bot_bo3145834390647256904nteger ) ) ) ).

% bot_empty_eq2
thf(fact_7519_bot__empty__eq2,axiom,
    ( bot_bo3000040243691356879eger_o
    = ( ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ bot_bo5443222936135328352nteger ) ) ) ).

% bot_empty_eq2
thf(fact_7520_bot__empty__eq2,axiom,
    ( bot_bo8662317086119403298_int_o
    = ( ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ bot_bo572930865798478029nt_int ) ) ) ).

% bot_empty_eq2
thf(fact_7521_bot__empty__eq2,axiom,
    ( bot_bo1403522918969695512_int_o
    = ( ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ bot_bo4508923176915781079nt_int ) ) ) ).

% bot_empty_eq2
thf(fact_7522_pred__equals__eq2,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ( ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 ) )
        = ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S2 ) ) )
      = ( R2 = S2 ) ) ).

% pred_equals_eq2
thf(fact_7523_pred__equals__eq2,axiom,
    ! [R2: set_Pr8056137968301705908nteger,S2: set_Pr8056137968301705908nteger] :
      ( ( ( ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ R2 ) )
        = ( ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ S2 ) ) )
      = ( R2 = S2 ) ) ).

% pred_equals_eq2
thf(fact_7524_pred__equals__eq2,axiom,
    ! [R2: set_Pr1281608226676607948nteger,S2: set_Pr1281608226676607948nteger] :
      ( ( ( ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ R2 ) )
        = ( ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ S2 ) ) )
      = ( R2 = S2 ) ) ).

% pred_equals_eq2
thf(fact_7525_pred__equals__eq2,axiom,
    ! [R2: set_Pr9222295170931077689nt_int,S2: set_Pr9222295170931077689nt_int] :
      ( ( ( ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ R2 ) )
        = ( ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ S2 ) ) )
      = ( R2 = S2 ) ) ).

% pred_equals_eq2
thf(fact_7526_pred__equals__eq2,axiom,
    ! [R2: set_Pr1872883991513573699nt_int,S2: set_Pr1872883991513573699nt_int] :
      ( ( ( ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ R2 ) )
        = ( ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ S2 ) ) )
      = ( R2 = S2 ) ) ).

% pred_equals_eq2
thf(fact_7527_summable__ratio__test,axiom,
    ! [C: real,N4: nat,F: nat > real] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N3: nat] :
            ( ( ord_less_eq_nat @ N4 @ N3 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N3 ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) ) ) )
       => ( summable_real @ F ) ) ) ).

% summable_ratio_test
thf(fact_7528_summable__ratio__test,axiom,
    ! [C: real,N4: nat,F: nat > complex] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N3: nat] :
            ( ( ord_less_eq_nat @ N4 @ N3 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N3 ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) ) ) )
       => ( summable_complex @ F ) ) ) ).

% summable_ratio_test
thf(fact_7529_pi__half__neq__zero,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% pi_half_neq_zero
thf(fact_7530_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_7531_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_7532_power__diff__1__eq,axiom,
    ! [X3: complex,N: nat] :
      ( ( minus_minus_complex @ ( power_power_complex @ X3 @ N ) @ one_one_complex )
      = ( times_times_complex @ ( minus_minus_complex @ X3 @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_1_eq
thf(fact_7533_power__diff__1__eq,axiom,
    ! [X3: rat,N: nat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X3 @ N ) @ one_one_rat )
      = ( times_times_rat @ ( minus_minus_rat @ X3 @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_1_eq
thf(fact_7534_power__diff__1__eq,axiom,
    ! [X3: int,N: nat] :
      ( ( minus_minus_int @ ( power_power_int @ X3 @ N ) @ one_one_int )
      = ( times_times_int @ ( minus_minus_int @ X3 @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_1_eq
thf(fact_7535_power__diff__1__eq,axiom,
    ! [X3: real,N: nat] :
      ( ( minus_minus_real @ ( power_power_real @ X3 @ N ) @ one_one_real )
      = ( times_times_real @ ( minus_minus_real @ X3 @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_1_eq
thf(fact_7536_one__diff__power__eq,axiom,
    ! [X3: complex,N: nat] :
      ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ N ) )
      = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X3 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq
thf(fact_7537_one__diff__power__eq,axiom,
    ! [X3: rat,N: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ N ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X3 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq
thf(fact_7538_one__diff__power__eq,axiom,
    ! [X3: int,N: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X3 @ N ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X3 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq
thf(fact_7539_one__diff__power__eq,axiom,
    ! [X3: real,N: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ N ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X3 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq
thf(fact_7540_geometric__sum,axiom,
    ! [X3: complex,N: nat] :
      ( ( X3 != one_one_complex )
     => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X3 @ N ) @ one_one_complex ) @ ( minus_minus_complex @ X3 @ one_one_complex ) ) ) ) ).

% geometric_sum
thf(fact_7541_geometric__sum,axiom,
    ! [X3: rat,N: nat] :
      ( ( X3 != one_one_rat )
     => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X3 @ N ) @ one_one_rat ) @ ( minus_minus_rat @ X3 @ one_one_rat ) ) ) ) ).

% geometric_sum
thf(fact_7542_geometric__sum,axiom,
    ! [X3: real,N: nat] :
      ( ( X3 != one_one_real )
     => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X3 @ N ) @ one_one_real ) @ ( minus_minus_real @ X3 @ one_one_real ) ) ) ) ).

% geometric_sum
thf(fact_7543_sum__gp__strict,axiom,
    ! [X3: complex,N: nat] :
      ( ( ( X3 = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
          = ( semiri8010041392384452111omplex @ N ) ) )
      & ( ( X3 != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ N ) ) @ ( minus_minus_complex @ one_one_complex @ X3 ) ) ) ) ) ).

% sum_gp_strict
thf(fact_7544_sum__gp__strict,axiom,
    ! [X3: rat,N: nat] :
      ( ( ( X3 = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
          = ( semiri681578069525770553at_rat @ N ) ) )
      & ( ( X3 != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ N ) ) @ ( minus_minus_rat @ one_one_rat @ X3 ) ) ) ) ) ).

% sum_gp_strict
thf(fact_7545_sum__gp__strict,axiom,
    ! [X3: real,N: nat] :
      ( ( ( X3 = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
          = ( semiri5074537144036343181t_real @ N ) ) )
      & ( ( X3 != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
          = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ N ) ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ).

% sum_gp_strict
thf(fact_7546_lemma__termdiff1,axiom,
    ! [Z: complex,H2: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [P5: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_complex @ Z @ P5 ) ) @ ( power_power_complex @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups2073611262835488442omplex
        @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ Z @ P5 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_7547_lemma__termdiff1,axiom,
    ! [Z: rat,H2: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [P5: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_rat @ Z @ P5 ) ) @ ( power_power_rat @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups2906978787729119204at_rat
        @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ Z @ P5 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_7548_lemma__termdiff1,axiom,
    ! [Z: int,H2: int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [P5: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_int @ Z @ P5 ) ) @ ( power_power_int @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups3539618377306564664at_int
        @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ Z @ P5 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_int @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_7549_lemma__termdiff1,axiom,
    ! [Z: real,H2: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [P5: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_real @ Z @ P5 ) ) @ ( power_power_real @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ Z @ P5 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_real @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_7550_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_7551_power__diff__sumr2,axiom,
    ! [X3: complex,N: nat,Y3: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X3 @ N ) @ ( power_power_complex @ Y3 @ N ) )
      = ( times_times_complex @ ( minus_minus_complex @ X3 @ Y3 )
        @ ( groups2073611262835488442omplex
          @ ^ [I4: nat] : ( times_times_complex @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_complex @ X3 @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_7552_power__diff__sumr2,axiom,
    ! [X3: rat,N: nat,Y3: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X3 @ N ) @ ( power_power_rat @ Y3 @ N ) )
      = ( times_times_rat @ ( minus_minus_rat @ X3 @ Y3 )
        @ ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( times_times_rat @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_rat @ X3 @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_7553_power__diff__sumr2,axiom,
    ! [X3: int,N: nat,Y3: int] :
      ( ( minus_minus_int @ ( power_power_int @ X3 @ N ) @ ( power_power_int @ Y3 @ N ) )
      = ( times_times_int @ ( minus_minus_int @ X3 @ Y3 )
        @ ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( times_times_int @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_int @ X3 @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_7554_power__diff__sumr2,axiom,
    ! [X3: real,N: nat,Y3: real] :
      ( ( minus_minus_real @ ( power_power_real @ X3 @ N ) @ ( power_power_real @ Y3 @ N ) )
      = ( times_times_real @ ( minus_minus_real @ X3 @ Y3 )
        @ ( groups6591440286371151544t_real
          @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_real @ X3 @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_7555_diff__power__eq__sum,axiom,
    ! [X3: complex,N: nat,Y3: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X3 @ ( suc @ N ) ) @ ( power_power_complex @ Y3 @ ( suc @ N ) ) )
      = ( times_times_complex @ ( minus_minus_complex @ X3 @ Y3 )
        @ ( groups2073611262835488442omplex
          @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ X3 @ P5 ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ N @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_7556_diff__power__eq__sum,axiom,
    ! [X3: rat,N: nat,Y3: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X3 @ ( suc @ N ) ) @ ( power_power_rat @ Y3 @ ( suc @ N ) ) )
      = ( times_times_rat @ ( minus_minus_rat @ X3 @ Y3 )
        @ ( groups2906978787729119204at_rat
          @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ X3 @ P5 ) @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ N @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_7557_diff__power__eq__sum,axiom,
    ! [X3: int,N: nat,Y3: int] :
      ( ( minus_minus_int @ ( power_power_int @ X3 @ ( suc @ N ) ) @ ( power_power_int @ Y3 @ ( suc @ N ) ) )
      = ( times_times_int @ ( minus_minus_int @ X3 @ Y3 )
        @ ( groups3539618377306564664at_int
          @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ X3 @ P5 ) @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_7558_diff__power__eq__sum,axiom,
    ! [X3: real,N: nat,Y3: real] :
      ( ( minus_minus_real @ ( power_power_real @ X3 @ ( suc @ N ) ) @ ( power_power_real @ Y3 @ ( suc @ N ) ) )
      = ( times_times_real @ ( minus_minus_real @ X3 @ Y3 )
        @ ( groups6591440286371151544t_real
          @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ X3 @ P5 ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ N @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_7559_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_7560_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_7561_arctan__ubound,axiom,
    ! [Y3: real] : ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arctan_ubound
thf(fact_7562_arctan__one,axiom,
    ( ( arctan @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% arctan_one
thf(fact_7563_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: nat > rat,K5: rat,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N )
         => ( ord_less_eq_rat @ ( F @ P7 ) @ K5 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ K5 )
       => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_7564_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: nat > int,K5: int,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N )
         => ( ord_less_eq_int @ ( F @ P7 ) @ K5 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ K5 )
       => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_7565_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: nat > nat,K5: nat,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N )
         => ( ord_less_eq_nat @ ( F @ P7 ) @ K5 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ K5 )
       => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_7566_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: nat > real,K5: real,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N )
         => ( ord_less_eq_real @ ( F @ P7 ) @ K5 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ K5 )
       => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_7567_one__diff__power__eq_H,axiom,
    ! [X3: complex,N: nat] :
      ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ N ) )
      = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X3 )
        @ ( groups2073611262835488442omplex
          @ ^ [I4: nat] : ( power_power_complex @ X3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_7568_one__diff__power__eq_H,axiom,
    ! [X3: rat,N: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ N ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X3 )
        @ ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( power_power_rat @ X3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_7569_one__diff__power__eq_H,axiom,
    ! [X3: int,N: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X3 @ N ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X3 )
        @ ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( power_power_int @ X3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_7570_one__diff__power__eq_H,axiom,
    ! [X3: real,N: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ N ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X3 )
        @ ( groups6591440286371151544t_real
          @ ^ [I4: nat] : ( power_power_real @ X3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_7571_subrelI,axiom,
    ! [R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ! [X5: int,Y4: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y4 ) @ R )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y4 ) @ S ) )
     => ( ord_le2843351958646193337nt_int @ R @ S ) ) ).

% subrelI
thf(fact_7572_subrelI,axiom,
    ! [R: set_Pr8056137968301705908nteger,S: set_Pr8056137968301705908nteger] :
      ( ! [X5: code_integer > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
          ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X5 @ Y4 ) @ R )
         => ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X5 @ Y4 ) @ S ) )
     => ( ord_le3216752416896350996nteger @ R @ S ) ) ).

% subrelI
thf(fact_7573_subrelI,axiom,
    ! [R: set_Pr1281608226676607948nteger,S: set_Pr1281608226676607948nteger] :
      ( ! [X5: produc6241069584506657477e_term > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
          ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X5 @ Y4 ) @ R )
         => ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X5 @ Y4 ) @ S ) )
     => ( ord_le653643898420964396nteger @ R @ S ) ) ).

% subrelI
thf(fact_7574_subrelI,axiom,
    ! [R: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
      ( ! [X5: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int] :
          ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X5 @ Y4 ) @ R )
         => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X5 @ Y4 ) @ S ) )
     => ( ord_le8725513860283290265nt_int @ R @ S ) ) ).

% subrelI
thf(fact_7575_subrelI,axiom,
    ! [R: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
      ( ! [X5: int > option6357759511663192854e_term,Y4: product_prod_int_int] :
          ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X5 @ Y4 ) @ R )
         => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X5 @ Y4 ) @ S ) )
     => ( ord_le135402666524580259nt_int @ R @ S ) ) ).

% subrelI
thf(fact_7576_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_7577_arctan__lbound,axiom,
    ! [Y3: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) ) ).

% arctan_lbound
thf(fact_7578_arctan__bounded,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
      & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% arctan_bounded
thf(fact_7579_unbounded__k__infinite,axiom,
    ! [K: nat,S2: set_nat] :
      ( ! [M4: nat] :
          ( ( ord_less_nat @ K @ M4 )
         => ? [N9: nat] :
              ( ( ord_less_nat @ M4 @ N9 )
              & ( member_nat @ N9 @ S2 ) ) )
     => ~ ( finite_finite_nat @ S2 ) ) ).

% unbounded_k_infinite
thf(fact_7580_infinite__nat__iff__unbounded,axiom,
    ! [S2: set_nat] :
      ( ( ~ ( finite_finite_nat @ S2 ) )
      = ( ! [M3: nat] :
          ? [N2: nat] :
            ( ( ord_less_nat @ M3 @ N2 )
            & ( member_nat @ N2 @ S2 ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_7581_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set_nat] :
      ( ( ~ ( finite_finite_nat @ S2 ) )
      = ( ! [M3: nat] :
          ? [N2: nat] :
            ( ( ord_less_eq_nat @ M3 @ N2 )
            & ( member_nat @ N2 @ S2 ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_7582_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_7583_pred__subset__eq,axiom,
    ! [R2: set_nat,S2: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X: nat] : ( member_nat @ X @ R2 )
        @ ^ [X: nat] : ( member_nat @ X @ S2 ) )
      = ( ord_less_eq_set_nat @ R2 @ S2 ) ) ).

% pred_subset_eq
thf(fact_7584_pred__subset__eq,axiom,
    ! [R2: set_real,S2: set_real] :
      ( ( ord_less_eq_real_o
        @ ^ [X: real] : ( member_real @ X @ R2 )
        @ ^ [X: real] : ( member_real @ X @ S2 ) )
      = ( ord_less_eq_set_real @ R2 @ S2 ) ) ).

% pred_subset_eq
thf(fact_7585_pred__subset__eq,axiom,
    ! [R2: set_complex,S2: set_complex] :
      ( ( ord_le4573692005234683329plex_o
        @ ^ [X: complex] : ( member_complex @ X @ R2 )
        @ ^ [X: complex] : ( member_complex @ X @ S2 ) )
      = ( ord_le211207098394363844omplex @ R2 @ S2 ) ) ).

% pred_subset_eq
thf(fact_7586_pred__subset__eq,axiom,
    ! [R2: set_int,S2: set_int] :
      ( ( ord_less_eq_int_o
        @ ^ [X: int] : ( member_int @ X @ R2 )
        @ ^ [X: int] : ( member_int @ X @ S2 ) )
      = ( ord_less_eq_set_int @ R2 @ S2 ) ) ).

% pred_subset_eq
thf(fact_7587_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_7588_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_7589_infinite__int__iff__unbounded__le,axiom,
    ! [S2: set_int] :
      ( ( ~ ( finite_finite_int @ S2 ) )
      = ( ! [M3: int] :
          ? [N2: int] :
            ( ( ord_less_eq_int @ M3 @ ( abs_abs_int @ N2 ) )
            & ( member_int @ N2 @ S2 ) ) ) ) ).

% infinite_int_iff_unbounded_le
thf(fact_7590_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_7591_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_7592_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_7593_accp__subset,axiom,
    ! [R1: product_prod_num_num > product_prod_num_num > $o,R22: product_prod_num_num > product_prod_num_num > $o] :
      ( ( ord_le2556027599737686990_num_o @ R1 @ R22 )
     => ( ord_le2239182809043710856_num_o @ ( accp_P3113834385874906142um_num @ R22 ) @ ( accp_P3113834385874906142um_num @ R1 ) ) ) ).

% accp_subset
thf(fact_7594_accp__subset,axiom,
    ! [R1: product_prod_nat_nat > product_prod_nat_nat > $o,R22: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( ord_le5604493270027003598_nat_o @ R1 @ R22 )
     => ( ord_le704812498762024988_nat_o @ ( accp_P4275260045618599050at_nat @ R22 ) @ ( accp_P4275260045618599050at_nat @ R1 ) ) ) ).

% accp_subset
thf(fact_7595_accp__subset,axiom,
    ! [R1: product_prod_int_int > product_prod_int_int > $o,R22: product_prod_int_int > product_prod_int_int > $o] :
      ( ( ord_le1598226405681992910_int_o @ R1 @ R22 )
     => ( ord_le8369615600986905444_int_o @ ( accp_P1096762738010456898nt_int @ R22 ) @ ( accp_P1096762738010456898nt_int @ R1 ) ) ) ).

% accp_subset
thf(fact_7596_accp__subset,axiom,
    ! [R1: list_nat > list_nat > $o,R22: list_nat > list_nat > $o] :
      ( ( ord_le6558929396352911974_nat_o @ R1 @ R22 )
     => ( ord_le1520216061033275535_nat_o @ ( accp_list_nat @ R22 ) @ ( accp_list_nat @ R1 ) ) ) ).

% accp_subset
thf(fact_7597_accp__subset,axiom,
    ! [R1: nat > nat > $o,R22: nat > nat > $o] :
      ( ( ord_le2646555220125990790_nat_o @ R1 @ R22 )
     => ( ord_less_eq_nat_o @ ( accp_nat @ R22 ) @ ( accp_nat @ R1 ) ) ) ).

% accp_subset
thf(fact_7598_geometric__deriv__sums,axiom,
    ! [Z: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
     => ( sums_real
        @ ^ [N2: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) )
        @ ( 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_7599_geometric__deriv__sums,axiom,
    ! [Z: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
     => ( sums_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) )
        @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_7600_monoseq__def,axiom,
    ( topolo6980174941875973593q_real
    = ( ^ [X7: nat > real] :
          ( ! [M3: nat,N2: nat] :
              ( ( ord_less_eq_nat @ M3 @ N2 )
             => ( ord_less_eq_real @ ( X7 @ M3 ) @ ( X7 @ N2 ) ) )
          | ! [M3: nat,N2: nat] :
              ( ( ord_less_eq_nat @ M3 @ N2 )
             => ( ord_less_eq_real @ ( X7 @ N2 ) @ ( X7 @ M3 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_7601_monoseq__def,axiom,
    ( topolo3100542954746470799et_int
    = ( ^ [X7: nat > set_int] :
          ( ! [M3: nat,N2: nat] :
              ( ( ord_less_eq_nat @ M3 @ N2 )
             => ( ord_less_eq_set_int @ ( X7 @ M3 ) @ ( X7 @ N2 ) ) )
          | ! [M3: nat,N2: nat] :
              ( ( ord_less_eq_nat @ M3 @ N2 )
             => ( ord_less_eq_set_int @ ( X7 @ N2 ) @ ( X7 @ M3 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_7602_monoseq__def,axiom,
    ( topolo4267028734544971653eq_rat
    = ( ^ [X7: nat > rat] :
          ( ! [M3: nat,N2: nat] :
              ( ( ord_less_eq_nat @ M3 @ N2 )
             => ( ord_less_eq_rat @ ( X7 @ M3 ) @ ( X7 @ N2 ) ) )
          | ! [M3: nat,N2: nat] :
              ( ( ord_less_eq_nat @ M3 @ N2 )
             => ( ord_less_eq_rat @ ( X7 @ N2 ) @ ( X7 @ M3 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_7603_monoseq__def,axiom,
    ( topolo1459490580787246023eq_num
    = ( ^ [X7: nat > num] :
          ( ! [M3: nat,N2: nat] :
              ( ( ord_less_eq_nat @ M3 @ N2 )
             => ( ord_less_eq_num @ ( X7 @ M3 ) @ ( X7 @ N2 ) ) )
          | ! [M3: nat,N2: nat] :
              ( ( ord_less_eq_nat @ M3 @ N2 )
             => ( ord_less_eq_num @ ( X7 @ N2 ) @ ( X7 @ M3 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_7604_monoseq__def,axiom,
    ( topolo4902158794631467389eq_nat
    = ( ^ [X7: nat > nat] :
          ( ! [M3: nat,N2: nat] :
              ( ( ord_less_eq_nat @ M3 @ N2 )
             => ( ord_less_eq_nat @ ( X7 @ M3 ) @ ( X7 @ N2 ) ) )
          | ! [M3: nat,N2: nat] :
              ( ( ord_less_eq_nat @ M3 @ N2 )
             => ( ord_less_eq_nat @ ( X7 @ N2 ) @ ( X7 @ M3 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_7605_monoseq__def,axiom,
    ( topolo4899668324122417113eq_int
    = ( ^ [X7: nat > int] :
          ( ! [M3: nat,N2: nat] :
              ( ( ord_less_eq_nat @ M3 @ N2 )
             => ( ord_less_eq_int @ ( X7 @ M3 ) @ ( X7 @ N2 ) ) )
          | ! [M3: nat,N2: nat] :
              ( ( ord_less_eq_nat @ M3 @ N2 )
             => ( ord_less_eq_int @ ( X7 @ N2 ) @ ( X7 @ M3 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_7606_sin__cos__squared__add3,axiom,
    ! [X3: complex] :
      ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ X3 ) ) @ ( times_times_complex @ ( sin_complex @ X3 ) @ ( sin_complex @ X3 ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add3
thf(fact_7607_sin__cos__squared__add3,axiom,
    ! [X3: real] :
      ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ X3 ) ) @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ X3 ) ) )
      = one_one_real ) ).

% sin_cos_squared_add3
thf(fact_7608_sin__npi2,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
      = zero_zero_real ) ).

% sin_npi2
thf(fact_7609_sin__npi,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% sin_npi
thf(fact_7610_powser__sums__zero__iff,axiom,
    ! [A: nat > complex,X3: complex] :
      ( ( sums_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( A @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
        @ X3 )
      = ( ( A @ zero_zero_nat )
        = X3 ) ) ).

% powser_sums_zero_iff
thf(fact_7611_powser__sums__zero__iff,axiom,
    ! [A: nat > real,X3: real] :
      ( ( sums_real
        @ ^ [N2: nat] : ( times_times_real @ ( A @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
        @ X3 )
      = ( ( A @ zero_zero_nat )
        = X3 ) ) ).

% powser_sums_zero_iff
thf(fact_7612_cos__pi__half,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = zero_zero_real ) ).

% cos_pi_half
thf(fact_7613_sin__two__pi,axiom,
    ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = zero_zero_real ) ).

% sin_two_pi
thf(fact_7614_sin__pi__half,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = one_one_real ) ).

% sin_pi_half
thf(fact_7615_cos__two__pi,axiom,
    ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_real ) ).

% cos_two_pi
thf(fact_7616_cos__periodic,axiom,
    ! [X3: real] :
      ( ( cos_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( cos_real @ X3 ) ) ).

% cos_periodic
thf(fact_7617_sin__periodic,axiom,
    ! [X3: real] :
      ( ( sin_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( sin_real @ X3 ) ) ).

% sin_periodic
thf(fact_7618_cos__2pi__minus,axiom,
    ! [X3: real] :
      ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X3 ) )
      = ( cos_real @ X3 ) ) ).

% cos_2pi_minus
thf(fact_7619_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_7620_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_7621_sin__cos__squared__add,axiom,
    ! [X3: real] :
      ( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_real ) ).

% sin_cos_squared_add
thf(fact_7622_sin__cos__squared__add,axiom,
    ! [X3: complex] :
      ( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add
thf(fact_7623_sin__cos__squared__add2,axiom,
    ! [X3: real] :
      ( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_real ) ).

% sin_cos_squared_add2
thf(fact_7624_sin__cos__squared__add2,axiom,
    ! [X3: complex] :
      ( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add2
thf(fact_7625_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_7626_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_7627_sin__2pi__minus,axiom,
    ! [X3: real] :
      ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X3 ) )
      = ( uminus_uminus_real @ ( sin_real @ X3 ) ) ) ).

% sin_2pi_minus
thf(fact_7628_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_7629_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_7630_sums__le,axiom,
    ! [F: nat > real,G: nat > real,S: real,T: real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
     => ( ( sums_real @ F @ S )
       => ( ( sums_real @ G @ T )
         => ( ord_less_eq_real @ S @ T ) ) ) ) ).

% sums_le
thf(fact_7631_sums__le,axiom,
    ! [F: nat > nat,G: nat > nat,S: nat,T: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ N3 ) )
     => ( ( sums_nat @ F @ S )
       => ( ( sums_nat @ G @ T )
         => ( ord_less_eq_nat @ S @ T ) ) ) ) ).

% sums_le
thf(fact_7632_sums__le,axiom,
    ! [F: nat > int,G: nat > int,S: int,T: int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G @ N3 ) )
     => ( ( sums_int @ F @ S )
       => ( ( sums_int @ G @ T )
         => ( ord_less_eq_int @ S @ T ) ) ) ) ).

% sums_le
thf(fact_7633_polar__Ex,axiom,
    ! [X3: real,Y3: real] :
    ? [R3: real,A3: real] :
      ( ( X3
        = ( times_times_real @ R3 @ ( cos_real @ A3 ) ) )
      & ( Y3
        = ( times_times_real @ R3 @ ( sin_real @ A3 ) ) ) ) ).

% polar_Ex
thf(fact_7634_sin__diff,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( sin_complex @ ( minus_minus_complex @ X3 @ Y3 ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( sin_complex @ X3 ) @ ( cos_complex @ Y3 ) ) @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( sin_complex @ Y3 ) ) ) ) ).

% sin_diff
thf(fact_7635_sin__diff,axiom,
    ! [X3: real,Y3: real] :
      ( ( sin_real @ ( minus_minus_real @ X3 @ Y3 ) )
      = ( minus_minus_real @ ( times_times_real @ ( sin_real @ X3 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( sin_real @ Y3 ) ) ) ) ).

% sin_diff
thf(fact_7636_sin__add,axiom,
    ! [X3: real,Y3: real] :
      ( ( sin_real @ ( plus_plus_real @ X3 @ Y3 ) )
      = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X3 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( sin_real @ Y3 ) ) ) ) ).

% sin_add
thf(fact_7637_sums__mult,axiom,
    ! [F: nat > real,A: real,C: real] :
      ( ( sums_real @ F @ A )
     => ( sums_real
        @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
        @ ( times_times_real @ C @ A ) ) ) ).

% sums_mult
thf(fact_7638_sums__mult2,axiom,
    ! [F: nat > real,A: real,C: real] :
      ( ( sums_real @ F @ A )
     => ( sums_real
        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C )
        @ ( times_times_real @ A @ C ) ) ) ).

% sums_mult2
thf(fact_7639_sums__add,axiom,
    ! [F: nat > real,A: real,G: nat > real,B2: real] :
      ( ( sums_real @ F @ A )
     => ( ( sums_real @ G @ B2 )
       => ( sums_real
          @ ^ [N2: nat] : ( plus_plus_real @ ( F @ N2 ) @ ( G @ N2 ) )
          @ ( plus_plus_real @ A @ B2 ) ) ) ) ).

% sums_add
thf(fact_7640_sums__add,axiom,
    ! [F: nat > nat,A: nat,G: nat > nat,B2: nat] :
      ( ( sums_nat @ F @ A )
     => ( ( sums_nat @ G @ B2 )
       => ( sums_nat
          @ ^ [N2: nat] : ( plus_plus_nat @ ( F @ N2 ) @ ( G @ N2 ) )
          @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).

% sums_add
thf(fact_7641_sums__add,axiom,
    ! [F: nat > int,A: int,G: nat > int,B2: int] :
      ( ( sums_int @ F @ A )
     => ( ( sums_int @ G @ B2 )
       => ( sums_int
          @ ^ [N2: nat] : ( plus_plus_int @ ( F @ N2 ) @ ( G @ N2 ) )
          @ ( plus_plus_int @ A @ B2 ) ) ) ) ).

% sums_add
thf(fact_7642_sums__diff,axiom,
    ! [F: nat > complex,A: complex,G: nat > complex,B2: complex] :
      ( ( sums_complex @ F @ A )
     => ( ( sums_complex @ G @ B2 )
       => ( sums_complex
          @ ^ [N2: nat] : ( minus_minus_complex @ ( F @ N2 ) @ ( G @ N2 ) )
          @ ( minus_minus_complex @ A @ B2 ) ) ) ) ).

% sums_diff
thf(fact_7643_sums__diff,axiom,
    ! [F: nat > real,A: real,G: nat > real,B2: real] :
      ( ( sums_real @ F @ A )
     => ( ( sums_real @ G @ B2 )
       => ( sums_real
          @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
          @ ( minus_minus_real @ A @ B2 ) ) ) ) ).

% sums_diff
thf(fact_7644_sums__divide,axiom,
    ! [F: nat > complex,A: complex,C: complex] :
      ( ( sums_complex @ F @ A )
     => ( sums_complex
        @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C )
        @ ( divide1717551699836669952omplex @ A @ C ) ) ) ).

% sums_divide
thf(fact_7645_sums__divide,axiom,
    ! [F: nat > real,A: real,C: real] :
      ( ( sums_real @ F @ A )
     => ( sums_real
        @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C )
        @ ( divide_divide_real @ A @ C ) ) ) ).

% sums_divide
thf(fact_7646_cos__add,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( cos_complex @ ( plus_plus_complex @ X3 @ Y3 ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ Y3 ) ) @ ( times_times_complex @ ( sin_complex @ X3 ) @ ( sin_complex @ Y3 ) ) ) ) ).

% cos_add
thf(fact_7647_cos__add,axiom,
    ! [X3: real,Y3: real] :
      ( ( cos_real @ ( plus_plus_real @ X3 @ Y3 ) )
      = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ Y3 ) ) ) ) ).

% cos_add
thf(fact_7648_cos__diff,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( cos_complex @ ( minus_minus_complex @ X3 @ Y3 ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ Y3 ) ) @ ( times_times_complex @ ( sin_complex @ X3 ) @ ( sin_complex @ Y3 ) ) ) ) ).

% cos_diff
thf(fact_7649_cos__diff,axiom,
    ! [X3: real,Y3: real] :
      ( ( cos_real @ ( minus_minus_real @ X3 @ Y3 ) )
      = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ Y3 ) ) ) ) ).

% cos_diff
thf(fact_7650_sin__double,axiom,
    ! [X3: complex] :
      ( ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ X3 ) ) @ ( cos_complex @ X3 ) ) ) ).

% sin_double
thf(fact_7651_sin__double,axiom,
    ! [X3: real] :
      ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X3 ) ) @ ( cos_real @ X3 ) ) ) ).

% sin_double
thf(fact_7652_sincos__principal__value,axiom,
    ! [X3: real] :
    ? [Y4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y4 )
      & ( ord_less_eq_real @ Y4 @ pi )
      & ( ( sin_real @ Y4 )
        = ( sin_real @ X3 ) )
      & ( ( cos_real @ Y4 )
        = ( cos_real @ X3 ) ) ) ).

% sincos_principal_value
thf(fact_7653_sin__x__le__x,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ord_less_eq_real @ ( sin_real @ X3 ) @ X3 ) ) ).

% sin_x_le_x
thf(fact_7654_sin__le__one,axiom,
    ! [X3: real] : ( ord_less_eq_real @ ( sin_real @ X3 ) @ one_one_real ) ).

% sin_le_one
thf(fact_7655_cos__le__one,axiom,
    ! [X3: real] : ( ord_less_eq_real @ ( cos_real @ X3 ) @ one_one_real ) ).

% cos_le_one
thf(fact_7656_abs__sin__x__le__abs__x,axiom,
    ! [X3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X3 ) ) @ ( abs_abs_real @ X3 ) ) ).

% abs_sin_x_le_abs_x
thf(fact_7657_sums__mult__iff,axiom,
    ! [C: complex,F: nat > complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( ( sums_complex
          @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) )
          @ ( times_times_complex @ C @ D ) )
        = ( sums_complex @ F @ D ) ) ) ).

% sums_mult_iff
thf(fact_7658_sums__mult__iff,axiom,
    ! [C: real,F: nat > real,D: real] :
      ( ( C != zero_zero_real )
     => ( ( sums_real
          @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
          @ ( times_times_real @ C @ D ) )
        = ( sums_real @ F @ D ) ) ) ).

% sums_mult_iff
thf(fact_7659_sums__mult2__iff,axiom,
    ! [C: complex,F: nat > complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( ( sums_complex
          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ C )
          @ ( times_times_complex @ D @ C ) )
        = ( sums_complex @ F @ D ) ) ) ).

% sums_mult2_iff
thf(fact_7660_sums__mult2__iff,axiom,
    ! [C: real,F: nat > real,D: real] :
      ( ( C != zero_zero_real )
     => ( ( sums_real
          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C )
          @ ( times_times_real @ D @ C ) )
        = ( sums_real @ F @ D ) ) ) ).

% sums_mult2_iff
thf(fact_7661_sin__cos__le1,axiom,
    ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) ) ) @ one_one_real ) ).

% sin_cos_le1
thf(fact_7662_cos__squared__eq,axiom,
    ! [X3: complex] :
      ( ( power_power_complex @ ( cos_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_squared_eq
thf(fact_7663_cos__squared__eq,axiom,
    ! [X3: real] :
      ( ( power_power_real @ ( cos_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_squared_eq
thf(fact_7664_sin__squared__eq,axiom,
    ! [X3: complex] :
      ( ( power_power_complex @ ( sin_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sin_squared_eq
thf(fact_7665_sin__squared__eq,axiom,
    ! [X3: real] :
      ( ( power_power_real @ ( sin_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sin_squared_eq
thf(fact_7666_sums__mult__D,axiom,
    ! [C: complex,F: nat > complex,A: complex] :
      ( ( sums_complex
        @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) )
        @ A )
     => ( ( C != zero_zero_complex )
       => ( sums_complex @ F @ ( divide1717551699836669952omplex @ A @ C ) ) ) ) ).

% sums_mult_D
thf(fact_7667_sums__mult__D,axiom,
    ! [C: real,F: nat > real,A: real] :
      ( ( sums_real
        @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
        @ A )
     => ( ( C != zero_zero_real )
       => ( sums_real @ F @ ( divide_divide_real @ A @ C ) ) ) ) ).

% sums_mult_D
thf(fact_7668_sums__Suc__iff,axiom,
    ! [F: nat > real,S: real] :
      ( ( sums_real
        @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
        @ S )
      = ( sums_real @ F @ ( plus_plus_real @ S @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc_iff
thf(fact_7669_sums__Suc,axiom,
    ! [F: nat > real,L: real] :
      ( ( sums_real
        @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
        @ L )
     => ( sums_real @ F @ ( plus_plus_real @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_7670_sums__Suc,axiom,
    ! [F: nat > nat,L: nat] :
      ( ( sums_nat
        @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
        @ L )
     => ( sums_nat @ F @ ( plus_plus_nat @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_7671_sums__Suc,axiom,
    ! [F: nat > int,L: int] :
      ( ( sums_int
        @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
        @ L )
     => ( sums_int @ F @ ( plus_plus_int @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_7672_sin__gt__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ pi )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X3 ) ) ) ) ).

% sin_gt_zero
thf(fact_7673_sin__x__ge__neg__x,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ X3 ) @ ( sin_real @ X3 ) ) ) ).

% sin_x_ge_neg_x
thf(fact_7674_sin__ge__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ X3 @ pi )
       => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X3 ) ) ) ) ).

% sin_ge_zero
thf(fact_7675_sums__zero__iff__shift,axiom,
    ! [N: nat,F: nat > complex,S: complex] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N )
         => ( ( F @ I3 )
            = zero_zero_complex ) )
     => ( ( sums_complex
          @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
          @ S )
        = ( sums_complex @ F @ S ) ) ) ).

% sums_zero_iff_shift
thf(fact_7676_sums__zero__iff__shift,axiom,
    ! [N: nat,F: nat > real,S: real] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N )
         => ( ( F @ I3 )
            = zero_zero_real ) )
     => ( ( sums_real
          @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
          @ S )
        = ( sums_real @ F @ S ) ) ) ).

% sums_zero_iff_shift
thf(fact_7677_sin__ge__minus__one,axiom,
    ! [X3: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X3 ) ) ).

% sin_ge_minus_one
thf(fact_7678_cos__monotone__0__pi__le,axiom,
    ! [Y3: real,X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ X3 )
       => ( ( ord_less_eq_real @ X3 @ pi )
         => ( ord_less_eq_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) ) ) ) ).

% cos_monotone_0_pi_le
thf(fact_7679_cos__mono__le__eq,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ X3 @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
         => ( ( ord_less_eq_real @ Y3 @ pi )
           => ( ( ord_less_eq_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) )
              = ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ) ) ).

% cos_mono_le_eq
thf(fact_7680_cos__inj__pi,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ X3 @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
         => ( ( ord_less_eq_real @ Y3 @ pi )
           => ( ( ( cos_real @ X3 )
                = ( cos_real @ Y3 ) )
             => ( X3 = Y3 ) ) ) ) ) ) ).

% cos_inj_pi
thf(fact_7681_cos__ge__minus__one,axiom,
    ! [X3: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X3 ) ) ).

% cos_ge_minus_one
thf(fact_7682_abs__sin__le__one,axiom,
    ! [X3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X3 ) ) @ one_one_real ) ).

% abs_sin_le_one
thf(fact_7683_abs__cos__le__one,axiom,
    ! [X3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X3 ) ) @ one_one_real ) ).

% abs_cos_le_one
thf(fact_7684_sin__times__sin,axiom,
    ! [W: complex,Z: complex] :
      ( ( times_times_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% sin_times_sin
thf(fact_7685_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_7686_sin__times__cos,axiom,
    ! [W: complex,Z: complex] :
      ( ( times_times_complex @ ( sin_complex @ W ) @ ( cos_complex @ Z ) )
      = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% sin_times_cos
thf(fact_7687_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_7688_cos__times__sin,axiom,
    ! [W: complex,Z: complex] :
      ( ( times_times_complex @ ( cos_complex @ W ) @ ( sin_complex @ Z ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% cos_times_sin
thf(fact_7689_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_7690_sin__plus__sin,axiom,
    ! [W: complex,Z: complex] :
      ( ( plus_plus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_plus_sin
thf(fact_7691_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_7692_sin__diff__sin,axiom,
    ! [W: complex,Z: complex] :
      ( ( minus_minus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_diff_sin
thf(fact_7693_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_7694_cos__diff__cos,axiom,
    ! [W: complex,Z: complex] :
      ( ( minus_minus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ Z @ W ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_diff_cos
thf(fact_7695_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_7696_cos__double,axiom,
    ! [X3: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) )
      = ( minus_minus_complex @ ( power_power_complex @ ( cos_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_double
thf(fact_7697_cos__double,axiom,
    ! [X3: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) )
      = ( minus_minus_real @ ( power_power_real @ ( cos_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_double
thf(fact_7698_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_7699_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_7700_powser__sums__if,axiom,
    ! [M: nat,Z: complex] :
      ( sums_complex
      @ ^ [N2: nat] : ( times_times_complex @ ( if_complex @ ( N2 = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z @ N2 ) )
      @ ( power_power_complex @ Z @ M ) ) ).

% powser_sums_if
thf(fact_7701_powser__sums__if,axiom,
    ! [M: nat,Z: real] :
      ( sums_real
      @ ^ [N2: nat] : ( times_times_real @ ( if_real @ ( N2 = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z @ N2 ) )
      @ ( power_power_real @ Z @ M ) ) ).

% powser_sums_if
thf(fact_7702_powser__sums__if,axiom,
    ! [M: nat,Z: int] :
      ( sums_int
      @ ^ [N2: nat] : ( times_times_int @ ( if_int @ ( N2 = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z @ N2 ) )
      @ ( power_power_int @ Z @ M ) ) ).

% powser_sums_if
thf(fact_7703_powser__sums__zero,axiom,
    ! [A: nat > complex] :
      ( sums_complex
      @ ^ [N2: nat] : ( times_times_complex @ ( A @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
      @ ( A @ zero_zero_nat ) ) ).

% powser_sums_zero
thf(fact_7704_powser__sums__zero,axiom,
    ! [A: nat > real] :
      ( sums_real
      @ ^ [N2: nat] : ( times_times_real @ ( A @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
      @ ( A @ zero_zero_nat ) ) ).

% powser_sums_zero
thf(fact_7705_cos__two__neq__zero,axiom,
    ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% cos_two_neq_zero
thf(fact_7706_cos__mono__less__eq,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ X3 @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
         => ( ( ord_less_eq_real @ Y3 @ pi )
           => ( ( ord_less_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) )
              = ( ord_less_real @ Y3 @ X3 ) ) ) ) ) ) ).

% cos_mono_less_eq
thf(fact_7707_cos__monotone__0__pi,axiom,
    ! [Y3: real,X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_real @ Y3 @ X3 )
       => ( ( ord_less_eq_real @ X3 @ pi )
         => ( ord_less_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) ) ) ) ).

% cos_monotone_0_pi
thf(fact_7708_sin__eq__0__pi,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X3 )
     => ( ( ord_less_real @ X3 @ pi )
       => ( ( ( sin_real @ X3 )
            = zero_zero_real )
         => ( X3 = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
thf(fact_7709_sums__iff__shift,axiom,
    ! [F: nat > real,N: nat,S: real] :
      ( ( sums_real
        @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
        @ S )
      = ( sums_real @ F @ ( plus_plus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% sums_iff_shift
thf(fact_7710_sums__split__initial__segment,axiom,
    ! [F: nat > complex,S: complex,N: nat] :
      ( ( sums_complex @ F @ S )
     => ( sums_complex
        @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
        @ ( minus_minus_complex @ S @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% sums_split_initial_segment
thf(fact_7711_sums__split__initial__segment,axiom,
    ! [F: nat > real,S: real,N: nat] :
      ( ( sums_real @ F @ S )
     => ( sums_real
        @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
        @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% sums_split_initial_segment
thf(fact_7712_sums__iff__shift_H,axiom,
    ! [F: nat > complex,N: nat,S: complex] :
      ( ( sums_complex
        @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
        @ ( minus_minus_complex @ S @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N ) ) ) )
      = ( sums_complex @ F @ S ) ) ).

% sums_iff_shift'
thf(fact_7713_sums__iff__shift_H,axiom,
    ! [F: nat > real,N: nat,S: real] :
      ( ( sums_real
        @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
        @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) )
      = ( sums_real @ F @ S ) ) ).

% sums_iff_shift'
thf(fact_7714_sin__zero__pi__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ pi )
     => ( ( ( sin_real @ X3 )
          = zero_zero_real )
        = ( X3 = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
thf(fact_7715_cos__monotone__minus__pi__0_H,axiom,
    ! [Y3: real,X3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ X3 )
       => ( ( ord_less_eq_real @ X3 @ zero_zero_real )
         => ( ord_less_eq_real @ ( cos_real @ Y3 ) @ ( cos_real @ X3 ) ) ) ) ) ).

% cos_monotone_minus_pi_0'
thf(fact_7716_sums__If__finite__set_H,axiom,
    ! [G: nat > complex,S2: complex,A4: set_nat,S5: complex,F: nat > complex] :
      ( ( sums_complex @ G @ S2 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( S5
            = ( plus_plus_complex @ S2
              @ ( groups2073611262835488442omplex
                @ ^ [N2: nat] : ( minus_minus_complex @ ( F @ N2 ) @ ( G @ N2 ) )
                @ A4 ) ) )
         => ( sums_complex
            @ ^ [N2: nat] : ( if_complex @ ( member_nat @ N2 @ A4 ) @ ( F @ N2 ) @ ( G @ N2 ) )
            @ S5 ) ) ) ) ).

% sums_If_finite_set'
thf(fact_7717_sums__If__finite__set_H,axiom,
    ! [G: nat > real,S2: real,A4: set_nat,S5: real,F: nat > real] :
      ( ( sums_real @ G @ S2 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( S5
            = ( plus_plus_real @ S2
              @ ( groups6591440286371151544t_real
                @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
                @ A4 ) ) )
         => ( sums_real
            @ ^ [N2: nat] : ( if_real @ ( member_nat @ N2 @ A4 ) @ ( F @ N2 ) @ ( G @ N2 ) )
            @ S5 ) ) ) ) ).

% sums_If_finite_set'
thf(fact_7718_sincos__total__pi,axiom,
    ! [Y3: real,X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = one_one_real )
       => ? [T4: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T4 )
            & ( ord_less_eq_real @ T4 @ pi )
            & ( X3
              = ( cos_real @ T4 ) )
            & ( Y3
              = ( sin_real @ T4 ) ) ) ) ) ).

% sincos_total_pi
thf(fact_7719_sin__expansion__lemma,axiom,
    ! [X3: real,M: nat] :
      ( ( sin_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( cos_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_expansion_lemma
thf(fact_7720_cos__expansion__lemma,axiom,
    ! [X3: real,M: nat] :
      ( ( cos_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_expansion_lemma
thf(fact_7721_sin__gt__zero__02,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X3 ) ) ) ) ).

% sin_gt_zero_02
thf(fact_7722_cos__two__less__zero,axiom,
    ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_less_zero
thf(fact_7723_cos__is__zero,axiom,
    ? [X5: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X5 )
      & ( ord_less_eq_real @ X5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
      & ( ( cos_real @ X5 )
        = zero_zero_real )
      & ! [Y5: real] :
          ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
            & ( ord_less_eq_real @ Y5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ Y5 )
              = zero_zero_real ) )
         => ( Y5 = X5 ) ) ) ).

% cos_is_zero
thf(fact_7724_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_7725_cos__monotone__minus__pi__0,axiom,
    ! [Y3: real,X3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
     => ( ( ord_less_real @ Y3 @ X3 )
       => ( ( ord_less_eq_real @ X3 @ zero_zero_real )
         => ( ord_less_real @ ( cos_real @ Y3 ) @ ( cos_real @ X3 ) ) ) ) ) ).

% cos_monotone_minus_pi_0
thf(fact_7726_cos__total,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ? [X5: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X5 )
            & ( ord_less_eq_real @ X5 @ pi )
            & ( ( cos_real @ X5 )
              = Y3 )
            & ! [Y5: real] :
                ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
                  & ( ord_less_eq_real @ Y5 @ pi )
                  & ( ( cos_real @ Y5 )
                    = Y3 ) )
               => ( Y5 = X5 ) ) ) ) ) ).

% cos_total
thf(fact_7727_sincos__total__pi__half,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
            = one_one_real )
         => ? [T4: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ T4 )
              & ( ord_less_eq_real @ T4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( X3
                = ( cos_real @ T4 ) )
              & ( Y3
                = ( sin_real @ T4 ) ) ) ) ) ) ).

% sincos_total_pi_half
thf(fact_7728_sincos__total__2pi__le,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ? [T4: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ T4 )
          & ( ord_less_eq_real @ T4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
          & ( X3
            = ( cos_real @ T4 ) )
          & ( Y3
            = ( sin_real @ T4 ) ) ) ) ).

% sincos_total_2pi_le
thf(fact_7729_sincos__total__2pi,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ~ ! [T4: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T4 )
           => ( ( ord_less_real @ T4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( ( X3
                  = ( cos_real @ T4 ) )
               => ( Y3
                 != ( sin_real @ T4 ) ) ) ) ) ) ).

% sincos_total_2pi
thf(fact_7730_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_7731_cos__times__cos,axiom,
    ! [W: complex,Z: complex] :
      ( ( times_times_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
      = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% cos_times_cos
thf(fact_7732_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_7733_cos__plus__cos,axiom,
    ! [W: complex,Z: complex] :
      ( ( plus_plus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_plus_cos
thf(fact_7734_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_7735_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_7736_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_7737_power__half__series,axiom,
    ( sums_real
    @ ^ [N2: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N2 ) )
    @ one_one_real ) ).

% power_half_series
thf(fact_7738_sin__gt__zero2,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X3 ) ) ) ) ).

% sin_gt_zero2
thf(fact_7739_sin__lt__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ pi @ X3 )
     => ( ( ord_less_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_real @ ( sin_real @ X3 ) @ zero_zero_real ) ) ) ).

% sin_lt_zero
thf(fact_7740_cos__double__less__one,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) @ one_one_real ) ) ) ).

% cos_double_less_one
thf(fact_7741_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_7742_cos__gt__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X3 ) ) ) ) ).

% cos_gt_zero
thf(fact_7743_sin__monotone__2pi__le,axiom,
    ! [Y3: real,X3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ X3 )
       => ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sin_real @ Y3 ) @ ( sin_real @ X3 ) ) ) ) ) ).

% sin_monotone_2pi_le
thf(fact_7744_sin__mono__le__eq,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
     => ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
         => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( sin_real @ X3 ) @ ( sin_real @ Y3 ) )
              = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ) ) ).

% sin_mono_le_eq
thf(fact_7745_sin__inj__pi,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
     => ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
         => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ( sin_real @ X3 )
                = ( sin_real @ Y3 ) )
             => ( X3 = Y3 ) ) ) ) ) ) ).

% sin_inj_pi
thf(fact_7746_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_7747_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_7748_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_7749_cos__treble__cos,axiom,
    ! [X3: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X3 ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ ( cos_complex @ X3 ) ) ) ) ).

% cos_treble_cos
thf(fact_7750_cos__treble__cos,axiom,
    ! [X3: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X3 ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X3 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( cos_real @ X3 ) ) ) ) ).

% cos_treble_cos
thf(fact_7751_sums__if_H,axiom,
    ! [G: nat > real,X3: real] :
      ( ( sums_real @ G @ X3 )
     => ( sums_real
        @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        @ X3 ) ) ).

% sums_if'
thf(fact_7752_sums__if,axiom,
    ! [G: nat > real,X3: real,F: nat > real,Y3: real] :
      ( ( sums_real @ G @ X3 )
     => ( ( sums_real @ F @ Y3 )
       => ( sums_real
          @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( F @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          @ ( plus_plus_real @ X3 @ Y3 ) ) ) ) ).

% sums_if
thf(fact_7753_sin__le__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ pi @ X3 )
     => ( ( ord_less_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_eq_real @ ( sin_real @ X3 ) @ zero_zero_real ) ) ) ).

% sin_le_zero
thf(fact_7754_sin__less__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 )
     => ( ( ord_less_real @ X3 @ zero_zero_real )
       => ( ord_less_real @ ( sin_real @ X3 ) @ zero_zero_real ) ) ) ).

% sin_less_zero
thf(fact_7755_sin__monotone__2pi,axiom,
    ! [Y3: real,X3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
     => ( ( ord_less_real @ Y3 @ X3 )
       => ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sin_real @ Y3 ) @ ( sin_real @ X3 ) ) ) ) ) ).

% sin_monotone_2pi
thf(fact_7756_sin__mono__less__eq,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
     => ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
         => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( sin_real @ X3 ) @ ( sin_real @ Y3 ) )
              = ( ord_less_real @ X3 @ Y3 ) ) ) ) ) ) ).

% sin_mono_less_eq
thf(fact_7757_sin__total,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ? [X5: real] :
            ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
            & ( ord_less_eq_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( sin_real @ X5 )
              = Y3 )
            & ! [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 )
                    = Y3 ) )
               => ( Y5 = X5 ) ) ) ) ) ).

% sin_total
thf(fact_7758_cos__gt__zero__pi,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
     => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X3 ) ) ) ) ).

% cos_gt_zero_pi
thf(fact_7759_cos__ge__zero,axiom,
    ! [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 ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X3 ) ) ) ) ).

% cos_ge_zero
thf(fact_7760_cos__one__2pi,axiom,
    ! [X3: real] :
      ( ( ( cos_real @ X3 )
        = one_one_real )
      = ( ? [X: nat] :
            ( X3
            = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
        | ? [X: nat] :
            ( X3
            = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).

% cos_one_2pi
thf(fact_7761_accp__subset__induct,axiom,
    ! [D4: product_prod_num_num > $o,R2: product_prod_num_num > product_prod_num_num > $o,X3: product_prod_num_num,P: product_prod_num_num > $o] :
      ( ( ord_le2239182809043710856_num_o @ D4 @ ( accp_P3113834385874906142um_num @ R2 ) )
     => ( ! [X5: product_prod_num_num,Z4: product_prod_num_num] :
            ( ( D4 @ X5 )
           => ( ( R2 @ Z4 @ X5 )
             => ( D4 @ Z4 ) ) )
       => ( ( D4 @ X3 )
         => ( ! [X5: product_prod_num_num] :
                ( ( D4 @ X5 )
               => ( ! [Z3: product_prod_num_num] :
                      ( ( R2 @ Z3 @ X5 )
                     => ( P @ Z3 ) )
                 => ( P @ X5 ) ) )
           => ( P @ X3 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_7762_accp__subset__induct,axiom,
    ! [D4: product_prod_nat_nat > $o,R2: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( ord_le704812498762024988_nat_o @ D4 @ ( accp_P4275260045618599050at_nat @ R2 ) )
     => ( ! [X5: product_prod_nat_nat,Z4: product_prod_nat_nat] :
            ( ( D4 @ X5 )
           => ( ( R2 @ Z4 @ X5 )
             => ( D4 @ Z4 ) ) )
       => ( ( D4 @ X3 )
         => ( ! [X5: product_prod_nat_nat] :
                ( ( D4 @ X5 )
               => ( ! [Z3: product_prod_nat_nat] :
                      ( ( R2 @ Z3 @ X5 )
                     => ( P @ Z3 ) )
                 => ( P @ X5 ) ) )
           => ( P @ X3 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_7763_accp__subset__induct,axiom,
    ! [D4: product_prod_int_int > $o,R2: product_prod_int_int > product_prod_int_int > $o,X3: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ D4 @ ( accp_P1096762738010456898nt_int @ R2 ) )
     => ( ! [X5: product_prod_int_int,Z4: product_prod_int_int] :
            ( ( D4 @ X5 )
           => ( ( R2 @ Z4 @ X5 )
             => ( D4 @ Z4 ) ) )
       => ( ( D4 @ X3 )
         => ( ! [X5: product_prod_int_int] :
                ( ( D4 @ X5 )
               => ( ! [Z3: product_prod_int_int] :
                      ( ( R2 @ Z3 @ X5 )
                     => ( P @ Z3 ) )
                 => ( P @ X5 ) ) )
           => ( P @ X3 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_7764_accp__subset__induct,axiom,
    ! [D4: list_nat > $o,R2: list_nat > list_nat > $o,X3: list_nat,P: list_nat > $o] :
      ( ( ord_le1520216061033275535_nat_o @ D4 @ ( accp_list_nat @ R2 ) )
     => ( ! [X5: list_nat,Z4: list_nat] :
            ( ( D4 @ X5 )
           => ( ( R2 @ Z4 @ X5 )
             => ( D4 @ Z4 ) ) )
       => ( ( D4 @ X3 )
         => ( ! [X5: list_nat] :
                ( ( D4 @ X5 )
               => ( ! [Z3: list_nat] :
                      ( ( R2 @ Z3 @ X5 )
                     => ( P @ Z3 ) )
                 => ( P @ X5 ) ) )
           => ( P @ X3 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_7765_accp__subset__induct,axiom,
    ! [D4: nat > $o,R2: nat > nat > $o,X3: nat,P: nat > $o] :
      ( ( ord_less_eq_nat_o @ D4 @ ( accp_nat @ R2 ) )
     => ( ! [X5: nat,Z4: nat] :
            ( ( D4 @ X5 )
           => ( ( R2 @ Z4 @ X5 )
             => ( D4 @ Z4 ) ) )
       => ( ( D4 @ X3 )
         => ( ! [X5: nat] :
                ( ( D4 @ X5 )
               => ( ! [Z3: nat] :
                      ( ( R2 @ Z3 @ X5 )
                     => ( P @ Z3 ) )
                 => ( P @ X5 ) ) )
           => ( P @ X3 ) ) ) ) ) ).

% accp_subset_induct
thf(fact_7766_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_7767_sin__zero__lemma,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ( sin_real @ X3 )
          = zero_zero_real )
       => ? [N3: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X3
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_lemma
thf(fact_7768_sin__zero__iff,axiom,
    ! [X3: real] :
      ( ( ( sin_real @ X3 )
        = zero_zero_real )
      = ( ? [N2: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X3
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N2: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X3
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% sin_zero_iff
thf(fact_7769_cos__zero__lemma,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ( cos_real @ X3 )
          = zero_zero_real )
       => ? [N3: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X3
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_lemma
thf(fact_7770_cos__zero__iff,axiom,
    ! [X3: real] :
      ( ( ( cos_real @ X3 )
        = zero_zero_real )
      = ( ? [N2: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X3
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N2: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X3
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% cos_zero_iff
thf(fact_7771_mono__SucI1,axiom,
    ! [X9: nat > real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( X9 @ N3 ) @ ( X9 @ ( suc @ N3 ) ) )
     => ( topolo6980174941875973593q_real @ X9 ) ) ).

% mono_SucI1
thf(fact_7772_mono__SucI1,axiom,
    ! [X9: nat > set_int] :
      ( ! [N3: nat] : ( ord_less_eq_set_int @ ( X9 @ N3 ) @ ( X9 @ ( suc @ N3 ) ) )
     => ( topolo3100542954746470799et_int @ X9 ) ) ).

% mono_SucI1
thf(fact_7773_mono__SucI1,axiom,
    ! [X9: nat > rat] :
      ( ! [N3: nat] : ( ord_less_eq_rat @ ( X9 @ N3 ) @ ( X9 @ ( suc @ N3 ) ) )
     => ( topolo4267028734544971653eq_rat @ X9 ) ) ).

% mono_SucI1
thf(fact_7774_mono__SucI1,axiom,
    ! [X9: nat > num] :
      ( ! [N3: nat] : ( ord_less_eq_num @ ( X9 @ N3 ) @ ( X9 @ ( suc @ N3 ) ) )
     => ( topolo1459490580787246023eq_num @ X9 ) ) ).

% mono_SucI1
thf(fact_7775_mono__SucI1,axiom,
    ! [X9: nat > nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( X9 @ N3 ) @ ( X9 @ ( suc @ N3 ) ) )
     => ( topolo4902158794631467389eq_nat @ X9 ) ) ).

% mono_SucI1
thf(fact_7776_mono__SucI1,axiom,
    ! [X9: nat > int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( X9 @ N3 ) @ ( X9 @ ( suc @ N3 ) ) )
     => ( topolo4899668324122417113eq_int @ X9 ) ) ).

% mono_SucI1
thf(fact_7777_mono__SucI2,axiom,
    ! [X9: nat > real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( X9 @ ( suc @ N3 ) ) @ ( X9 @ N3 ) )
     => ( topolo6980174941875973593q_real @ X9 ) ) ).

% mono_SucI2
thf(fact_7778_mono__SucI2,axiom,
    ! [X9: nat > set_int] :
      ( ! [N3: nat] : ( ord_less_eq_set_int @ ( X9 @ ( suc @ N3 ) ) @ ( X9 @ N3 ) )
     => ( topolo3100542954746470799et_int @ X9 ) ) ).

% mono_SucI2
thf(fact_7779_mono__SucI2,axiom,
    ! [X9: nat > rat] :
      ( ! [N3: nat] : ( ord_less_eq_rat @ ( X9 @ ( suc @ N3 ) ) @ ( X9 @ N3 ) )
     => ( topolo4267028734544971653eq_rat @ X9 ) ) ).

% mono_SucI2
thf(fact_7780_mono__SucI2,axiom,
    ! [X9: nat > num] :
      ( ! [N3: nat] : ( ord_less_eq_num @ ( X9 @ ( suc @ N3 ) ) @ ( X9 @ N3 ) )
     => ( topolo1459490580787246023eq_num @ X9 ) ) ).

% mono_SucI2
thf(fact_7781_mono__SucI2,axiom,
    ! [X9: nat > nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( X9 @ ( suc @ N3 ) ) @ ( X9 @ N3 ) )
     => ( topolo4902158794631467389eq_nat @ X9 ) ) ).

% mono_SucI2
thf(fact_7782_mono__SucI2,axiom,
    ! [X9: nat > int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( X9 @ ( suc @ N3 ) ) @ ( X9 @ N3 ) )
     => ( topolo4899668324122417113eq_int @ X9 ) ) ).

% mono_SucI2
thf(fact_7783_monoseq__Suc,axiom,
    ( topolo6980174941875973593q_real
    = ( ^ [X7: nat > real] :
          ( ! [N2: nat] : ( ord_less_eq_real @ ( X7 @ N2 ) @ ( X7 @ ( suc @ N2 ) ) )
          | ! [N2: nat] : ( ord_less_eq_real @ ( X7 @ ( suc @ N2 ) ) @ ( X7 @ N2 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7784_monoseq__Suc,axiom,
    ( topolo3100542954746470799et_int
    = ( ^ [X7: nat > set_int] :
          ( ! [N2: nat] : ( ord_less_eq_set_int @ ( X7 @ N2 ) @ ( X7 @ ( suc @ N2 ) ) )
          | ! [N2: nat] : ( ord_less_eq_set_int @ ( X7 @ ( suc @ N2 ) ) @ ( X7 @ N2 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7785_monoseq__Suc,axiom,
    ( topolo4267028734544971653eq_rat
    = ( ^ [X7: nat > rat] :
          ( ! [N2: nat] : ( ord_less_eq_rat @ ( X7 @ N2 ) @ ( X7 @ ( suc @ N2 ) ) )
          | ! [N2: nat] : ( ord_less_eq_rat @ ( X7 @ ( suc @ N2 ) ) @ ( X7 @ N2 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7786_monoseq__Suc,axiom,
    ( topolo1459490580787246023eq_num
    = ( ^ [X7: nat > num] :
          ( ! [N2: nat] : ( ord_less_eq_num @ ( X7 @ N2 ) @ ( X7 @ ( suc @ N2 ) ) )
          | ! [N2: nat] : ( ord_less_eq_num @ ( X7 @ ( suc @ N2 ) ) @ ( X7 @ N2 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7787_monoseq__Suc,axiom,
    ( topolo4902158794631467389eq_nat
    = ( ^ [X7: nat > nat] :
          ( ! [N2: nat] : ( ord_less_eq_nat @ ( X7 @ N2 ) @ ( X7 @ ( suc @ N2 ) ) )
          | ! [N2: nat] : ( ord_less_eq_nat @ ( X7 @ ( suc @ N2 ) ) @ ( X7 @ N2 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7788_monoseq__Suc,axiom,
    ( topolo4899668324122417113eq_int
    = ( ^ [X7: nat > int] :
          ( ! [N2: nat] : ( ord_less_eq_int @ ( X7 @ N2 ) @ ( X7 @ ( suc @ N2 ) ) )
          | ! [N2: nat] : ( ord_less_eq_int @ ( X7 @ ( suc @ N2 ) ) @ ( X7 @ N2 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_7789_monoI1,axiom,
    ! [X9: nat > real] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
         => ( ord_less_eq_real @ ( X9 @ M4 ) @ ( X9 @ N3 ) ) )
     => ( topolo6980174941875973593q_real @ X9 ) ) ).

% monoI1
thf(fact_7790_monoI1,axiom,
    ! [X9: nat > set_int] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
         => ( ord_less_eq_set_int @ ( X9 @ M4 ) @ ( X9 @ N3 ) ) )
     => ( topolo3100542954746470799et_int @ X9 ) ) ).

% monoI1
thf(fact_7791_monoI1,axiom,
    ! [X9: nat > rat] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
         => ( ord_less_eq_rat @ ( X9 @ M4 ) @ ( X9 @ N3 ) ) )
     => ( topolo4267028734544971653eq_rat @ X9 ) ) ).

% monoI1
thf(fact_7792_monoI1,axiom,
    ! [X9: nat > num] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
         => ( ord_less_eq_num @ ( X9 @ M4 ) @ ( X9 @ N3 ) ) )
     => ( topolo1459490580787246023eq_num @ X9 ) ) ).

% monoI1
thf(fact_7793_monoI1,axiom,
    ! [X9: nat > nat] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
         => ( ord_less_eq_nat @ ( X9 @ M4 ) @ ( X9 @ N3 ) ) )
     => ( topolo4902158794631467389eq_nat @ X9 ) ) ).

% monoI1
thf(fact_7794_monoI1,axiom,
    ! [X9: nat > int] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
         => ( ord_less_eq_int @ ( X9 @ M4 ) @ ( X9 @ N3 ) ) )
     => ( topolo4899668324122417113eq_int @ X9 ) ) ).

% monoI1
thf(fact_7795_monoI2,axiom,
    ! [X9: nat > real] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
         => ( ord_less_eq_real @ ( X9 @ N3 ) @ ( X9 @ M4 ) ) )
     => ( topolo6980174941875973593q_real @ X9 ) ) ).

% monoI2
thf(fact_7796_monoI2,axiom,
    ! [X9: nat > set_int] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
         => ( ord_less_eq_set_int @ ( X9 @ N3 ) @ ( X9 @ M4 ) ) )
     => ( topolo3100542954746470799et_int @ X9 ) ) ).

% monoI2
thf(fact_7797_monoI2,axiom,
    ! [X9: nat > rat] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
         => ( ord_less_eq_rat @ ( X9 @ N3 ) @ ( X9 @ M4 ) ) )
     => ( topolo4267028734544971653eq_rat @ X9 ) ) ).

% monoI2
thf(fact_7798_monoI2,axiom,
    ! [X9: nat > num] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
         => ( ord_less_eq_num @ ( X9 @ N3 ) @ ( X9 @ M4 ) ) )
     => ( topolo1459490580787246023eq_num @ X9 ) ) ).

% monoI2
thf(fact_7799_monoI2,axiom,
    ! [X9: nat > nat] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
         => ( ord_less_eq_nat @ ( X9 @ N3 ) @ ( X9 @ M4 ) ) )
     => ( topolo4902158794631467389eq_nat @ X9 ) ) ).

% monoI2
thf(fact_7800_monoI2,axiom,
    ! [X9: nat > int] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
         => ( ord_less_eq_int @ ( X9 @ N3 ) @ ( X9 @ M4 ) ) )
     => ( topolo4899668324122417113eq_int @ X9 ) ) ).

% monoI2
thf(fact_7801_vebt__maxt_Opelims,axiom,
    ! [X3: vEBT_VEBT,Y3: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X3 )
        = Y3 )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X3 )
       => ( ! [A3: $o,B3: $o] :
              ( ( X3
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( ( B3
                   => ( Y3
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B3
                   => ( ( A3
                       => ( Y3
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A3
                       => ( Y3 = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y3 = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y3
                      = ( some_nat @ Ma2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_maxt.pelims
thf(fact_7802_vebt__mint_Opelims,axiom,
    ! [X3: vEBT_VEBT,Y3: option_nat] :
      ( ( ( vEBT_vebt_mint @ X3 )
        = Y3 )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X3 )
       => ( ! [A3: $o,B3: $o] :
              ( ( X3
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( ( A3
                   => ( Y3
                      = ( some_nat @ zero_zero_nat ) ) )
                  & ( ~ A3
                   => ( ( B3
                       => ( Y3
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B3
                       => ( Y3 = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y3 = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y3
                      = ( some_nat @ Mi2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_mint.pelims
thf(fact_7803_Maclaurin__cos__expansion2,axiom,
    ! [X3: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ? [T4: real] :
            ( ( ord_less_real @ zero_zero_real @ T4 )
            & ( ord_less_real @ T4 @ X3 )
            & ( ( cos_real @ X3 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X3 @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T4 @ ( 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 @ X3 @ N ) ) ) ) ) ) ) ).

% Maclaurin_cos_expansion2
thf(fact_7804_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ X3 @ zero_zero_real )
       => ? [T4: real] :
            ( ( ord_less_real @ X3 @ T4 )
            & ( ord_less_real @ T4 @ zero_zero_real )
            & ( ( cos_real @ X3 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X3 @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T4 @ ( 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 @ X3 @ N ) ) ) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
thf(fact_7805_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% fact_Suc
thf(fact_7806_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).

% fact_Suc
thf(fact_7807_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri2265585572941072030t_real @ ( suc @ N ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).

% fact_Suc
thf(fact_7808_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% fact_Suc
thf(fact_7809_fact__2,axiom,
    ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_7810_fact__2,axiom,
    ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_7811_fact__2,axiom,
    ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_7812_fact__2,axiom,
    ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_7813_fact__2,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_7814_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).

% fact_ge_zero
thf(fact_7815_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).

% fact_ge_zero
thf(fact_7816_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).

% fact_ge_zero
thf(fact_7817_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_zero
thf(fact_7818_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).

% fact_not_neg
thf(fact_7819_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).

% fact_not_neg
thf(fact_7820_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).

% fact_not_neg
thf(fact_7821_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).

% fact_not_neg
thf(fact_7822_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).

% fact_gt_zero
thf(fact_7823_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).

% fact_gt_zero
thf(fact_7824_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).

% fact_gt_zero
thf(fact_7825_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_gt_zero
thf(fact_7826_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).

% fact_ge_1
thf(fact_7827_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).

% fact_ge_1
thf(fact_7828_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).

% fact_ge_1
thf(fact_7829_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_1
thf(fact_7830_fact__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% fact_mono
thf(fact_7831_fact__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).

% fact_mono
thf(fact_7832_fact__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).

% fact_mono
thf(fact_7833_fact__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% fact_mono
thf(fact_7834_fact__dvd,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).

% fact_dvd
thf(fact_7835_fact__dvd,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).

% fact_dvd
thf(fact_7836_fact__dvd,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).

% fact_dvd
thf(fact_7837_fact__dvd,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).

% fact_dvd
thf(fact_7838_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_7839_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_7840_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_7841_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_7842_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_7843_fact__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) )
        = zero_z3403309356797280102nteger ) ) ).

% fact_mod
thf(fact_7844_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_7845_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_7846_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_7847_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_7848_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_7849_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ K ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_7850_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_7851_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_7852_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_7853_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_7854_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_7855_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_7856_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_7857_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_7858_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_7859_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_7860_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_7861_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_7862_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_7863_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_7864_Maclaurin__zero,axiom,
    ! [X3: real,N: nat,Diff: nat > complex > real] :
      ( ( X3 = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_complex ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          = ( Diff @ zero_zero_nat @ zero_zero_complex ) ) ) ) ).

% Maclaurin_zero
thf(fact_7865_Maclaurin__zero,axiom,
    ! [X3: real,N: nat,Diff: nat > real > real] :
      ( ( X3 = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          = ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).

% Maclaurin_zero
thf(fact_7866_Maclaurin__zero,axiom,
    ! [X3: real,N: nat,Diff: nat > rat > real] :
      ( ( X3 = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_rat ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          = ( Diff @ zero_zero_nat @ zero_zero_rat ) ) ) ) ).

% Maclaurin_zero
thf(fact_7867_Maclaurin__zero,axiom,
    ! [X3: real,N: nat,Diff: nat > nat > real] :
      ( ( X3 = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_nat ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          = ( Diff @ zero_zero_nat @ zero_zero_nat ) ) ) ) ).

% Maclaurin_zero
thf(fact_7868_Maclaurin__zero,axiom,
    ! [X3: real,N: nat,Diff: nat > int > real] :
      ( ( X3 = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_int ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          = ( Diff @ zero_zero_nat @ zero_zero_int ) ) ) ) ).

% Maclaurin_zero
thf(fact_7869_Maclaurin__lemma,axiom,
    ! [H2: real,F: real > real,J: nat > real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ? [B9: 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 @ B9 @ ( divide_divide_real @ ( power_power_real @ H2 @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).

% Maclaurin_lemma
thf(fact_7870_cos__paired,axiom,
    ! [X3: real] :
      ( sums_real
      @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( power_power_real @ X3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      @ ( cos_real @ X3 ) ) ).

% cos_paired
thf(fact_7871_cos__coeff__def,axiom,
    ( cos_coeff
    = ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ zero_zero_real ) ) ) ).

% cos_coeff_def
thf(fact_7872_sin__paired,axiom,
    ! [X3: real] :
      ( sums_real
      @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
      @ ( sin_real @ X3 ) ) ).

% sin_paired
thf(fact_7873_Maclaurin__cos__expansion,axiom,
    ! [X3: real,N: nat] :
    ? [T4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X3 ) )
      & ( ( cos_real @ X3 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X3 @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T4 @ ( 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 @ X3 @ N ) ) ) ) ) ).

% Maclaurin_cos_expansion
thf(fact_7874_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ? [T4: real] :
            ( ( ord_less_real @ zero_zero_real @ T4 )
            & ( ord_less_real @ T4 @ X3 )
            & ( ( sin_real @ X3 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X3 @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( 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 @ X3 @ N ) ) ) ) ) ) ) ).

% Maclaurin_sin_expansion3
thf(fact_7875_Maclaurin__sin__expansion4,axiom,
    ! [X3: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ? [T4: real] :
          ( ( ord_less_real @ zero_zero_real @ T4 )
          & ( ord_less_eq_real @ T4 @ X3 )
          & ( ( sin_real @ X3 )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X3 @ M3 ) )
                @ ( set_ord_lessThan_nat @ N ) )
              @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( 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 @ X3 @ N ) ) ) ) ) ) ).

% Maclaurin_sin_expansion4
thf(fact_7876_Maclaurin__sin__expansion2,axiom,
    ! [X3: real,N: nat] :
    ? [T4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X3 ) )
      & ( ( sin_real @ X3 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X3 @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( 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 @ X3 @ N ) ) ) ) ) ).

% Maclaurin_sin_expansion2
thf(fact_7877_Maclaurin__sin__expansion,axiom,
    ! [X3: real,N: nat] :
    ? [T4: real] :
      ( ( sin_real @ X3 )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X3 @ M3 ) )
          @ ( set_ord_lessThan_nat @ N ) )
        @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( 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 @ X3 @ N ) ) ) ) ).

% Maclaurin_sin_expansion
thf(fact_7878_sin__coeff__def,axiom,
    ( sin_coeff
    = ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ).

% sin_coeff_def
thf(fact_7879_fact__ge__self,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_self
thf(fact_7880_fact__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% fact_mono_nat
thf(fact_7881_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_7882_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_7883_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_7884_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_7885_fact__div__fact__le__pow,axiom,
    ! [R: nat,N: nat] :
      ( ( ord_less_eq_nat @ R @ N )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R ) ) ) @ ( power_power_nat @ N @ R ) ) ) ).

% fact_div_fact_le_pow
thf(fact_7886_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).

% choose_dvd
thf(fact_7887_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% choose_dvd
thf(fact_7888_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).

% choose_dvd
thf(fact_7889_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).

% choose_dvd
thf(fact_7890_choose__dvd,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% choose_dvd
thf(fact_7891_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [F: nat > nat > nat,A: nat,B2: nat,Acc: nat] :
      ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A @ ( product_Pair_nat_nat @ B2 @ Acc ) ) ) )
     => ( ( ( ord_less_nat @ B2 @ A )
         => ( ( set_fo2584398358068434914at_nat @ F @ A @ B2 @ Acc )
            = Acc ) )
        & ( ~ ( ord_less_nat @ B2 @ A )
         => ( ( set_fo2584398358068434914at_nat @ F @ A @ B2 @ Acc )
            = ( set_fo2584398358068434914at_nat @ F @ ( plus_plus_nat @ A @ one_one_nat ) @ B2 @ ( F @ A @ Acc ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.psimps
thf(fact_7892_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [X3: nat > nat > nat,Xa2: nat,Xb2: nat,Xc: nat,Y3: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X3 @ Xa2 @ Xb2 @ Xc )
        = Y3 )
     => ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X3 @ ( produc487386426758144856at_nat @ Xa2 @ ( product_Pair_nat_nat @ Xb2 @ Xc ) ) ) )
       => ~ ( ( ( ( ord_less_nat @ Xb2 @ Xa2 )
               => ( Y3 = Xc ) )
              & ( ~ ( ord_less_nat @ Xb2 @ Xa2 )
               => ( Y3
                  = ( set_fo2584398358068434914at_nat @ X3 @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb2 @ ( X3 @ Xa2 @ Xc ) ) ) ) )
           => ~ ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X3 @ ( produc487386426758144856at_nat @ Xa2 @ ( product_Pair_nat_nat @ Xb2 @ Xc ) ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
thf(fact_7893_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [X3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb2: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
      ( ( ( vEBT_V1502963449132264192at_nat @ X3 @ Xa2 @ Xb2 )
        = Y3 )
     => ( ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X3 @ ( produc488173922507101015at_nat @ Xa2 @ Xb2 ) ) )
       => ( ( ( Xa2 = none_P5556105721700978146at_nat )
           => ( ( Y3 = none_P5556105721700978146at_nat )
             => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X3 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Xb2 ) ) ) ) )
         => ( ! [V2: product_prod_nat_nat] :
                ( ( Xa2
                  = ( some_P7363390416028606310at_nat @ V2 ) )
               => ( ( Xb2 = none_P5556105721700978146at_nat )
                 => ( ( Y3 = none_P5556105721700978146at_nat )
                   => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X3 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) ) ) ) )
           => ~ ! [A3: product_prod_nat_nat] :
                  ( ( Xa2
                    = ( some_P7363390416028606310at_nat @ A3 ) )
                 => ! [B3: product_prod_nat_nat] :
                      ( ( Xb2
                        = ( some_P7363390416028606310at_nat @ B3 ) )
                     => ( ( Y3
                          = ( some_P7363390416028606310at_nat @ ( X3 @ A3 @ B3 ) ) )
                       => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X3 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A3 ) @ ( some_P7363390416028606310at_nat @ B3 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
thf(fact_7894_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [X3: num > num > num,Xa2: option_num,Xb2: option_num,Y3: option_num] :
      ( ( ( vEBT_V819420779217536731ft_num @ X3 @ Xa2 @ Xb2 )
        = Y3 )
     => ( ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X3 @ ( produc8585076106096196333on_num @ Xa2 @ Xb2 ) ) )
       => ( ( ( Xa2 = none_num )
           => ( ( Y3 = none_num )
             => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X3 @ ( produc8585076106096196333on_num @ none_num @ Xb2 ) ) ) ) )
         => ( ! [V2: num] :
                ( ( Xa2
                  = ( some_num @ V2 ) )
               => ( ( Xb2 = none_num )
                 => ( ( Y3 = none_num )
                   => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X3 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) ) ) ) )
           => ~ ! [A3: num] :
                  ( ( Xa2
                    = ( some_num @ A3 ) )
                 => ! [B3: num] :
                      ( ( Xb2
                        = ( some_num @ B3 ) )
                     => ( ( Y3
                          = ( some_num @ ( X3 @ A3 @ B3 ) ) )
                       => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X3 @ ( produc8585076106096196333on_num @ ( some_num @ A3 ) @ ( some_num @ B3 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
thf(fact_7895_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [X3: nat > nat > nat,Xa2: option_nat,Xb2: option_nat,Y3: option_nat] :
      ( ( ( vEBT_V4262088993061758097ft_nat @ X3 @ Xa2 @ Xb2 )
        = Y3 )
     => ( ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X3 @ ( produc5098337634421038937on_nat @ Xa2 @ Xb2 ) ) )
       => ( ( ( Xa2 = none_nat )
           => ( ( Y3 = none_nat )
             => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X3 @ ( produc5098337634421038937on_nat @ none_nat @ Xb2 ) ) ) ) )
         => ( ! [V2: nat] :
                ( ( Xa2
                  = ( some_nat @ V2 ) )
               => ( ( Xb2 = none_nat )
                 => ( ( Y3 = none_nat )
                   => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X3 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) ) ) ) )
           => ~ ! [A3: nat] :
                  ( ( Xa2
                    = ( some_nat @ A3 ) )
                 => ! [B3: nat] :
                      ( ( Xb2
                        = ( some_nat @ B3 ) )
                     => ( ( Y3
                          = ( some_nat @ ( X3 @ A3 @ B3 ) ) )
                       => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X3 @ ( produc5098337634421038937on_nat @ ( some_nat @ A3 ) @ ( some_nat @ B3 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
thf(fact_7896_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_7897_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_7898_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_7899_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_7900_fact__fact__dvd__fact,axiom,
    ! [K: nat,N: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_7901_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X3: nat > nat > nat,Xa2: nat,Xb2: nat,Xc: nat,Y3: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X3 @ Xa2 @ Xb2 @ Xc )
        = Y3 )
     => ( ( ( ord_less_nat @ Xb2 @ Xa2 )
         => ( Y3 = Xc ) )
        & ( ~ ( ord_less_nat @ Xb2 @ Xa2 )
         => ( Y3
            = ( set_fo2584398358068434914at_nat @ X3 @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb2 @ ( X3 @ Xa2 @ Xc ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
thf(fact_7902_fold__atLeastAtMost__nat_Osimps,axiom,
    ( set_fo2584398358068434914at_nat
    = ( ^ [F3: nat > nat > nat,A5: nat,B4: nat,Acc2: nat] : ( if_nat @ ( ord_less_nat @ B4 @ A5 ) @ Acc2 @ ( set_fo2584398358068434914at_nat @ F3 @ ( plus_plus_nat @ A5 @ one_one_nat ) @ B4 @ ( F3 @ A5 @ Acc2 ) ) ) ) ) ).

% fold_atLeastAtMost_nat.simps
thf(fact_7903_sum__atLeastAtMost__code,axiom,
    ! [F: nat > complex,A: nat,B2: nat] :
      ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_fo1517530859248394432omplex
        @ ^ [A5: nat] : ( plus_plus_complex @ ( F @ A5 ) )
        @ A
        @ B2
        @ zero_zero_complex ) ) ).

% sum_atLeastAtMost_code
thf(fact_7904_sum__atLeastAtMost__code,axiom,
    ! [F: nat > rat,A: nat,B2: nat] :
      ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_fo1949268297981939178at_rat
        @ ^ [A5: nat] : ( plus_plus_rat @ ( F @ A5 ) )
        @ A
        @ B2
        @ zero_zero_rat ) ) ).

% sum_atLeastAtMost_code
thf(fact_7905_sum__atLeastAtMost__code,axiom,
    ! [F: nat > int,A: nat,B2: nat] :
      ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A5: nat] : ( plus_plus_int @ ( F @ A5 ) )
        @ A
        @ B2
        @ zero_zero_int ) ) ).

% sum_atLeastAtMost_code
thf(fact_7906_sum__atLeastAtMost__code,axiom,
    ! [F: nat > nat,A: nat,B2: nat] :
      ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A5: nat] : ( plus_plus_nat @ ( F @ A5 ) )
        @ A
        @ B2
        @ zero_zero_nat ) ) ).

% sum_atLeastAtMost_code
thf(fact_7907_sum__atLeastAtMost__code,axiom,
    ! [F: nat > real,A: nat,B2: nat] :
      ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_fo3111899725591712190t_real
        @ ^ [A5: nat] : ( plus_plus_real @ ( F @ A5 ) )
        @ A
        @ B2
        @ zero_zero_real ) ) ).

% sum_atLeastAtMost_code
thf(fact_7908_fact__code,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [N2: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_7909_fact__code,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [N2: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_7910_fact__code,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [N2: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_7911_diffs__equiv,axiom,
    ! [C: nat > complex,X3: complex] :
      ( ( summable_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X3 @ N2 ) ) )
     => ( sums_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( C @ N2 ) ) @ ( power_power_complex @ X3 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
        @ ( suminf_complex
          @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X3 @ N2 ) ) ) ) ) ).

% diffs_equiv
thf(fact_7912_diffs__equiv,axiom,
    ! [C: nat > real,X3: real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X3 @ N2 ) ) )
     => ( sums_real
        @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( C @ N2 ) ) @ ( power_power_real @ X3 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
        @ ( suminf_real
          @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ).

% diffs_equiv
thf(fact_7913_tan__double,axiom,
    ! [X3: complex] :
      ( ( ( cos_complex @ X3 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) )
         != zero_zero_complex )
       => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) )
          = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X3 ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% tan_double
thf(fact_7914_tan__double,axiom,
    ! [X3: real] :
      ( ( ( cos_real @ X3 )
       != zero_zero_real )
     => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) )
         != zero_zero_real )
       => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) )
          = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X3 ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% tan_double
thf(fact_7915_in__measure,axiom,
    ! [X3: int,Y3: int,F: int > nat] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ ( measure_int @ F ) )
      = ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) ) ).

% in_measure
thf(fact_7916_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( ( real_V1022390504157884413omplex @ Z )
        = one_one_real )
     => ~ ! [T4: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T4 )
           => ( ( ord_less_real @ T4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( Z
               != ( complex2 @ ( cos_real @ T4 ) @ ( sin_real @ T4 ) ) ) ) ) ) ).

% complex_unimodular_polar
thf(fact_7917_Maclaurin__exp__lt,axiom,
    ! [X3: real,N: nat] :
      ( ( X3 != zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ? [T4: real] :
            ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T4 ) )
            & ( ord_less_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X3 ) )
            & ( ( exp_real @ X3 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X3 @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ) ).

% Maclaurin_exp_lt
thf(fact_7918_exp__less__mono,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ( ord_less_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) ) ) ).

% exp_less_mono
thf(fact_7919_exp__less__cancel__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) )
      = ( ord_less_real @ X3 @ Y3 ) ) ).

% exp_less_cancel_iff
thf(fact_7920_exp__le__cancel__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) )
      = ( ord_less_eq_real @ X3 @ Y3 ) ) ).

% exp_le_cancel_iff
thf(fact_7921_tan__npi,axiom,
    ! [N: nat] :
      ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% tan_npi
thf(fact_7922_one__less__exp__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ one_one_real @ ( exp_real @ X3 ) )
      = ( ord_less_real @ zero_zero_real @ X3 ) ) ).

% one_less_exp_iff
thf(fact_7923_exp__less__one__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( exp_real @ X3 ) @ one_one_real )
      = ( ord_less_real @ X3 @ zero_zero_real ) ) ).

% exp_less_one_iff
thf(fact_7924_one__le__exp__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X3 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).

% one_le_exp_iff
thf(fact_7925_exp__le__one__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X3 ) @ one_one_real )
      = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).

% exp_le_one_iff
thf(fact_7926_tan__periodic__n,axiom,
    ! [X3: real,N: num] :
      ( ( tan_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ N ) @ pi ) ) )
      = ( tan_real @ X3 ) ) ).

% tan_periodic_n
thf(fact_7927_tan__periodic__nat,axiom,
    ! [X3: real,N: nat] :
      ( ( tan_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) ) )
      = ( tan_real @ X3 ) ) ).

% tan_periodic_nat
thf(fact_7928_exp__ln,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( exp_real @ ( ln_ln_real @ X3 ) )
        = X3 ) ) ).

% exp_ln
thf(fact_7929_exp__ln__iff,axiom,
    ! [X3: real] :
      ( ( ( exp_real @ ( ln_ln_real @ X3 ) )
        = X3 )
      = ( ord_less_real @ zero_zero_real @ X3 ) ) ).

% exp_ln_iff
thf(fact_7930_tan__periodic,axiom,
    ! [X3: real] :
      ( ( tan_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( tan_real @ X3 ) ) ).

% tan_periodic
thf(fact_7931_norm__exp,axiom,
    ! [X3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ X3 ) ) @ ( exp_real @ ( real_V7735802525324610683m_real @ X3 ) ) ) ).

% norm_exp
thf(fact_7932_norm__exp,axiom,
    ! [X3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ X3 ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ X3 ) ) ) ).

% norm_exp
thf(fact_7933_exp__less__cancel,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) )
     => ( ord_less_real @ X3 @ Y3 ) ) ).

% exp_less_cancel
thf(fact_7934_exp__times__arg__commute,axiom,
    ! [A4: complex] :
      ( ( times_times_complex @ ( exp_complex @ A4 ) @ A4 )
      = ( times_times_complex @ A4 @ ( exp_complex @ A4 ) ) ) ).

% exp_times_arg_commute
thf(fact_7935_exp__times__arg__commute,axiom,
    ! [A4: real] :
      ( ( times_times_real @ ( exp_real @ A4 ) @ A4 )
      = ( times_times_real @ A4 @ ( exp_real @ A4 ) ) ) ).

% exp_times_arg_commute
thf(fact_7936_exp__total,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ? [X5: real] :
          ( ( exp_real @ X5 )
          = Y3 ) ) ).

% exp_total
thf(fact_7937_exp__gt__zero,axiom,
    ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X3 ) ) ).

% exp_gt_zero
thf(fact_7938_not__exp__less__zero,axiom,
    ! [X3: real] :
      ~ ( ord_less_real @ ( exp_real @ X3 ) @ zero_zero_real ) ).

% not_exp_less_zero
thf(fact_7939_exp__ge__zero,axiom,
    ! [X3: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X3 ) ) ).

% exp_ge_zero
thf(fact_7940_not__exp__le__zero,axiom,
    ! [X3: real] :
      ~ ( ord_less_eq_real @ ( exp_real @ X3 ) @ zero_zero_real ) ).

% not_exp_le_zero
thf(fact_7941_mult__exp__exp,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( times_times_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ Y3 ) )
      = ( exp_complex @ ( plus_plus_complex @ X3 @ Y3 ) ) ) ).

% mult_exp_exp
thf(fact_7942_mult__exp__exp,axiom,
    ! [X3: real,Y3: real] :
      ( ( times_times_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) )
      = ( exp_real @ ( plus_plus_real @ X3 @ Y3 ) ) ) ).

% mult_exp_exp
thf(fact_7943_exp__add__commuting,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( ( times_times_complex @ X3 @ Y3 )
        = ( times_times_complex @ Y3 @ X3 ) )
     => ( ( exp_complex @ ( plus_plus_complex @ X3 @ Y3 ) )
        = ( times_times_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ Y3 ) ) ) ) ).

% exp_add_commuting
thf(fact_7944_exp__add__commuting,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( times_times_real @ X3 @ Y3 )
        = ( times_times_real @ Y3 @ X3 ) )
     => ( ( exp_real @ ( plus_plus_real @ X3 @ Y3 ) )
        = ( times_times_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) ) ) ) ).

% exp_add_commuting
thf(fact_7945_exp__diff,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( exp_complex @ ( minus_minus_complex @ X3 @ Y3 ) )
      = ( divide1717551699836669952omplex @ ( exp_complex @ X3 ) @ ( exp_complex @ Y3 ) ) ) ).

% exp_diff
thf(fact_7946_exp__diff,axiom,
    ! [X3: real,Y3: real] :
      ( ( exp_real @ ( minus_minus_real @ X3 @ Y3 ) )
      = ( divide_divide_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) ) ) ).

% exp_diff
thf(fact_7947_Complex__eq__numeral,axiom,
    ! [A: real,B2: real,W: num] :
      ( ( ( complex2 @ A @ B2 )
        = ( numera6690914467698888265omplex @ W ) )
      = ( ( A
          = ( numeral_numeral_real @ W ) )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_numeral
thf(fact_7948_exp__gt__one,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ord_less_real @ one_one_real @ ( exp_real @ X3 ) ) ) ).

% exp_gt_one
thf(fact_7949_exp__ge__add__one__self,axiom,
    ! [X3: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( exp_real @ X3 ) ) ).

% exp_ge_add_one_self
thf(fact_7950_exp__minus__inverse,axiom,
    ! [X3: real] :
      ( ( times_times_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) )
      = one_one_real ) ).

% exp_minus_inverse
thf(fact_7951_exp__minus__inverse,axiom,
    ! [X3: complex] :
      ( ( times_times_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) )
      = one_one_complex ) ).

% exp_minus_inverse
thf(fact_7952_exp__of__nat2__mult,axiom,
    ! [X3: complex,N: nat] :
      ( ( exp_complex @ ( times_times_complex @ X3 @ ( semiri8010041392384452111omplex @ N ) ) )
      = ( power_power_complex @ ( exp_complex @ X3 ) @ N ) ) ).

% exp_of_nat2_mult
thf(fact_7953_exp__of__nat2__mult,axiom,
    ! [X3: real,N: nat] :
      ( ( exp_real @ ( times_times_real @ X3 @ ( semiri5074537144036343181t_real @ N ) ) )
      = ( power_power_real @ ( exp_real @ X3 ) @ N ) ) ).

% exp_of_nat2_mult
thf(fact_7954_exp__of__nat__mult,axiom,
    ! [N: nat,X3: complex] :
      ( ( exp_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ X3 ) )
      = ( power_power_complex @ ( exp_complex @ X3 ) @ N ) ) ).

% exp_of_nat_mult
thf(fact_7955_exp__of__nat__mult,axiom,
    ! [N: nat,X3: real] :
      ( ( exp_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X3 ) )
      = ( power_power_real @ ( exp_real @ X3 ) @ N ) ) ).

% exp_of_nat_mult
thf(fact_7956_Complex__eq__neg__numeral,axiom,
    ! [A: real,B2: real,W: num] :
      ( ( ( complex2 @ A @ B2 )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( A
          = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
thf(fact_7957_complex__mult,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( times_times_complex @ ( complex2 @ A @ B2 ) @ ( complex2 @ C @ D ) )
      = ( complex2 @ ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) @ ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B2 @ C ) ) ) ) ).

% complex_mult
thf(fact_7958_tan__def,axiom,
    ( tan_complex
    = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ X ) @ ( cos_complex @ X ) ) ) ) ).

% tan_def
thf(fact_7959_tan__def,axiom,
    ( tan_real
    = ( ^ [X: real] : ( divide_divide_real @ ( sin_real @ X ) @ ( cos_real @ X ) ) ) ) ).

% tan_def
thf(fact_7960_exp__ge__add__one__self__aux,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( exp_real @ X3 ) ) ) ).

% exp_ge_add_one_self_aux
thf(fact_7961_lemma__exp__total,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ one_one_real @ Y3 )
     => ? [X5: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X5 )
          & ( ord_less_eq_real @ X5 @ ( minus_minus_real @ Y3 @ one_one_real ) )
          & ( ( exp_real @ X5 )
            = Y3 ) ) ) ).

% lemma_exp_total
thf(fact_7962_ln__ge__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ Y3 @ ( ln_ln_real @ X3 ) )
        = ( ord_less_eq_real @ ( exp_real @ Y3 ) @ X3 ) ) ) ).

% ln_ge_iff
thf(fact_7963_ln__x__over__x__mono,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X3 )
     => ( ( ord_less_eq_real @ X3 @ Y3 )
       => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y3 ) @ Y3 ) @ ( divide_divide_real @ ( ln_ln_real @ X3 ) @ X3 ) ) ) ) ).

% ln_x_over_x_mono
thf(fact_7964_diffs__def,axiom,
    ( diffs_rat
    = ( ^ [C3: nat > rat,N2: nat] : ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) @ ( C3 @ ( suc @ N2 ) ) ) ) ) ).

% diffs_def
thf(fact_7965_diffs__def,axiom,
    ( diffs_int
    = ( ^ [C3: nat > int,N2: nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) @ ( C3 @ ( suc @ N2 ) ) ) ) ) ).

% diffs_def
thf(fact_7966_diffs__def,axiom,
    ( diffs_real
    = ( ^ [C3: nat > real,N2: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( C3 @ ( suc @ N2 ) ) ) ) ) ).

% diffs_def
thf(fact_7967_exp__le,axiom,
    ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).

% exp_le
thf(fact_7968_termdiff__converges__all,axiom,
    ! [C: nat > complex,X3: complex] :
      ( ! [X5: complex] :
          ( summable_complex
          @ ^ [N2: nat] : ( times_times_complex @ ( C @ N2 ) @ ( power_power_complex @ X5 @ N2 ) ) )
     => ( summable_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X3 @ N2 ) ) ) ) ).

% termdiff_converges_all
thf(fact_7969_termdiff__converges__all,axiom,
    ! [C: nat > real,X3: real] :
      ( ! [X5: real] :
          ( summable_real
          @ ^ [N2: nat] : ( times_times_real @ ( C @ N2 ) @ ( power_power_real @ X5 @ N2 ) ) )
     => ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ).

% termdiff_converges_all
thf(fact_7970_exp__divide__power__eq,axiom,
    ! [N: nat,X3: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X3 @ ( semiri8010041392384452111omplex @ N ) ) ) @ N )
        = ( exp_complex @ X3 ) ) ) ).

% exp_divide_power_eq
thf(fact_7971_exp__divide__power__eq,axiom,
    ! [N: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X3 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
        = ( exp_real @ X3 ) ) ) ).

% exp_divide_power_eq
thf(fact_7972_tanh__altdef,axiom,
    ( tanh_real
    = ( ^ [X: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ) ).

% tanh_altdef
thf(fact_7973_tanh__altdef,axiom,
    ( tanh_complex
    = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ) ) ) ).

% tanh_altdef
thf(fact_7974_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_7975_tan__45,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = one_one_real ) ).

% tan_45
thf(fact_7976_exp__double,axiom,
    ! [Z: complex] :
      ( ( exp_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) )
      = ( power_power_complex @ ( exp_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% exp_double
thf(fact_7977_exp__double,axiom,
    ! [Z: real] :
      ( ( exp_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) )
      = ( power_power_real @ ( exp_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% exp_double
thf(fact_7978_lemma__tan__total,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ? [X5: real] :
          ( ( ord_less_real @ zero_zero_real @ X5 )
          & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ord_less_real @ Y3 @ ( tan_real @ X5 ) ) ) ) ).

% lemma_tan_total
thf(fact_7979_tan__gt__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( tan_real @ X3 ) ) ) ) ).

% tan_gt_zero
thf(fact_7980_lemma__tan__total1,axiom,
    ! [Y3: real] :
    ? [X5: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
      & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X5 )
        = Y3 ) ) ).

% lemma_tan_total1
thf(fact_7981_tan__mono__lt__eq,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
     => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
         => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) )
              = ( ord_less_real @ X3 @ Y3 ) ) ) ) ) ) ).

% tan_mono_lt_eq
thf(fact_7982_tan__monotone_H,axiom,
    ! [Y3: real,X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
     => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
         => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ Y3 @ X3 )
              = ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X3 ) ) ) ) ) ) ) ).

% tan_monotone'
thf(fact_7983_tan__monotone,axiom,
    ! [Y3: real,X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
     => ( ( ord_less_real @ Y3 @ X3 )
       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X3 ) ) ) ) ) ).

% tan_monotone
thf(fact_7984_tan__total,axiom,
    ! [Y3: real] :
    ? [X5: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
      & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X5 )
        = Y3 )
      & ! [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 )
              = Y3 ) )
         => ( Y5 = X5 ) ) ) ).

% tan_total
thf(fact_7985_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_7986_tan__inverse,axiom,
    ! [Y3: real] :
      ( ( divide_divide_real @ one_one_real @ ( tan_real @ Y3 ) )
      = ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 ) ) ) ).

% tan_inverse
thf(fact_7987_exp__bound__half,axiom,
    ! [Z: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% exp_bound_half
thf(fact_7988_exp__bound__half,axiom,
    ! [Z: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% exp_bound_half
thf(fact_7989_add__tan__eq,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( ( cos_complex @ X3 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y3 )
         != zero_zero_complex )
       => ( ( plus_plus_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y3 ) )
          = ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ Y3 ) ) ) ) ) ) ).

% add_tan_eq
thf(fact_7990_add__tan__eq,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( cos_real @ X3 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y3 )
         != zero_zero_real )
       => ( ( plus_plus_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) )
          = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) ) ) ) ) ).

% add_tan_eq
thf(fact_7991_termdiff__converges,axiom,
    ! [X3: real,K5: real,C: nat > real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ K5 )
     => ( ! [X5: real] :
            ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X5 ) @ K5 )
           => ( summable_real
              @ ^ [N2: nat] : ( times_times_real @ ( C @ N2 ) @ ( power_power_real @ X5 @ N2 ) ) ) )
       => ( summable_real
          @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ).

% termdiff_converges
thf(fact_7992_termdiff__converges,axiom,
    ! [X3: complex,K5: real,C: nat > complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ K5 )
     => ( ! [X5: complex] :
            ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X5 ) @ K5 )
           => ( summable_complex
              @ ^ [N2: nat] : ( times_times_complex @ ( C @ N2 ) @ ( power_power_complex @ X5 @ N2 ) ) ) )
       => ( summable_complex
          @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X3 @ N2 ) ) ) ) ) ).

% termdiff_converges
thf(fact_7993_exp__bound,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ X3 @ one_one_real )
       => ( ord_less_eq_real @ ( exp_real @ X3 ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% exp_bound
thf(fact_7994_tan__pos__pi2__le,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X3 ) ) ) ) ).

% tan_pos_pi2_le
thf(fact_7995_tan__total__pos,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ? [X5: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X5 )
          & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( tan_real @ X5 )
            = Y3 ) ) ) ).

% tan_total_pos
thf(fact_7996_tan__less__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 )
     => ( ( ord_less_real @ X3 @ zero_zero_real )
       => ( ord_less_real @ ( tan_real @ X3 ) @ zero_zero_real ) ) ) ).

% tan_less_zero
thf(fact_7997_tan__mono__le__eq,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
     => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
         => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) )
              = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ) ) ).

% tan_mono_le_eq
thf(fact_7998_tan__mono__le,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
     => ( ( ord_less_eq_real @ X3 @ Y3 )
       => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) ) ) ) ) ).

% tan_mono_le
thf(fact_7999_tan__bound__pi2,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
     => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X3 ) ) @ one_one_real ) ) ).

% tan_bound_pi2
thf(fact_8000_arctan__unique,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
     => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ( tan_real @ X3 )
            = Y3 )
         => ( ( arctan @ Y3 )
            = X3 ) ) ) ) ).

% arctan_unique
thf(fact_8001_arctan__tan,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
     => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( arctan @ ( tan_real @ X3 ) )
          = X3 ) ) ) ).

% arctan_tan
thf(fact_8002_arctan,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
      & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ ( arctan @ Y3 ) )
        = Y3 ) ) ).

% arctan
thf(fact_8003_lemma__tan__add1,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( ( cos_complex @ X3 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y3 )
         != zero_zero_complex )
       => ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y3 ) ) )
          = ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ Y3 ) ) ) ) ) ) ).

% lemma_tan_add1
thf(fact_8004_lemma__tan__add1,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( cos_real @ X3 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y3 )
         != zero_zero_real )
       => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) ) )
          = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) ) ) ) ) ).

% lemma_tan_add1
thf(fact_8005_tan__diff,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( ( cos_complex @ X3 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y3 )
         != zero_zero_complex )
       => ( ( ( cos_complex @ ( minus_minus_complex @ X3 @ Y3 ) )
           != zero_zero_complex )
         => ( ( tan_complex @ ( minus_minus_complex @ X3 @ Y3 ) )
            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y3 ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y3 ) ) ) ) ) ) ) ) ).

% tan_diff
thf(fact_8006_tan__diff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( cos_real @ X3 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y3 )
         != zero_zero_real )
       => ( ( ( cos_real @ ( minus_minus_real @ X3 @ Y3 ) )
           != zero_zero_real )
         => ( ( tan_real @ ( minus_minus_real @ X3 @ Y3 ) )
            = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) ) ) ) ) ) ) ) ).

% tan_diff
thf(fact_8007_tan__add,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( ( cos_complex @ X3 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y3 )
         != zero_zero_complex )
       => ( ( ( cos_complex @ ( plus_plus_complex @ X3 @ Y3 ) )
           != zero_zero_complex )
         => ( ( tan_complex @ ( plus_plus_complex @ X3 @ Y3 ) )
            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y3 ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y3 ) ) ) ) ) ) ) ) ).

% tan_add
thf(fact_8008_tan__add,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( cos_real @ X3 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y3 )
         != zero_zero_real )
       => ( ( ( cos_real @ ( plus_plus_real @ X3 @ Y3 ) )
           != zero_zero_real )
         => ( ( tan_real @ ( plus_plus_real @ X3 @ Y3 ) )
            = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) ) ) ) ) ) ) ) ).

% tan_add
thf(fact_8009_real__exp__bound__lemma,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( exp_real @ X3 ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) ) ) ) ).

% real_exp_bound_lemma
thf(fact_8010_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,X3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X3 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X3 ) ) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
thf(fact_8011_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X3: real,N: nat] :
      ( ( ord_less_eq_real @ X3 @ ( 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 @ X3 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
thf(fact_8012_tan__total__pi4,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ? [Z4: real] :
          ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z4 )
          & ( ord_less_real @ Z4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
          & ( ( tan_real @ Z4 )
            = X3 ) ) ) ).

% tan_total_pi4
thf(fact_8013_exp__bound__lemma,axiom,
    ! [Z: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V7735802525324610683m_real @ Z ) ) ) ) ) ).

% exp_bound_lemma
thf(fact_8014_exp__bound__lemma,axiom,
    ! [Z: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ) ).

% exp_bound_lemma
thf(fact_8015_Maclaurin__exp__le,axiom,
    ! [X3: real,N: nat] :
    ? [T4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X3 ) )
      & ( ( exp_real @ X3 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X3 @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ).

% Maclaurin_exp_le
thf(fact_8016_exp__lower__Taylor__quadratic,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( divide_divide_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X3 ) ) ) ).

% exp_lower_Taylor_quadratic
thf(fact_8017_tanh__real__altdef,axiom,
    ( tanh_real
    = ( ^ [X: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) ) ) ) ).

% tanh_real_altdef
thf(fact_8018_tan__half,axiom,
    ( tan_complex
    = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) ) @ one_one_complex ) ) ) ) ).

% tan_half
thf(fact_8019_tan__half,axiom,
    ( tan_real
    = ( ^ [X: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ) ).

% tan_half
thf(fact_8020_binomial__code,axiom,
    ( binomial
    = ( ^ [N2: nat,K2: nat] : ( if_nat @ ( ord_less_nat @ N2 @ K2 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) ) @ ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K2 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N2 @ K2 ) @ one_one_nat ) @ N2 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K2 ) ) ) ) ) ) ).

% binomial_code
thf(fact_8021_in__finite__psubset,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A4 @ B5 ) @ finite_psubset_nat )
      = ( ( ord_less_set_nat @ A4 @ B5 )
        & ( finite_finite_nat @ B5 ) ) ) ).

% in_finite_psubset
thf(fact_8022_in__finite__psubset,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ A4 @ B5 ) @ finite_psubset_int )
      = ( ( ord_less_set_int @ A4 @ B5 )
        & ( finite_finite_int @ B5 ) ) ) ).

% in_finite_psubset
thf(fact_8023_in__finite__psubset,axiom,
    ! [A4: set_complex,B5: set_complex] :
      ( ( member351165363924911826omplex @ ( produc3790773574474814305omplex @ A4 @ B5 ) @ finite8643634255014194347omplex )
      = ( ( ord_less_set_complex @ A4 @ B5 )
        & ( finite3207457112153483333omplex @ B5 ) ) ) ).

% in_finite_psubset
thf(fact_8024_sin__tan,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( sin_real @ X3 )
        = ( divide_divide_real @ ( tan_real @ X3 ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_tan
thf(fact_8025_cos__tan,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( cos_real @ X3 )
        = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_tan
thf(fact_8026_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_8027_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_8028_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_8029_real__sqrt__eq__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( sqrt @ X3 )
        = ( sqrt @ Y3 ) )
      = ( X3 = Y3 ) ) ).

% real_sqrt_eq_iff
thf(fact_8030_real__sqrt__zero,axiom,
    ( ( sqrt @ zero_zero_real )
    = zero_zero_real ) ).

% real_sqrt_zero
thf(fact_8031_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [X3: real] :
      ( ( ( sqrt @ X3 )
        = zero_zero_real )
      = ( X3 = zero_zero_real ) ) ).

% real_sqrt_eq_zero_cancel_iff
thf(fact_8032_real__sqrt__less__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) )
      = ( ord_less_real @ X3 @ Y3 ) ) ).

% real_sqrt_less_iff
thf(fact_8033_real__sqrt__le__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) )
      = ( ord_less_eq_real @ X3 @ Y3 ) ) ).

% real_sqrt_le_iff
thf(fact_8034_real__sqrt__one,axiom,
    ( ( sqrt @ one_one_real )
    = one_one_real ) ).

% real_sqrt_one
thf(fact_8035_real__sqrt__eq__1__iff,axiom,
    ! [X3: real] :
      ( ( ( sqrt @ X3 )
        = one_one_real )
      = ( X3 = one_one_real ) ) ).

% real_sqrt_eq_1_iff
thf(fact_8036_real__sqrt__gt__0__iff,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y3 ) )
      = ( ord_less_real @ zero_zero_real @ Y3 ) ) ).

% real_sqrt_gt_0_iff
thf(fact_8037_real__sqrt__lt__0__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( sqrt @ X3 ) @ zero_zero_real )
      = ( ord_less_real @ X3 @ zero_zero_real ) ) ).

% real_sqrt_lt_0_iff
thf(fact_8038_real__sqrt__ge__0__iff,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y3 ) )
      = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ).

% real_sqrt_ge_0_iff
thf(fact_8039_real__sqrt__le__0__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X3 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).

% real_sqrt_le_0_iff
thf(fact_8040_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_8041_binomial__Suc__Suc,axiom,
    ! [N: nat,K: nat] :
      ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
      = ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).

% binomial_Suc_Suc
thf(fact_8042_real__sqrt__lt__1__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( sqrt @ X3 ) @ one_one_real )
      = ( ord_less_real @ X3 @ one_one_real ) ) ).

% real_sqrt_lt_1_iff
thf(fact_8043_real__sqrt__gt__1__iff,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ one_one_real @ ( sqrt @ Y3 ) )
      = ( ord_less_real @ one_one_real @ Y3 ) ) ).

% real_sqrt_gt_1_iff
thf(fact_8044_real__sqrt__ge__1__iff,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y3 ) )
      = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ).

% real_sqrt_ge_1_iff
thf(fact_8045_real__sqrt__le__1__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X3 ) @ one_one_real )
      = ( ord_less_eq_real @ X3 @ one_one_real ) ) ).

% real_sqrt_le_1_iff
thf(fact_8046_real__sqrt__mult__self,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
      = ( abs_abs_real @ A ) ) ).

% real_sqrt_mult_self
thf(fact_8047_real__sqrt__abs2,axiom,
    ! [X3: real] :
      ( ( sqrt @ ( times_times_real @ X3 @ X3 ) )
      = ( abs_abs_real @ X3 ) ) ).

% real_sqrt_abs2
thf(fact_8048_real__sqrt__four,axiom,
    ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% real_sqrt_four
thf(fact_8049_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_8050_real__sqrt__abs,axiom,
    ! [X3: real] :
      ( ( sqrt @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( abs_abs_real @ X3 ) ) ).

% real_sqrt_abs
thf(fact_8051_real__sqrt__pow2,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( power_power_real @ ( sqrt @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X3 ) ) ).

% real_sqrt_pow2
thf(fact_8052_real__sqrt__pow2__iff,axiom,
    ! [X3: real] :
      ( ( ( power_power_real @ ( sqrt @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X3 )
      = ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).

% real_sqrt_pow2_iff
thf(fact_8053_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X3: real,Y3: real,Xa2: real,Ya: real] :
      ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_squared_eq
thf(fact_8054_real__sqrt__less__mono,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ( ord_less_real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) ) ) ).

% real_sqrt_less_mono
thf(fact_8055_real__sqrt__le__mono,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ X3 @ Y3 )
     => ( ord_less_eq_real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) ) ) ).

% real_sqrt_le_mono
thf(fact_8056_real__sqrt__minus,axiom,
    ! [X3: real] :
      ( ( sqrt @ ( uminus_uminus_real @ X3 ) )
      = ( uminus_uminus_real @ ( sqrt @ X3 ) ) ) ).

% real_sqrt_minus
thf(fact_8057_real__sqrt__power,axiom,
    ! [X3: real,K: nat] :
      ( ( sqrt @ ( power_power_real @ X3 @ K ) )
      = ( power_power_real @ ( sqrt @ X3 ) @ K ) ) ).

% real_sqrt_power
thf(fact_8058_real__sqrt__mult,axiom,
    ! [X3: real,Y3: real] :
      ( ( sqrt @ ( times_times_real @ X3 @ Y3 ) )
      = ( times_times_real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) ) ) ).

% real_sqrt_mult
thf(fact_8059_real__sqrt__divide,axiom,
    ! [X3: real,Y3: real] :
      ( ( sqrt @ ( divide_divide_real @ X3 @ Y3 ) )
      = ( divide_divide_real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) ) ) ).

% real_sqrt_divide
thf(fact_8060_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( binomial @ N @ K )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_8061_Suc__times__binomial,axiom,
    ! [K: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
      = ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).

% Suc_times_binomial
thf(fact_8062_Suc__times__binomial__eq,axiom,
    ! [N: nat,K: nat] :
      ( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
      = ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).

% Suc_times_binomial_eq
thf(fact_8063_binomial__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( binomial @ N @ K )
        = ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% binomial_symmetric
thf(fact_8064_real__sqrt__gt__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ord_less_real @ zero_zero_real @ ( sqrt @ X3 ) ) ) ).

% real_sqrt_gt_zero
thf(fact_8065_real__sqrt__eq__zero__cancel,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ( sqrt @ X3 )
          = zero_zero_real )
       => ( X3 = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
thf(fact_8066_real__sqrt__ge__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X3 ) ) ) ).

% real_sqrt_ge_zero
thf(fact_8067_choose__mult__lemma,axiom,
    ! [M: nat,R: nat,K: nat] :
      ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
      = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R ) @ M ) ) ) ).

% choose_mult_lemma
thf(fact_8068_real__sqrt__ge__one,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ one_one_real @ X3 )
     => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X3 ) ) ) ).

% real_sqrt_ge_one
thf(fact_8069_binomial__le__pow,axiom,
    ! [R: nat,N: nat] :
      ( ( ord_less_eq_nat @ R @ N )
     => ( ord_less_eq_nat @ ( binomial @ N @ R ) @ ( power_power_nat @ N @ R ) ) ) ).

% binomial_le_pow
thf(fact_8070_pochhammer__pos,axiom,
    ! [X3: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X3 @ N ) ) ) ).

% pochhammer_pos
thf(fact_8071_pochhammer__pos,axiom,
    ! [X3: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X3 )
     => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X3 @ N ) ) ) ).

% pochhammer_pos
thf(fact_8072_pochhammer__pos,axiom,
    ! [X3: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X3 )
     => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X3 @ N ) ) ) ).

% pochhammer_pos
thf(fact_8073_pochhammer__pos,axiom,
    ! [X3: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ X3 )
     => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X3 @ N ) ) ) ).

% pochhammer_pos
thf(fact_8074_pochhammer__neq__0__mono,axiom,
    ! [A: complex,M: nat,N: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ M )
       != zero_zero_complex )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( comm_s2602460028002588243omplex @ A @ N )
         != zero_zero_complex ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_8075_pochhammer__neq__0__mono,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ M )
       != zero_zero_real )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( comm_s7457072308508201937r_real @ A @ N )
         != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_8076_pochhammer__neq__0__mono,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ M )
       != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( comm_s4028243227959126397er_rat @ A @ N )
         != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_8077_pochhammer__eq__0__mono,axiom,
    ! [A: complex,N: nat,M: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ N )
        = zero_zero_complex )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( comm_s2602460028002588243omplex @ A @ M )
          = zero_zero_complex ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_8078_pochhammer__eq__0__mono,axiom,
    ! [A: real,N: nat,M: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ N )
        = zero_zero_real )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( comm_s7457072308508201937r_real @ A @ M )
          = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_8079_pochhammer__eq__0__mono,axiom,
    ! [A: rat,N: nat,M: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ N )
        = zero_zero_rat )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( comm_s4028243227959126397er_rat @ A @ M )
          = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_8080_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_8081_Suc__times__binomial__add,axiom,
    ! [A: nat,B2: nat] :
      ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B2 ) ) @ ( suc @ A ) ) )
      = ( times_times_nat @ ( suc @ B2 ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B2 ) ) @ A ) ) ) ).

% Suc_times_binomial_add
thf(fact_8082_binomial__Suc__Suc__eq__times,axiom,
    ! [N: nat,K: nat] :
      ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).

% binomial_Suc_Suc_eq_times
thf(fact_8083_choose__mult,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
          = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).

% choose_mult
thf(fact_8084_real__div__sqrt,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( divide_divide_real @ X3 @ ( sqrt @ X3 ) )
        = ( sqrt @ X3 ) ) ) ).

% real_div_sqrt
thf(fact_8085_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_8086_sqrt__add__le__add__sqrt,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X3 @ Y3 ) ) @ ( plus_plus_real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) ) ) ) ) ).

% sqrt_add_le_add_sqrt
thf(fact_8087_le__real__sqrt__sumsq,axiom,
    ! [X3: real,Y3: real] : ( ord_less_eq_real @ X3 @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ) ).

% le_real_sqrt_sumsq
thf(fact_8088_pochhammer__nonneg,axiom,
    ! [X3: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X3 @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_8089_pochhammer__nonneg,axiom,
    ! [X3: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X3 )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X3 @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_8090_pochhammer__nonneg,axiom,
    ! [X3: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X3 )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X3 @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_8091_pochhammer__nonneg,axiom,
    ! [X3: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ X3 )
     => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X3 @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_8092_sqrt2__less__2,axiom,
    ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% sqrt2_less_2
thf(fact_8093_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_8094_binomial__fact__lemma,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
        = ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% binomial_fact_lemma
thf(fact_8095_pochhammer__rec,axiom,
    ! [A: complex,N: nat] :
      ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
      = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_8096_pochhammer__rec,axiom,
    ! [A: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
      = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_8097_pochhammer__rec,axiom,
    ! [A: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
      = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_8098_pochhammer__rec,axiom,
    ! [A: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
      = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_8099_pochhammer__rec,axiom,
    ! [A: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
      = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_8100_pochhammer__Suc,axiom,
    ! [A: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_8101_pochhammer__Suc,axiom,
    ! [A: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_8102_pochhammer__Suc,axiom,
    ! [A: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
      = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_8103_pochhammer__Suc,axiom,
    ! [A: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).

% pochhammer_Suc
thf(fact_8104_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_8105_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_8106_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_8107_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_8108_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_8109_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_8110_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_8111_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_8112_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_8113_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_8114_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_8115_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_8116_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_8117_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_8118_pochhammer__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ N )
        = zero_zero_complex )
      = ( ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N )
            & ( A
              = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K2 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_8119_pochhammer__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ N )
        = zero_zero_rat )
      = ( ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N )
            & ( A
              = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K2 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_8120_pochhammer__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ N )
        = zero_zero_real )
      = ( ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N )
            & ( A
              = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K2 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_8121_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_8122_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_8123_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_8124_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_8125_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_8126_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_8127_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_8128_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_8129_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_8130_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_8131_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_8132_binomial__mono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
       => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).

% binomial_mono
thf(fact_8133_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_8134_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_8135_binomial__antimono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
       => ( ( ord_less_eq_nat @ K6 @ N )
         => ( ord_less_eq_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).

% binomial_antimono
thf(fact_8136_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_8137_real__less__rsqrt,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
     => ( ord_less_real @ X3 @ ( sqrt @ Y3 ) ) ) ).

% real_less_rsqrt
thf(fact_8138_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_8139_sqrt__le__D,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X3 ) @ Y3 )
     => ( ord_less_eq_real @ X3 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sqrt_le_D
thf(fact_8140_real__le__rsqrt,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
     => ( ord_less_eq_real @ X3 @ ( sqrt @ Y3 ) ) ) ).

% real_le_rsqrt
thf(fact_8141_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_8142_tan__60,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
    = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).

% tan_60
thf(fact_8143_binomial__altdef__nat,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( binomial @ N @ K )
        = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).

% binomial_altdef_nat
thf(fact_8144_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_8145_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_8146_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_8147_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_8148_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_8149_binomial__strict__mono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( ( ord_less_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
       => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).

% binomial_strict_mono
thf(fact_8150_binomial__strict__antimono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( ( ord_less_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
       => ( ( ord_less_eq_nat @ K6 @ N )
         => ( ord_less_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).

% binomial_strict_antimono
thf(fact_8151_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_8152_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_8153_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_8154_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_8155_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_8156_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_8157_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_8158_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_8159_real__le__lsqrt,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_eq_real @ X3 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sqrt @ X3 ) @ Y3 ) ) ) ) ).

% real_le_lsqrt
thf(fact_8160_real__sqrt__unique,axiom,
    ! [Y3: real,X3: real] :
      ( ( ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( sqrt @ X3 )
          = Y3 ) ) ) ).

% real_sqrt_unique
thf(fact_8161_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( ( ord_less_real @ zero_zero_real @ U )
     => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).

% lemma_real_divide_sqrt_less
thf(fact_8162_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = X3 )
     => ( Y3 = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
thf(fact_8163_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = Y3 )
     => ( X3 = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
thf(fact_8164_real__sqrt__sum__squares__ge1,axiom,
    ! [X3: real,Y3: real] : ( ord_less_eq_real @ X3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge1
thf(fact_8165_real__sqrt__sum__squares__ge2,axiom,
    ! [Y3: real,X3: real] : ( ord_less_eq_real @ Y3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge2
thf(fact_8166_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C: real,B2: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B2 @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ 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_8167_sqrt__ge__absD,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ ( sqrt @ Y3 ) )
     => ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 ) ) ).

% sqrt_ge_absD
thf(fact_8168_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_8169_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_8170_pochhammer__absorb__comp,axiom,
    ! [R: complex,K: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ R @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R ) @ K ) )
      = ( times_times_complex @ R @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R ) @ one_one_complex ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_8171_pochhammer__absorb__comp,axiom,
    ! [R: code_integer,K: nat] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R ) @ K ) )
      = ( times_3573771949741848930nteger @ R @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R ) @ one_one_Code_integer ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_8172_pochhammer__absorb__comp,axiom,
    ! [R: rat,K: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ R @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R ) @ K ) )
      = ( times_times_rat @ R @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R ) @ one_one_rat ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_8173_pochhammer__absorb__comp,axiom,
    ! [R: int,K: nat] :
      ( ( times_times_int @ ( minus_minus_int @ R @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R ) @ K ) )
      = ( times_times_int @ R @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R ) @ one_one_int ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_8174_pochhammer__absorb__comp,axiom,
    ! [R: real,K: nat] :
      ( ( times_times_real @ ( minus_minus_real @ R @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R ) @ K ) )
      = ( times_times_real @ R @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R ) @ one_one_real ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_8175_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_8176_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_8177_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_8178_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_8179_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_8180_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_8181_real__less__lsqrt,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_real @ X3 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sqrt @ X3 ) @ Y3 ) ) ) ) ).

% real_less_lsqrt
thf(fact_8182_sqrt__sum__squares__le__sum,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X3 @ Y3 ) ) ) ) ).

% sqrt_sum_squares_le_sum
thf(fact_8183_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_8184_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_8185_real__sqrt__ge__abs1,axiom,
    ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs1
thf(fact_8186_real__sqrt__ge__abs2,axiom,
    ! [Y3: real,X3: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs2
thf(fact_8187_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X3 ) @ ( abs_abs_real @ Y3 ) ) ) ).

% sqrt_sum_squares_le_sum_abs
thf(fact_8188_ln__sqrt,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ln_ln_real @ ( sqrt @ X3 ) )
        = ( divide_divide_real @ ( ln_ln_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% ln_sqrt
thf(fact_8189_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_8190_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_8191_complex__norm,axiom,
    ! [X3: real,Y3: real] :
      ( ( real_V1022390504157884413omplex @ ( complex2 @ X3 @ Y3 ) )
      = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_norm
thf(fact_8192_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_8193_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_8194_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_8195_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_8196_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_8197_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_8198_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_8199_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_8200_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_8201_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_8202_finite__psubset__def,axiom,
    ( finite_psubset_nat
    = ( collec6662362479098859352et_nat
      @ ( produc6247414631856714078_nat_o
        @ ^ [A6: set_nat,B6: set_nat] :
            ( ( ord_less_set_nat @ A6 @ B6 )
            & ( finite_finite_nat @ B6 ) ) ) ) ) ).

% finite_psubset_def
thf(fact_8203_finite__psubset__def,axiom,
    ( finite_psubset_int
    = ( collec957716948307931664et_int
      @ ( produc4109468873575309990_int_o
        @ ^ [A6: set_int,B6: set_int] :
            ( ( ord_less_set_int @ A6 @ B6 )
            & ( finite_finite_int @ B6 ) ) ) ) ) ).

% finite_psubset_def
thf(fact_8204_finite__psubset__def,axiom,
    ( finite8643634255014194347omplex
    = ( collec5108298041176329748omplex
      @ ( produc3914248068834153634plex_o
        @ ^ [A6: set_complex,B6: set_complex] :
            ( ( ord_less_set_complex @ A6 @ B6 )
            & ( finite3207457112153483333omplex @ B6 ) ) ) ) ) ).

% finite_psubset_def
thf(fact_8205_arsinh__real__aux,axiom,
    ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% arsinh_real_aux
thf(fact_8206_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X3: real,Y3: real,Xa2: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_ge_zero
thf(fact_8207_real__sqrt__power__even,axiom,
    ! [N: nat,X3: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
       => ( ( power_power_real @ ( sqrt @ X3 ) @ N )
          = ( power_power_real @ X3 @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_power_even
thf(fact_8208_arith__geo__mean__sqrt,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X3 @ Y3 ) ) @ ( divide_divide_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arith_geo_mean_sqrt
thf(fact_8209_cos__x__y__le__one,axiom,
    ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).

% cos_x_y_le_one
thf(fact_8210_real__sqrt__sum__squares__less,axiom,
    ! [X3: real,U: real,Y3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
     => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).

% real_sqrt_sum_squares_less
thf(fact_8211_cos__arctan,axiom,
    ! [X3: real] :
      ( ( cos_real @ ( arctan @ X3 ) )
      = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_arctan
thf(fact_8212_sin__arctan,axiom,
    ! [X3: real] :
      ( ( sin_real @ ( arctan @ X3 ) )
      = ( divide_divide_real @ X3 @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_arctan
thf(fact_8213_pochhammer__code,axiom,
    ( comm_s2602460028002588243omplex
    = ( ^ [A5: complex,N2: nat] :
          ( if_complex @ ( N2 = zero_zero_nat ) @ one_one_complex
          @ ( set_fo1517530859248394432omplex
            @ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A5 @ ( semiri8010041392384452111omplex @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N2 @ one_one_nat )
            @ one_one_complex ) ) ) ) ).

% pochhammer_code
thf(fact_8214_pochhammer__code,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A5: rat,N2: nat] :
          ( if_rat @ ( N2 = zero_zero_nat ) @ one_one_rat
          @ ( set_fo1949268297981939178at_rat
            @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A5 @ ( semiri681578069525770553at_rat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N2 @ one_one_nat )
            @ one_one_rat ) ) ) ) ).

% pochhammer_code
thf(fact_8215_pochhammer__code,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A5: int,N2: nat] :
          ( if_int @ ( N2 = zero_zero_nat ) @ one_one_int
          @ ( set_fo2581907887559384638at_int
            @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A5 @ ( semiri1314217659103216013at_int @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N2 @ one_one_nat )
            @ one_one_int ) ) ) ) ).

% pochhammer_code
thf(fact_8216_pochhammer__code,axiom,
    ( comm_s7457072308508201937r_real
    = ( ^ [A5: real,N2: nat] :
          ( if_real @ ( N2 = zero_zero_nat ) @ one_one_real
          @ ( set_fo3111899725591712190t_real
            @ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A5 @ ( semiri5074537144036343181t_real @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N2 @ one_one_nat )
            @ one_one_real ) ) ) ) ).

% pochhammer_code
thf(fact_8217_pochhammer__code,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A5: nat,N2: nat] :
          ( if_nat @ ( N2 = zero_zero_nat ) @ one_one_nat
          @ ( set_fo2584398358068434914at_nat
            @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A5 @ ( semiri1316708129612266289at_nat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N2 @ one_one_nat )
            @ one_one_nat ) ) ) ) ).

% pochhammer_code
thf(fact_8218_sqrt__sum__squares__half__less,axiom,
    ! [X3: real,U: real,Y3: real] :
      ( ( ord_less_real @ X3 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
         => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
           => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).

% sqrt_sum_squares_half_less
thf(fact_8219_sin__cos__sqrt,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X3 ) )
     => ( ( sin_real @ X3 )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_cos_sqrt
thf(fact_8220_arctan__half,axiom,
    ( arctan
    = ( ^ [X: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% arctan_half
thf(fact_8221_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_8222_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_8223_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_8224_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_8225_arcosh__real__def,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ one_one_real @ X3 )
     => ( ( arcosh_real @ X3 )
        = ( ln_ln_real @ ( plus_plus_real @ X3 @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arcosh_real_def
thf(fact_8226_pochhammer__times__pochhammer__half,axiom,
    ! [Z: complex,N: nat] :
      ( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ ( suc @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
      = ( groups6464643781859351333omplex
        @ ^ [K2: nat] : ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_8227_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,N: nat] :
      ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [K2: nat] : ( plus_plus_rat @ Z @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K2 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_8228_pochhammer__times__pochhammer__half,axiom,
    ! [Z: real,N: nat] :
      ( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
      = ( groups129246275422532515t_real
        @ ^ [K2: nat] : ( plus_plus_real @ Z @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_8229_arsinh__real__def,axiom,
    ( arsinh_real
    = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arsinh_real_def
thf(fact_8230_choose__odd__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I4: nat] :
                ( if_complex
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
                @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) )
                @ zero_zero_complex )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N ) ) ) ).

% choose_odd_sum
thf(fact_8231_choose__odd__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I4: nat] :
                ( if_rat
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
                @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) )
                @ zero_zero_rat )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).

% choose_odd_sum
thf(fact_8232_choose__odd__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I4: nat] :
                ( if_int
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
                @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) )
                @ zero_zero_int )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% choose_odd_sum
thf(fact_8233_choose__odd__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I4: nat] :
                ( if_real
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
                @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) )
                @ zero_zero_real )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).

% choose_odd_sum
thf(fact_8234_choose__even__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I4: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) ) @ zero_zero_complex )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N ) ) ) ).

% choose_even_sum
thf(fact_8235_choose__even__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( if_rat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) ) @ zero_zero_rat )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).

% choose_even_sum
thf(fact_8236_choose__even__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I4: nat] : ( if_int @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) ) @ zero_zero_int )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% choose_even_sum
thf(fact_8237_choose__even__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) ) @ zero_zero_real )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).

% choose_even_sum
thf(fact_8238_atMost__iff,axiom,
    ! [I: real,K: real] :
      ( ( member_real @ I @ ( set_ord_atMost_real @ K ) )
      = ( ord_less_eq_real @ I @ K ) ) ).

% atMost_iff
thf(fact_8239_atMost__iff,axiom,
    ! [I: set_int,K: set_int] :
      ( ( member_set_int @ I @ ( set_or58775011639299419et_int @ K ) )
      = ( ord_less_eq_set_int @ I @ K ) ) ).

% atMost_iff
thf(fact_8240_atMost__iff,axiom,
    ! [I: rat,K: rat] :
      ( ( member_rat @ I @ ( set_ord_atMost_rat @ K ) )
      = ( ord_less_eq_rat @ I @ K ) ) ).

% atMost_iff
thf(fact_8241_atMost__iff,axiom,
    ! [I: num,K: num] :
      ( ( member_num @ I @ ( set_ord_atMost_num @ K ) )
      = ( ord_less_eq_num @ I @ K ) ) ).

% atMost_iff
thf(fact_8242_atMost__iff,axiom,
    ! [I: nat,K: nat] :
      ( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
      = ( ord_less_eq_nat @ I @ K ) ) ).

% atMost_iff
thf(fact_8243_atMost__iff,axiom,
    ! [I: int,K: int] :
      ( ( member_int @ I @ ( set_ord_atMost_int @ K ) )
      = ( ord_less_eq_int @ I @ K ) ) ).

% atMost_iff
thf(fact_8244_atMost__subset__iff,axiom,
    ! [X3: set_int,Y3: set_int] :
      ( ( ord_le4403425263959731960et_int @ ( set_or58775011639299419et_int @ X3 ) @ ( set_or58775011639299419et_int @ Y3 ) )
      = ( ord_less_eq_set_int @ X3 @ Y3 ) ) ).

% atMost_subset_iff
thf(fact_8245_atMost__subset__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ X3 ) @ ( set_ord_atMost_rat @ Y3 ) )
      = ( ord_less_eq_rat @ X3 @ Y3 ) ) ).

% atMost_subset_iff
thf(fact_8246_atMost__subset__iff,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ X3 ) @ ( set_ord_atMost_num @ Y3 ) )
      = ( ord_less_eq_num @ X3 @ Y3 ) ) ).

% atMost_subset_iff
thf(fact_8247_atMost__subset__iff,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X3 ) @ ( set_ord_atMost_nat @ Y3 ) )
      = ( ord_less_eq_nat @ X3 @ Y3 ) ) ).

% atMost_subset_iff
thf(fact_8248_atMost__subset__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X3 ) @ ( set_ord_atMost_int @ Y3 ) )
      = ( ord_less_eq_int @ X3 @ Y3 ) ) ).

% atMost_subset_iff
thf(fact_8249_Icc__subset__Iic__iff,axiom,
    ! [L: set_int,H2: set_int,H3: set_int] :
      ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ L @ H2 ) @ ( set_or58775011639299419et_int @ H3 ) )
      = ( ~ ( ord_less_eq_set_int @ L @ H2 )
        | ( ord_less_eq_set_int @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_8250_Icc__subset__Iic__iff,axiom,
    ! [L: rat,H2: rat,H3: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ L @ H2 ) @ ( set_ord_atMost_rat @ H3 ) )
      = ( ~ ( ord_less_eq_rat @ L @ H2 )
        | ( ord_less_eq_rat @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_8251_Icc__subset__Iic__iff,axiom,
    ! [L: num,H2: num,H3: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ L @ H2 ) @ ( set_ord_atMost_num @ H3 ) )
      = ( ~ ( ord_less_eq_num @ L @ H2 )
        | ( ord_less_eq_num @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_8252_Icc__subset__Iic__iff,axiom,
    ! [L: nat,H2: nat,H3: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H2 ) @ ( set_ord_atMost_nat @ H3 ) )
      = ( ~ ( ord_less_eq_nat @ L @ H2 )
        | ( ord_less_eq_nat @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_8253_Icc__subset__Iic__iff,axiom,
    ! [L: int,H2: int,H3: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L @ H2 ) @ ( set_ord_atMost_int @ H3 ) )
      = ( ~ ( ord_less_eq_int @ L @ H2 )
        | ( ord_less_eq_int @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_8254_Icc__subset__Iic__iff,axiom,
    ! [L: real,H2: real,H3: real] :
      ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L @ H2 ) @ ( set_ord_atMost_real @ H3 ) )
      = ( ~ ( ord_less_eq_real @ L @ H2 )
        | ( ord_less_eq_real @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_8255_sum_OatMost__Suc,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% sum.atMost_Suc
thf(fact_8256_sum_OatMost__Suc,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% sum.atMost_Suc
thf(fact_8257_sum_OatMost__Suc,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% sum.atMost_Suc
thf(fact_8258_sum_OatMost__Suc,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% sum.atMost_Suc
thf(fact_8259_prod_OlessThan__Suc,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).

% prod.lessThan_Suc
thf(fact_8260_prod_OlessThan__Suc,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).

% prod.lessThan_Suc
thf(fact_8261_prod_OlessThan__Suc,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).

% prod.lessThan_Suc
thf(fact_8262_prod_OlessThan__Suc,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).

% prod.lessThan_Suc
thf(fact_8263_prod_OatMost__Suc,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% prod.atMost_Suc
thf(fact_8264_prod_OatMost__Suc,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% prod.atMost_Suc
thf(fact_8265_prod_OatMost__Suc,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% prod.atMost_Suc
thf(fact_8266_prod_OatMost__Suc,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% prod.atMost_Suc
thf(fact_8267_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > complex] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = one_one_complex ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_8268_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = one_one_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_8269_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = one_one_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_8270_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_8271_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = one_one_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_8272_prod_Odistrib,axiom,
    ! [G: nat > nat,H2: nat > nat,A4: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [X: nat] : ( times_times_nat @ ( G @ X ) @ ( H2 @ X ) )
        @ A4 )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ A4 ) @ ( groups708209901874060359at_nat @ H2 @ A4 ) ) ) ).

% prod.distrib
thf(fact_8273_prod_Odistrib,axiom,
    ! [G: nat > int,H2: nat > int,A4: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [X: nat] : ( times_times_int @ ( G @ X ) @ ( H2 @ X ) )
        @ A4 )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ A4 ) @ ( groups705719431365010083at_int @ H2 @ A4 ) ) ) ).

% prod.distrib
thf(fact_8274_prod_Odistrib,axiom,
    ! [G: int > int,H2: int > int,A4: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [X: int] : ( times_times_int @ ( G @ X ) @ ( H2 @ X ) )
        @ A4 )
      = ( times_times_int @ ( groups1705073143266064639nt_int @ G @ A4 ) @ ( groups1705073143266064639nt_int @ H2 @ A4 ) ) ) ).

% prod.distrib
thf(fact_8275_prod__power__distrib,axiom,
    ! [F: nat > nat,A4: set_nat,N: nat] :
      ( ( power_power_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ N )
      = ( groups708209901874060359at_nat
        @ ^ [X: nat] : ( power_power_nat @ ( F @ X ) @ N )
        @ A4 ) ) ).

% prod_power_distrib
thf(fact_8276_prod__power__distrib,axiom,
    ! [F: nat > int,A4: set_nat,N: nat] :
      ( ( power_power_int @ ( groups705719431365010083at_int @ F @ A4 ) @ N )
      = ( groups705719431365010083at_int
        @ ^ [X: nat] : ( power_power_int @ ( F @ X ) @ N )
        @ A4 ) ) ).

% prod_power_distrib
thf(fact_8277_prod__power__distrib,axiom,
    ! [F: int > int,A4: set_int,N: nat] :
      ( ( power_power_int @ ( groups1705073143266064639nt_int @ F @ A4 ) @ N )
      = ( groups1705073143266064639nt_int
        @ ^ [X: int] : ( power_power_int @ ( F @ X ) @ N )
        @ A4 ) ) ).

% prod_power_distrib
thf(fact_8278_mod__prod__eq,axiom,
    ! [F: nat > nat,A: nat,A4: set_nat] :
      ( ( modulo_modulo_nat
        @ ( groups708209901874060359at_nat
          @ ^ [I4: nat] : ( modulo_modulo_nat @ ( F @ I4 ) @ A )
          @ A4 )
        @ A )
      = ( modulo_modulo_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ A ) ) ).

% mod_prod_eq
thf(fact_8279_mod__prod__eq,axiom,
    ! [F: nat > int,A: int,A4: set_nat] :
      ( ( modulo_modulo_int
        @ ( groups705719431365010083at_int
          @ ^ [I4: nat] : ( modulo_modulo_int @ ( F @ I4 ) @ A )
          @ A4 )
        @ A )
      = ( modulo_modulo_int @ ( groups705719431365010083at_int @ F @ A4 ) @ A ) ) ).

% mod_prod_eq
thf(fact_8280_mod__prod__eq,axiom,
    ! [F: int > int,A: int,A4: set_int] :
      ( ( modulo_modulo_int
        @ ( groups1705073143266064639nt_int
          @ ^ [I4: int] : ( modulo_modulo_int @ ( F @ I4 ) @ A )
          @ A4 )
        @ A )
      = ( modulo_modulo_int @ ( groups1705073143266064639nt_int @ F @ A4 ) @ A ) ) ).

% mod_prod_eq
thf(fact_8281_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_real @ ( G @ zero_zero_nat )
        @ ( groups129246275422532515t_real
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_8282_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_rat @ ( G @ zero_zero_nat )
        @ ( groups73079841787564623at_rat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_8283_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_8284_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_8285_atMost__def,axiom,
    ( set_ord_atMost_real
    = ( ^ [U2: real] :
          ( collect_real
          @ ^ [X: real] : ( ord_less_eq_real @ X @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8286_atMost__def,axiom,
    ( set_or58775011639299419et_int
    = ( ^ [U2: set_int] :
          ( collect_set_int
          @ ^ [X: set_int] : ( ord_less_eq_set_int @ X @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8287_atMost__def,axiom,
    ( set_ord_atMost_rat
    = ( ^ [U2: rat] :
          ( collect_rat
          @ ^ [X: rat] : ( ord_less_eq_rat @ X @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8288_atMost__def,axiom,
    ( set_ord_atMost_num
    = ( ^ [U2: num] :
          ( collect_num
          @ ^ [X: num] : ( ord_less_eq_num @ X @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8289_atMost__def,axiom,
    ( set_ord_atMost_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X: nat] : ( ord_less_eq_nat @ X @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8290_atMost__def,axiom,
    ( set_ord_atMost_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X: int] : ( ord_less_eq_int @ X @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8291_prod__mono,axiom,
    ! [A4: set_nat,F: nat > real,G: nat > real] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
            & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A4 ) @ ( groups129246275422532515t_real @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_8292_prod__mono,axiom,
    ! [A4: set_real,F: real > real,G: real > real] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
            & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ ( groups1681761925125756287l_real @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_8293_prod__mono,axiom,
    ! [A4: set_int,F: int > real,G: int > real] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
            & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ ( groups2316167850115554303t_real @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_8294_prod__mono,axiom,
    ! [A4: set_complex,F: complex > real,G: complex > real] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
            & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A4 ) @ ( groups766887009212190081x_real @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_8295_prod__mono,axiom,
    ! [A4: set_nat,F: nat > rat,G: nat > rat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
            & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ ( groups73079841787564623at_rat @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_8296_prod__mono,axiom,
    ! [A4: set_real,F: real > rat,G: real > rat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
            & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ ( groups4061424788464935467al_rat @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_8297_prod__mono,axiom,
    ! [A4: set_int,F: int > rat,G: int > rat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
            & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ ( groups1072433553688619179nt_rat @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_8298_prod__mono,axiom,
    ! [A4: set_complex,F: complex > rat,G: complex > rat] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
            & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) @ ( groups225925009352817453ex_rat @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_8299_prod__mono,axiom,
    ! [A4: set_real,F: real > nat,G: real > nat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ A4 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
            & ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) @ ( groups4696554848551431203al_nat @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_8300_prod__mono,axiom,
    ! [A4: set_int,F: int > nat,G: int > nat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A4 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
            & ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ ( groups1707563613775114915nt_nat @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_8301_prod__nonneg,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A4 ) ) ) ).

% prod_nonneg
thf(fact_8302_prod__nonneg,axiom,
    ! [A4: set_nat,F: nat > int] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A4 ) ) ) ).

% prod_nonneg
thf(fact_8303_prod__nonneg,axiom,
    ! [A4: set_int,F: int > int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A4 ) ) ) ).

% prod_nonneg
thf(fact_8304_prod__pos,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A4 ) ) ) ).

% prod_pos
thf(fact_8305_prod__pos,axiom,
    ! [A4: set_nat,F: nat > int] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( ord_less_int @ zero_zero_int @ ( F @ X5 ) ) )
     => ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A4 ) ) ) ).

% prod_pos
thf(fact_8306_prod__pos,axiom,
    ! [A4: set_int,F: int > int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ord_less_int @ zero_zero_int @ ( F @ X5 ) ) )
     => ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A4 ) ) ) ).

% prod_pos
thf(fact_8307_prod__ge__1,axiom,
    ! [A4: set_nat,F: nat > real] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_8308_prod__ge__1,axiom,
    ! [A4: set_real,F: real > real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_8309_prod__ge__1,axiom,
    ! [A4: set_int,F: int > real] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_8310_prod__ge__1,axiom,
    ! [A4: set_complex,F: complex > real] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A4 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups766887009212190081x_real @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_8311_prod__ge__1,axiom,
    ! [A4: set_nat,F: nat > rat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X5 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_8312_prod__ge__1,axiom,
    ! [A4: set_real,F: real > rat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X5 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_8313_prod__ge__1,axiom,
    ! [A4: set_int,F: int > rat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X5 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_8314_prod__ge__1,axiom,
    ! [A4: set_complex,F: complex > rat] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A4 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X5 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_8315_prod__ge__1,axiom,
    ! [A4: set_real,F: real > nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_8316_prod__ge__1,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_8317_prod_OatMost__shift,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N ) )
      = ( times_times_real @ ( G @ zero_zero_nat )
        @ ( groups129246275422532515t_real
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% prod.atMost_shift
thf(fact_8318_prod_OatMost__shift,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N ) )
      = ( times_times_rat @ ( G @ zero_zero_nat )
        @ ( groups73079841787564623at_rat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% prod.atMost_shift
thf(fact_8319_prod_OatMost__shift,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% prod.atMost_shift
thf(fact_8320_prod_OatMost__shift,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% prod.atMost_shift
thf(fact_8321_not__Iic__le__Icc,axiom,
    ! [H2: int,L3: int,H3: int] :
      ~ ( ord_less_eq_set_int @ ( set_ord_atMost_int @ H2 ) @ ( set_or1266510415728281911st_int @ L3 @ H3 ) ) ).

% not_Iic_le_Icc
thf(fact_8322_not__Iic__le__Icc,axiom,
    ! [H2: real,L3: real,H3: real] :
      ~ ( ord_less_eq_set_real @ ( set_ord_atMost_real @ H2 ) @ ( set_or1222579329274155063t_real @ L3 @ H3 ) ) ).

% not_Iic_le_Icc
thf(fact_8323_power__sum,axiom,
    ! [C: real,F: nat > nat,A4: set_nat] :
      ( ( power_power_real @ C @ ( groups3542108847815614940at_nat @ F @ A4 ) )
      = ( groups129246275422532515t_real
        @ ^ [A5: nat] : ( power_power_real @ C @ ( F @ A5 ) )
        @ A4 ) ) ).

% power_sum
thf(fact_8324_power__sum,axiom,
    ! [C: complex,F: nat > nat,A4: set_nat] :
      ( ( power_power_complex @ C @ ( groups3542108847815614940at_nat @ F @ A4 ) )
      = ( groups6464643781859351333omplex
        @ ^ [A5: nat] : ( power_power_complex @ C @ ( F @ A5 ) )
        @ A4 ) ) ).

% power_sum
thf(fact_8325_power__sum,axiom,
    ! [C: nat,F: nat > nat,A4: set_nat] :
      ( ( power_power_nat @ C @ ( groups3542108847815614940at_nat @ F @ A4 ) )
      = ( groups708209901874060359at_nat
        @ ^ [A5: nat] : ( power_power_nat @ C @ ( F @ A5 ) )
        @ A4 ) ) ).

% power_sum
thf(fact_8326_power__sum,axiom,
    ! [C: int,F: nat > nat,A4: set_nat] :
      ( ( power_power_int @ C @ ( groups3542108847815614940at_nat @ F @ A4 ) )
      = ( groups705719431365010083at_int
        @ ^ [A5: nat] : ( power_power_int @ C @ ( F @ A5 ) )
        @ A4 ) ) ).

% power_sum
thf(fact_8327_power__sum,axiom,
    ! [C: int,F: int > nat,A4: set_int] :
      ( ( power_power_int @ C @ ( groups4541462559716669496nt_nat @ F @ A4 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [A5: int] : ( power_power_int @ C @ ( F @ A5 ) )
        @ A4 ) ) ).

% power_sum
thf(fact_8328_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K: nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_8329_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > int,M: nat,K: nat,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_8330_finite__nat__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [S4: set_nat] :
        ? [K2: nat] : ( ord_less_eq_set_nat @ S4 @ ( set_ord_atMost_nat @ K2 ) ) ) ) ).

% finite_nat_iff_bounded_le
thf(fact_8331_prod__le__1,axiom,
    ! [A4: set_nat,F: nat > real] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
            & ( ord_less_eq_real @ ( F @ X5 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A4 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_8332_prod__le__1,axiom,
    ! [A4: set_real,F: real > real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
            & ( ord_less_eq_real @ ( F @ X5 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_8333_prod__le__1,axiom,
    ! [A4: set_int,F: int > real] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
            & ( ord_less_eq_real @ ( F @ X5 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_8334_prod__le__1,axiom,
    ! [A4: set_complex,F: complex > real] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
            & ( ord_less_eq_real @ ( F @ X5 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A4 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_8335_prod__le__1,axiom,
    ! [A4: set_nat,F: nat > rat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) )
            & ( ord_less_eq_rat @ ( F @ X5 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_8336_prod__le__1,axiom,
    ! [A4: set_real,F: real > rat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) )
            & ( ord_less_eq_rat @ ( F @ X5 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_8337_prod__le__1,axiom,
    ! [A4: set_int,F: int > rat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) )
            & ( ord_less_eq_rat @ ( F @ X5 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_8338_prod__le__1,axiom,
    ! [A4: set_complex,F: complex > rat] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) )
            & ( ord_less_eq_rat @ ( F @ X5 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_8339_prod__le__1,axiom,
    ! [A4: set_real,F: real > nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) )
            & ( ord_less_eq_nat @ ( F @ X5 ) @ one_one_nat ) ) )
     => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_8340_prod__le__1,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) )
            & ( ord_less_eq_nat @ ( F @ X5 ) @ one_one_nat ) ) )
     => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_8341_prod_Orelated,axiom,
    ! [R2: complex > complex > $o,S2: set_nat,H2: nat > complex,G: nat > complex] :
      ( ( R2 @ one_one_complex @ one_one_complex )
     => ( ! [X15: complex,Y15: complex,X22: complex,Y22: complex] :
            ( ( ( R2 @ X15 @ X22 )
              & ( R2 @ Y15 @ Y22 ) )
           => ( R2 @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X22 @ Y22 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S2 )
               => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R2 @ ( groups6464643781859351333omplex @ H2 @ S2 ) @ ( groups6464643781859351333omplex @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_8342_prod_Orelated,axiom,
    ! [R2: complex > complex > $o,S2: set_int,H2: int > complex,G: int > complex] :
      ( ( R2 @ one_one_complex @ one_one_complex )
     => ( ! [X15: complex,Y15: complex,X22: complex,Y22: complex] :
            ( ( ( R2 @ X15 @ X22 )
              & ( R2 @ Y15 @ Y22 ) )
           => ( R2 @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X22 @ Y22 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S2 )
               => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R2 @ ( groups7440179247065528705omplex @ H2 @ S2 ) @ ( groups7440179247065528705omplex @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_8343_prod_Orelated,axiom,
    ! [R2: complex > complex > $o,S2: set_complex,H2: complex > complex,G: complex > complex] :
      ( ( R2 @ one_one_complex @ one_one_complex )
     => ( ! [X15: complex,Y15: complex,X22: complex,Y22: complex] :
            ( ( ( R2 @ X15 @ X22 )
              & ( R2 @ Y15 @ Y22 ) )
           => ( R2 @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X22 @ Y22 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R2 @ ( groups3708469109370488835omplex @ H2 @ S2 ) @ ( groups3708469109370488835omplex @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_8344_prod_Orelated,axiom,
    ! [R2: real > real > $o,S2: set_nat,H2: nat > real,G: nat > real] :
      ( ( R2 @ one_one_real @ one_one_real )
     => ( ! [X15: real,Y15: real,X22: real,Y22: real] :
            ( ( ( R2 @ X15 @ X22 )
              & ( R2 @ Y15 @ Y22 ) )
           => ( R2 @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X22 @ Y22 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S2 )
               => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R2 @ ( groups129246275422532515t_real @ H2 @ S2 ) @ ( groups129246275422532515t_real @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_8345_prod_Orelated,axiom,
    ! [R2: real > real > $o,S2: set_int,H2: int > real,G: int > real] :
      ( ( R2 @ one_one_real @ one_one_real )
     => ( ! [X15: real,Y15: real,X22: real,Y22: real] :
            ( ( ( R2 @ X15 @ X22 )
              & ( R2 @ Y15 @ Y22 ) )
           => ( R2 @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X22 @ Y22 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S2 )
               => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R2 @ ( groups2316167850115554303t_real @ H2 @ S2 ) @ ( groups2316167850115554303t_real @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_8346_prod_Orelated,axiom,
    ! [R2: real > real > $o,S2: set_complex,H2: complex > real,G: complex > real] :
      ( ( R2 @ one_one_real @ one_one_real )
     => ( ! [X15: real,Y15: real,X22: real,Y22: real] :
            ( ( ( R2 @ X15 @ X22 )
              & ( R2 @ Y15 @ Y22 ) )
           => ( R2 @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X22 @ Y22 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R2 @ ( groups766887009212190081x_real @ H2 @ S2 ) @ ( groups766887009212190081x_real @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_8347_prod_Orelated,axiom,
    ! [R2: rat > rat > $o,S2: set_nat,H2: nat > rat,G: nat > rat] :
      ( ( R2 @ one_one_rat @ one_one_rat )
     => ( ! [X15: rat,Y15: rat,X22: rat,Y22: rat] :
            ( ( ( R2 @ X15 @ X22 )
              & ( R2 @ Y15 @ Y22 ) )
           => ( R2 @ ( times_times_rat @ X15 @ Y15 ) @ ( times_times_rat @ X22 @ Y22 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S2 )
               => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R2 @ ( groups73079841787564623at_rat @ H2 @ S2 ) @ ( groups73079841787564623at_rat @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_8348_prod_Orelated,axiom,
    ! [R2: rat > rat > $o,S2: set_int,H2: int > rat,G: int > rat] :
      ( ( R2 @ one_one_rat @ one_one_rat )
     => ( ! [X15: rat,Y15: rat,X22: rat,Y22: rat] :
            ( ( ( R2 @ X15 @ X22 )
              & ( R2 @ Y15 @ Y22 ) )
           => ( R2 @ ( times_times_rat @ X15 @ Y15 ) @ ( times_times_rat @ X22 @ Y22 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S2 )
               => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R2 @ ( groups1072433553688619179nt_rat @ H2 @ S2 ) @ ( groups1072433553688619179nt_rat @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_8349_prod_Orelated,axiom,
    ! [R2: rat > rat > $o,S2: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( R2 @ one_one_rat @ one_one_rat )
     => ( ! [X15: rat,Y15: rat,X22: rat,Y22: rat] :
            ( ( ( R2 @ X15 @ X22 )
              & ( R2 @ Y15 @ Y22 ) )
           => ( R2 @ ( times_times_rat @ X15 @ Y15 ) @ ( times_times_rat @ X22 @ Y22 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R2 @ ( groups225925009352817453ex_rat @ H2 @ S2 ) @ ( groups225925009352817453ex_rat @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_8350_prod_Orelated,axiom,
    ! [R2: nat > nat > $o,S2: set_int,H2: int > nat,G: int > nat] :
      ( ( R2 @ one_one_nat @ one_one_nat )
     => ( ! [X15: nat,Y15: nat,X22: nat,Y22: nat] :
            ( ( ( R2 @ X15 @ X22 )
              & ( R2 @ Y15 @ Y22 ) )
           => ( R2 @ ( times_times_nat @ X15 @ Y15 ) @ ( times_times_nat @ X22 @ Y22 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S2 )
               => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R2 @ ( groups1707563613775114915nt_nat @ H2 @ S2 ) @ ( groups1707563613775114915nt_nat @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_8351_prod__dvd__prod__subset,axiom,
    ! [B5: set_complex,A4: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) @ ( groups861055069439313189ex_nat @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8352_prod__dvd__prod__subset,axiom,
    ! [B5: set_complex,A4: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A4 ) @ ( groups858564598930262913ex_int @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8353_prod__dvd__prod__subset,axiom,
    ! [B5: set_nat,A4: set_nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A4 ) @ ( groups3455450783089532116nteger @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8354_prod__dvd__prod__subset,axiom,
    ! [B5: set_complex,A4: set_complex,F: complex > code_integer] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A4 ) @ ( groups8682486955453173170nteger @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8355_prod__dvd__prod__subset,axiom,
    ! [B5: set_int,A4: set_int,F: int > nat] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ ( groups1707563613775114915nt_nat @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8356_prod__dvd__prod__subset,axiom,
    ! [B5: set_int,A4: set_int,F: int > code_integer] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( dvd_dvd_Code_integer @ ( groups3827104343326376752nteger @ F @ A4 ) @ ( groups3827104343326376752nteger @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8357_prod__dvd__prod__subset,axiom,
    ! [B5: set_nat,A4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ ( groups708209901874060359at_nat @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8358_prod__dvd__prod__subset,axiom,
    ! [B5: set_nat,A4: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A4 ) @ ( groups705719431365010083at_int @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8359_prod__dvd__prod__subset,axiom,
    ! [B5: set_int,A4: set_int,F: int > int] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( dvd_dvd_int @ ( groups1705073143266064639nt_int @ F @ A4 ) @ ( groups1705073143266064639nt_int @ F @ B5 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_8360_prod__dvd__prod__subset2,axiom,
    ! [B5: set_real,A4: set_real,F: real > nat,G: real > nat] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A4 @ B5 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ A4 )
             => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) @ ( groups4696554848551431203al_nat @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8361_prod__dvd__prod__subset2,axiom,
    ! [B5: set_complex,A4: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( ! [A3: complex] :
              ( ( member_complex @ A3 @ A4 )
             => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) @ ( groups861055069439313189ex_nat @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8362_prod__dvd__prod__subset2,axiom,
    ! [B5: set_real,A4: set_real,F: real > int,G: real > int] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A4 @ B5 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ A4 )
             => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_int @ ( groups4694064378042380927al_int @ F @ A4 ) @ ( groups4694064378042380927al_int @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8363_prod__dvd__prod__subset2,axiom,
    ! [B5: set_complex,A4: set_complex,F: complex > int,G: complex > int] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( ! [A3: complex] :
              ( ( member_complex @ A3 @ A4 )
             => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A4 ) @ ( groups858564598930262913ex_int @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8364_prod__dvd__prod__subset2,axiom,
    ! [B5: set_real,A4: set_real,F: real > code_integer,G: real > code_integer] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A4 @ B5 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ A4 )
             => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_Code_integer @ ( groups6225526099057966256nteger @ F @ A4 ) @ ( groups6225526099057966256nteger @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8365_prod__dvd__prod__subset2,axiom,
    ! [B5: set_nat,A4: set_nat,F: nat > code_integer,G: nat > code_integer] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( ! [A3: nat] :
              ( ( member_nat @ A3 @ A4 )
             => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A4 ) @ ( groups3455450783089532116nteger @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8366_prod__dvd__prod__subset2,axiom,
    ! [B5: set_complex,A4: set_complex,F: complex > code_integer,G: complex > code_integer] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( ! [A3: complex] :
              ( ( member_complex @ A3 @ A4 )
             => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A4 ) @ ( groups8682486955453173170nteger @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8367_prod__dvd__prod__subset2,axiom,
    ! [B5: set_int,A4: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ A4 )
             => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ ( groups1707563613775114915nt_nat @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8368_prod__dvd__prod__subset2,axiom,
    ! [B5: set_int,A4: set_int,F: int > code_integer,G: int > code_integer] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ A4 )
             => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_Code_integer @ ( groups3827104343326376752nteger @ F @ A4 ) @ ( groups3827104343326376752nteger @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8369_prod__dvd__prod__subset2,axiom,
    ! [B5: set_nat,A4: set_nat,F: nat > nat,G: nat > nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( ! [A3: nat] :
              ( ( member_nat @ A3 @ A4 )
             => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ ( groups708209901874060359at_nat @ G @ B5 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_8370_prod_Oin__pairs__0,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups129246275422532515t_real
        @ ^ [I4: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% prod.in_pairs_0
thf(fact_8371_prod_Oin__pairs__0,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [I4: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% prod.in_pairs_0
thf(fact_8372_prod_Oin__pairs__0,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% prod.in_pairs_0
thf(fact_8373_prod_Oin__pairs__0,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% prod.in_pairs_0
thf(fact_8374_prod_Onat__diff__reindex,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
        @ ( set_ord_lessThan_nat @ N ) )
      = ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).

% prod.nat_diff_reindex
thf(fact_8375_prod_Onat__diff__reindex,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
        @ ( set_ord_lessThan_nat @ N ) )
      = ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).

% prod.nat_diff_reindex
thf(fact_8376_Iic__subset__Iio__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ A ) @ ( set_ord_lessThan_rat @ B2 ) )
      = ( ord_less_rat @ A @ B2 ) ) ).

% Iic_subset_Iio_iff
thf(fact_8377_Iic__subset__Iio__iff,axiom,
    ! [A: num,B2: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ A ) @ ( set_ord_lessThan_num @ B2 ) )
      = ( ord_less_num @ A @ B2 ) ) ).

% Iic_subset_Iio_iff
thf(fact_8378_Iic__subset__Iio__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B2 ) )
      = ( ord_less_nat @ A @ B2 ) ) ).

% Iic_subset_Iio_iff
thf(fact_8379_Iic__subset__Iio__iff,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ A ) @ ( set_ord_lessThan_int @ B2 ) )
      = ( ord_less_int @ A @ B2 ) ) ).

% Iic_subset_Iio_iff
thf(fact_8380_Iic__subset__Iio__iff,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ A ) @ ( set_or5984915006950818249n_real @ B2 ) )
      = ( ord_less_real @ A @ B2 ) ) ).

% Iic_subset_Iio_iff
thf(fact_8381_prod_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N: nat,M: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
      = ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_8382_prod_OatLeastAtMost__rev,axiom,
    ! [G: nat > int,N: nat,M: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
      = ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_8383_prod_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > complex,H2: nat > complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups6464643781859351333omplex
            @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_complex @ ( J3 = K ) @ one_one_complex @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups6464643781859351333omplex
            @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_8384_prod_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > real,H2: nat > real] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups129246275422532515t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K ) @ one_one_real @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups129246275422532515t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_8385_prod_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > rat,H2: nat > rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups73079841787564623at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K ) @ one_one_rat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups73079841787564623at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_8386_prod_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > nat,H2: nat > nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups708209901874060359at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K ) @ one_one_nat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups708209901874060359at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_8387_prod_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > int,H2: nat > int] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups705719431365010083at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K ) @ one_one_int @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups705719431365010083at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_8388_less__1__prod2,axiom,
    ! [I5: set_real,I: real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I3: real] :
                ( ( member_real @ I3 @ I5 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_8389_less__1__prod2,axiom,
    ! [I5: set_nat,I: nat,F: nat > real] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I @ I5 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_8390_less__1__prod2,axiom,
    ! [I5: set_int,I: int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ I5 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_8391_less__1__prod2,axiom,
    ! [I5: set_complex,I: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_8392_less__1__prod2,axiom,
    ! [I5: set_real,I: real,F: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I3: real] :
                ( ( member_real @ I3 @ I5 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_8393_less__1__prod2,axiom,
    ! [I5: set_nat,I: nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I @ I5 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_8394_less__1__prod2,axiom,
    ! [I5: set_int,I: int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ I5 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_8395_less__1__prod2,axiom,
    ! [I5: set_complex,I: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_8396_less__1__prod2,axiom,
    ! [I5: set_real,I: real,F: real > int] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_int @ one_one_int @ ( F @ I ) )
         => ( ! [I3: real] :
                ( ( member_real @ I3 @ I5 )
               => ( ord_less_eq_int @ one_one_int @ ( F @ I3 ) ) )
           => ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_8397_less__1__prod2,axiom,
    ! [I5: set_complex,I: complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_int @ one_one_int @ ( F @ I ) )
         => ( ! [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
               => ( ord_less_eq_int @ one_one_int @ ( F @ I3 ) ) )
           => ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_8398_less__1__prod,axiom,
    ! [I5: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
             => ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_8399_less__1__prod,axiom,
    ! [I5: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ I5 )
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
             => ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_8400_less__1__prod,axiom,
    ! [I5: set_int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_8401_less__1__prod,axiom,
    ! [I5: set_real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ I5 )
             => ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_8402_less__1__prod,axiom,
    ! [I5: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_8403_less__1__prod,axiom,
    ! [I5: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_8404_less__1__prod,axiom,
    ! [I5: set_int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_8405_less__1__prod,axiom,
    ! [I5: set_real,F: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ I5 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_8406_less__1__prod,axiom,
    ! [I5: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
             => ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
         => ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_8407_less__1__prod,axiom,
    ! [I5: set_real,F: real > int] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ I5 )
             => ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
         => ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_8408_prod_Osubset__diff,axiom,
    ! [B5: set_nat,A4: set_nat,G: nat > real] :
      ( ( ord_less_eq_set_nat @ B5 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups129246275422532515t_real @ G @ A4 )
          = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups129246275422532515t_real @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8409_prod_Osubset__diff,axiom,
    ! [B5: set_complex,A4: set_complex,G: complex > real] :
      ( ( ord_le211207098394363844omplex @ B5 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups766887009212190081x_real @ G @ A4 )
          = ( times_times_real @ ( groups766887009212190081x_real @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups766887009212190081x_real @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8410_prod_Osubset__diff,axiom,
    ! [B5: set_nat,A4: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ B5 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups73079841787564623at_rat @ G @ A4 )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups73079841787564623at_rat @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8411_prod_Osubset__diff,axiom,
    ! [B5: set_complex,A4: set_complex,G: complex > rat] :
      ( ( ord_le211207098394363844omplex @ B5 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups225925009352817453ex_rat @ G @ A4 )
          = ( times_times_rat @ ( groups225925009352817453ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups225925009352817453ex_rat @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8412_prod_Osubset__diff,axiom,
    ! [B5: set_complex,A4: set_complex,G: complex > nat] :
      ( ( ord_le211207098394363844omplex @ B5 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups861055069439313189ex_nat @ G @ A4 )
          = ( times_times_nat @ ( groups861055069439313189ex_nat @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups861055069439313189ex_nat @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8413_prod_Osubset__diff,axiom,
    ! [B5: set_complex,A4: set_complex,G: complex > int] :
      ( ( ord_le211207098394363844omplex @ B5 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups858564598930262913ex_int @ G @ A4 )
          = ( times_times_int @ ( groups858564598930262913ex_int @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups858564598930262913ex_int @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8414_prod_Osubset__diff,axiom,
    ! [B5: set_int,A4: set_int,G: int > real] :
      ( ( ord_less_eq_set_int @ B5 @ A4 )
     => ( ( finite_finite_int @ A4 )
       => ( ( groups2316167850115554303t_real @ G @ A4 )
          = ( times_times_real @ ( groups2316167850115554303t_real @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups2316167850115554303t_real @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8415_prod_Osubset__diff,axiom,
    ! [B5: set_int,A4: set_int,G: int > rat] :
      ( ( ord_less_eq_set_int @ B5 @ A4 )
     => ( ( finite_finite_int @ A4 )
       => ( ( groups1072433553688619179nt_rat @ G @ A4 )
          = ( times_times_rat @ ( groups1072433553688619179nt_rat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups1072433553688619179nt_rat @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8416_prod_Osubset__diff,axiom,
    ! [B5: set_int,A4: set_int,G: int > nat] :
      ( ( ord_less_eq_set_int @ B5 @ A4 )
     => ( ( finite_finite_int @ A4 )
       => ( ( groups1707563613775114915nt_nat @ G @ A4 )
          = ( times_times_nat @ ( groups1707563613775114915nt_nat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups1707563613775114915nt_nat @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8417_prod_Osubset__diff,axiom,
    ! [B5: set_nat,A4: set_nat,G: nat > nat] :
      ( ( ord_less_eq_set_nat @ B5 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups708209901874060359at_nat @ G @ A4 )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups708209901874060359at_nat @ G @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_8418_prod_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_complex ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = one_one_complex ) )
             => ( ( ( groups713298508707869441omplex @ G @ A4 )
                  = ( groups713298508707869441omplex @ H2 @ B5 ) )
                = ( ( groups713298508707869441omplex @ G @ C4 )
                  = ( groups713298508707869441omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8419_prod_Osame__carrier,axiom,
    ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > complex,H2: nat > complex] :
      ( ( finite_finite_nat @ C4 )
     => ( ( ord_less_eq_set_nat @ A4 @ C4 )
       => ( ( ord_less_eq_set_nat @ B5 @ C4 )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_complex ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = one_one_complex ) )
             => ( ( ( groups6464643781859351333omplex @ G @ A4 )
                  = ( groups6464643781859351333omplex @ H2 @ B5 ) )
                = ( ( groups6464643781859351333omplex @ G @ C4 )
                  = ( groups6464643781859351333omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8420_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > complex,H2: complex > complex] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_complex ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = one_one_complex ) )
             => ( ( ( groups3708469109370488835omplex @ G @ A4 )
                  = ( groups3708469109370488835omplex @ H2 @ B5 ) )
                = ( ( groups3708469109370488835omplex @ G @ C4 )
                  = ( groups3708469109370488835omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8421_prod_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = one_one_real ) )
             => ( ( ( groups1681761925125756287l_real @ G @ A4 )
                  = ( groups1681761925125756287l_real @ H2 @ B5 ) )
                = ( ( groups1681761925125756287l_real @ G @ C4 )
                  = ( groups1681761925125756287l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8422_prod_Osame__carrier,axiom,
    ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > real,H2: nat > real] :
      ( ( finite_finite_nat @ C4 )
     => ( ( ord_less_eq_set_nat @ A4 @ C4 )
       => ( ( ord_less_eq_set_nat @ B5 @ C4 )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = one_one_real ) )
             => ( ( ( groups129246275422532515t_real @ G @ A4 )
                  = ( groups129246275422532515t_real @ H2 @ B5 ) )
                = ( ( groups129246275422532515t_real @ G @ C4 )
                  = ( groups129246275422532515t_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8423_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = one_one_real ) )
             => ( ( ( groups766887009212190081x_real @ G @ A4 )
                  = ( groups766887009212190081x_real @ H2 @ B5 ) )
                = ( ( groups766887009212190081x_real @ G @ C4 )
                  = ( groups766887009212190081x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8424_prod_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups4061424788464935467al_rat @ G @ A4 )
                  = ( groups4061424788464935467al_rat @ H2 @ B5 ) )
                = ( ( groups4061424788464935467al_rat @ G @ C4 )
                  = ( groups4061424788464935467al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8425_prod_Osame__carrier,axiom,
    ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > rat,H2: nat > rat] :
      ( ( finite_finite_nat @ C4 )
     => ( ( ord_less_eq_set_nat @ A4 @ C4 )
       => ( ( ord_less_eq_set_nat @ B5 @ C4 )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups73079841787564623at_rat @ G @ A4 )
                  = ( groups73079841787564623at_rat @ H2 @ B5 ) )
                = ( ( groups73079841787564623at_rat @ G @ C4 )
                  = ( groups73079841787564623at_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8426_prod_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups225925009352817453ex_rat @ G @ A4 )
                  = ( groups225925009352817453ex_rat @ H2 @ B5 ) )
                = ( ( groups225925009352817453ex_rat @ G @ C4 )
                  = ( groups225925009352817453ex_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8427_prod_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_nat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = one_one_nat ) )
             => ( ( ( groups4696554848551431203al_nat @ G @ A4 )
                  = ( groups4696554848551431203al_nat @ H2 @ B5 ) )
                = ( ( groups4696554848551431203al_nat @ G @ C4 )
                  = ( groups4696554848551431203al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_8428_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_complex ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = one_one_complex ) )
             => ( ( ( groups713298508707869441omplex @ G @ C4 )
                  = ( groups713298508707869441omplex @ H2 @ C4 ) )
               => ( ( groups713298508707869441omplex @ G @ A4 )
                  = ( groups713298508707869441omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8429_prod_Osame__carrierI,axiom,
    ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > complex,H2: nat > complex] :
      ( ( finite_finite_nat @ C4 )
     => ( ( ord_less_eq_set_nat @ A4 @ C4 )
       => ( ( ord_less_eq_set_nat @ B5 @ C4 )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_complex ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = one_one_complex ) )
             => ( ( ( groups6464643781859351333omplex @ G @ C4 )
                  = ( groups6464643781859351333omplex @ H2 @ C4 ) )
               => ( ( groups6464643781859351333omplex @ G @ A4 )
                  = ( groups6464643781859351333omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8430_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > complex,H2: complex > complex] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_complex ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = one_one_complex ) )
             => ( ( ( groups3708469109370488835omplex @ G @ C4 )
                  = ( groups3708469109370488835omplex @ H2 @ C4 ) )
               => ( ( groups3708469109370488835omplex @ G @ A4 )
                  = ( groups3708469109370488835omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8431_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = one_one_real ) )
             => ( ( ( groups1681761925125756287l_real @ G @ C4 )
                  = ( groups1681761925125756287l_real @ H2 @ C4 ) )
               => ( ( groups1681761925125756287l_real @ G @ A4 )
                  = ( groups1681761925125756287l_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8432_prod_Osame__carrierI,axiom,
    ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > real,H2: nat > real] :
      ( ( finite_finite_nat @ C4 )
     => ( ( ord_less_eq_set_nat @ A4 @ C4 )
       => ( ( ord_less_eq_set_nat @ B5 @ C4 )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = one_one_real ) )
             => ( ( ( groups129246275422532515t_real @ G @ C4 )
                  = ( groups129246275422532515t_real @ H2 @ C4 ) )
               => ( ( groups129246275422532515t_real @ G @ A4 )
                  = ( groups129246275422532515t_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8433_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = one_one_real ) )
             => ( ( ( groups766887009212190081x_real @ G @ C4 )
                  = ( groups766887009212190081x_real @ H2 @ C4 ) )
               => ( ( groups766887009212190081x_real @ G @ A4 )
                  = ( groups766887009212190081x_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8434_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups4061424788464935467al_rat @ G @ C4 )
                  = ( groups4061424788464935467al_rat @ H2 @ C4 ) )
               => ( ( groups4061424788464935467al_rat @ G @ A4 )
                  = ( groups4061424788464935467al_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8435_prod_Osame__carrierI,axiom,
    ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > rat,H2: nat > rat] :
      ( ( finite_finite_nat @ C4 )
     => ( ( ord_less_eq_set_nat @ A4 @ C4 )
       => ( ( ord_less_eq_set_nat @ B5 @ C4 )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups73079841787564623at_rat @ G @ C4 )
                  = ( groups73079841787564623at_rat @ H2 @ C4 ) )
               => ( ( groups73079841787564623at_rat @ G @ A4 )
                  = ( groups73079841787564623at_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8436_prod_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups225925009352817453ex_rat @ G @ C4 )
                  = ( groups225925009352817453ex_rat @ H2 @ C4 ) )
               => ( ( groups225925009352817453ex_rat @ G @ A4 )
                  = ( groups225925009352817453ex_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8437_prod_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_nat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H2 @ B3 )
                    = one_one_nat ) )
             => ( ( ( groups4696554848551431203al_nat @ G @ C4 )
                  = ( groups4696554848551431203al_nat @ H2 @ C4 ) )
               => ( ( groups4696554848551431203al_nat @ G @ A4 )
                  = ( groups4696554848551431203al_nat @ H2 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_8438_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ( groups6464643781859351333omplex @ G @ S2 )
            = ( groups6464643781859351333omplex @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8439_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ( groups3708469109370488835omplex @ G @ S2 )
            = ( groups3708469109370488835omplex @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8440_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > real] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ( groups129246275422532515t_real @ G @ S2 )
            = ( groups129246275422532515t_real @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8441_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ( groups766887009212190081x_real @ G @ S2 )
            = ( groups766887009212190081x_real @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8442_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_rat ) )
         => ( ( groups73079841787564623at_rat @ G @ S2 )
            = ( groups73079841787564623at_rat @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8443_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_rat ) )
         => ( ( groups225925009352817453ex_rat @ G @ S2 )
            = ( groups225925009352817453ex_rat @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8444_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ( groups861055069439313189ex_nat @ G @ S2 )
            = ( groups861055069439313189ex_nat @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8445_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_int ) )
         => ( ( groups858564598930262913ex_int @ G @ S2 )
            = ( groups858564598930262913ex_int @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8446_prod_Omono__neutral__left,axiom,
    ! [T3: set_int,S2: set_int,G: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S2 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ( groups7440179247065528705omplex @ G @ S2 )
            = ( groups7440179247065528705omplex @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8447_prod_Omono__neutral__left,axiom,
    ! [T3: set_int,S2: set_int,G: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S2 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ( groups2316167850115554303t_real @ G @ S2 )
            = ( groups2316167850115554303t_real @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_8448_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ( groups6464643781859351333omplex @ G @ T3 )
            = ( groups6464643781859351333omplex @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8449_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ( groups3708469109370488835omplex @ G @ T3 )
            = ( groups3708469109370488835omplex @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8450_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > real] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ( groups129246275422532515t_real @ G @ T3 )
            = ( groups129246275422532515t_real @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8451_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ( groups766887009212190081x_real @ G @ T3 )
            = ( groups766887009212190081x_real @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8452_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_rat ) )
         => ( ( groups73079841787564623at_rat @ G @ T3 )
            = ( groups73079841787564623at_rat @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8453_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_rat ) )
         => ( ( groups225925009352817453ex_rat @ G @ T3 )
            = ( groups225925009352817453ex_rat @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8454_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ( groups861055069439313189ex_nat @ G @ T3 )
            = ( groups861055069439313189ex_nat @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8455_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_int ) )
         => ( ( groups858564598930262913ex_int @ G @ T3 )
            = ( groups858564598930262913ex_int @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8456_prod_Omono__neutral__right,axiom,
    ! [T3: set_int,S2: set_int,G: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S2 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ( groups7440179247065528705omplex @ G @ T3 )
            = ( groups7440179247065528705omplex @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8457_prod_Omono__neutral__right,axiom,
    ! [T3: set_int,S2: set_int,G: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S2 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ( groups2316167850115554303t_real @ G @ T3 )
            = ( groups2316167850115554303t_real @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_8458_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H2: real > complex,G: real > complex] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = one_one_complex ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups713298508707869441omplex @ G @ S2 )
              = ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8459_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_nat,S2: set_nat,H2: nat > complex,G: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = one_one_complex ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups6464643781859351333omplex @ G @ S2 )
              = ( groups6464643781859351333omplex @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8460_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H2: complex > complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = one_one_complex ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups3708469109370488835omplex @ G @ S2 )
              = ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8461_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H2: real > real,G: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = one_one_real ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1681761925125756287l_real @ G @ S2 )
              = ( groups1681761925125756287l_real @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8462_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_nat,S2: set_nat,H2: nat > real,G: nat > real] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = one_one_real ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups129246275422532515t_real @ G @ S2 )
              = ( groups129246275422532515t_real @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8463_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H2: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = one_one_real ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups766887009212190081x_real @ G @ S2 )
              = ( groups766887009212190081x_real @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8464_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H2: real > rat,G: real > rat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = one_one_rat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4061424788464935467al_rat @ G @ S2 )
              = ( groups4061424788464935467al_rat @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8465_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_nat,S2: set_nat,H2: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = one_one_rat ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups73079841787564623at_rat @ G @ S2 )
              = ( groups73079841787564623at_rat @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8466_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = one_one_rat ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups225925009352817453ex_rat @ G @ S2 )
              = ( groups225925009352817453ex_rat @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8467_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H2: real > nat,G: real > nat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( H2 @ X5 )
                = one_one_nat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4696554848551431203al_nat @ G @ S2 )
              = ( groups4696554848551431203al_nat @ H2 @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_8468_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups713298508707869441omplex @ G @ T3 )
              = ( groups713298508707869441omplex @ H2 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8469_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > complex,H2: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups6464643781859351333omplex @ G @ T3 )
              = ( groups6464643781859351333omplex @ H2 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8470_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > complex,H2: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups3708469109370488835omplex @ G @ T3 )
              = ( groups3708469109370488835omplex @ H2 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8471_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1681761925125756287l_real @ G @ T3 )
              = ( groups1681761925125756287l_real @ H2 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8472_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > real,H2: nat > real] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups129246275422532515t_real @ G @ T3 )
              = ( groups129246275422532515t_real @ H2 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8473_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups766887009212190081x_real @ G @ T3 )
              = ( groups766887009212190081x_real @ H2 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8474_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_rat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4061424788464935467al_rat @ G @ T3 )
              = ( groups4061424788464935467al_rat @ H2 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8475_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > rat,H2: nat > rat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_rat ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups73079841787564623at_rat @ G @ T3 )
              = ( groups73079841787564623at_rat @ H2 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8476_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_rat ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups225925009352817453ex_rat @ G @ T3 )
              = ( groups225925009352817453ex_rat @ H2 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8477_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S2 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4696554848551431203al_nat @ G @ T3 )
              = ( groups4696554848551431203al_nat @ H2 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_8478_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_8479_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_8480_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_8481_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_8482_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( times_times_real @ ( G @ M ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_8483_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( times_times_rat @ ( G @ M ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_8484_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( times_times_nat @ ( G @ M ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_8485_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( times_times_int @ ( G @ M ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_8486_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( times_times_real @ ( G @ ( suc @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_8487_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( times_times_rat @ ( G @ ( suc @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_8488_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( times_times_nat @ ( G @ ( suc @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_8489_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( times_times_int @ ( G @ ( suc @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_8490_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( times_times_real @ ( G @ zero_zero_nat )
        @ ( groups129246275422532515t_real
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_8491_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( times_times_rat @ ( G @ zero_zero_nat )
        @ ( groups73079841787564623at_rat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_8492_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_8493_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_8494_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( times_times_real @ ( G @ M )
          @ ( groups129246275422532515t_real
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_8495_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( times_times_rat @ ( G @ M )
          @ ( groups73079841787564623at_rat
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_8496_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( times_times_nat @ ( G @ M )
          @ ( groups708209901874060359at_nat
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_8497_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( times_times_int @ ( G @ M )
          @ ( groups705719431365010083at_int
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_8498_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_8499_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_8500_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_8501_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_8502_sum__telescope,axiom,
    ! [F: nat > complex,I: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [I4: nat] : ( minus_minus_complex @ ( F @ I4 ) @ ( F @ ( suc @ I4 ) ) )
        @ ( set_ord_atMost_nat @ I ) )
      = ( minus_minus_complex @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).

% sum_telescope
thf(fact_8503_sum__telescope,axiom,
    ! [F: nat > rat,I: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ I4 ) @ ( F @ ( suc @ I4 ) ) )
        @ ( set_ord_atMost_nat @ I ) )
      = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).

% sum_telescope
thf(fact_8504_sum__telescope,axiom,
    ! [F: nat > int,I: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I4: nat] : ( minus_minus_int @ ( F @ I4 ) @ ( F @ ( suc @ I4 ) ) )
        @ ( set_ord_atMost_nat @ I ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).

% sum_telescope
thf(fact_8505_sum__telescope,axiom,
    ! [F: nat > real,I: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( minus_minus_real @ ( F @ I4 ) @ ( F @ ( suc @ I4 ) ) )
        @ ( set_ord_atMost_nat @ I ) )
      = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).

% sum_telescope
thf(fact_8506_polyfun__eq__coeffs,axiom,
    ! [C: nat > complex,N: nat,D: nat > complex] :
      ( ( ! [X: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X @ I4 ) )
              @ ( set_ord_atMost_nat @ N ) )
            = ( groups2073611262835488442omplex
              @ ^ [I4: nat] : ( times_times_complex @ ( D @ I4 ) @ ( power_power_complex @ X @ I4 ) )
              @ ( set_ord_atMost_nat @ N ) ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_eq_nat @ I4 @ N )
           => ( ( C @ I4 )
              = ( D @ I4 ) ) ) ) ) ).

% polyfun_eq_coeffs
thf(fact_8507_polyfun__eq__coeffs,axiom,
    ! [C: nat > real,N: nat,D: nat > real] :
      ( ( ! [X: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X @ I4 ) )
              @ ( set_ord_atMost_nat @ N ) )
            = ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( D @ I4 ) @ ( power_power_real @ X @ I4 ) )
              @ ( set_ord_atMost_nat @ N ) ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_eq_nat @ I4 @ N )
           => ( ( C @ I4 )
              = ( D @ I4 ) ) ) ) ) ).

% polyfun_eq_coeffs
thf(fact_8508_bounded__imp__summable,axiom,
    ! [A: nat > int,B5: int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( A @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B5 )
       => ( summable_int @ A ) ) ) ).

% bounded_imp_summable
thf(fact_8509_bounded__imp__summable,axiom,
    ! [A: nat > nat,B5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( A @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B5 )
       => ( summable_nat @ A ) ) ) ).

% bounded_imp_summable
thf(fact_8510_bounded__imp__summable,axiom,
    ! [A: nat > real,B5: real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B5 )
       => ( summable_real @ A ) ) ) ).

% bounded_imp_summable
thf(fact_8511_prod__atLeastAtMost__code,axiom,
    ! [F: nat > complex,A: nat,B2: nat] :
      ( ( groups6464643781859351333omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_fo1517530859248394432omplex
        @ ^ [A5: nat] : ( times_times_complex @ ( F @ A5 ) )
        @ A
        @ B2
        @ one_one_complex ) ) ).

% prod_atLeastAtMost_code
thf(fact_8512_prod__atLeastAtMost__code,axiom,
    ! [F: nat > real,A: nat,B2: nat] :
      ( ( groups129246275422532515t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_fo3111899725591712190t_real
        @ ^ [A5: nat] : ( times_times_real @ ( F @ A5 ) )
        @ A
        @ B2
        @ one_one_real ) ) ).

% prod_atLeastAtMost_code
thf(fact_8513_prod__atLeastAtMost__code,axiom,
    ! [F: nat > rat,A: nat,B2: nat] :
      ( ( groups73079841787564623at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_fo1949268297981939178at_rat
        @ ^ [A5: nat] : ( times_times_rat @ ( F @ A5 ) )
        @ A
        @ B2
        @ one_one_rat ) ) ).

% prod_atLeastAtMost_code
thf(fact_8514_prod__atLeastAtMost__code,axiom,
    ! [F: nat > nat,A: nat,B2: nat] :
      ( ( groups708209901874060359at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A5: nat] : ( times_times_nat @ ( F @ A5 ) )
        @ A
        @ B2
        @ one_one_nat ) ) ).

% prod_atLeastAtMost_code
thf(fact_8515_prod__atLeastAtMost__code,axiom,
    ! [F: nat > int,A: nat,B2: nat] :
      ( ( groups705719431365010083at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A5: nat] : ( times_times_int @ ( F @ A5 ) )
        @ A
        @ B2
        @ one_one_int ) ) ).

% prod_atLeastAtMost_code
thf(fact_8516_prod__mono__strict,axiom,
    ! [A4: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ A4 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
              & ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_complex )
         => ( ord_less_real @ ( groups766887009212190081x_real @ F @ A4 ) @ ( groups766887009212190081x_real @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8517_prod__mono__strict,axiom,
    ! [A4: set_nat,F: nat > real,G: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ A4 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
              & ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_nat )
         => ( ord_less_real @ ( groups129246275422532515t_real @ F @ A4 ) @ ( groups129246275422532515t_real @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8518_prod__mono__strict,axiom,
    ! [A4: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ A4 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
              & ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_int )
         => ( ord_less_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ ( groups2316167850115554303t_real @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8519_prod__mono__strict,axiom,
    ! [A4: set_real,F: real > real,G: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ A4 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
              & ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_real )
         => ( ord_less_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ ( groups1681761925125756287l_real @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8520_prod__mono__strict,axiom,
    ! [A4: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ A4 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
              & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_complex )
         => ( ord_less_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) @ ( groups225925009352817453ex_rat @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8521_prod__mono__strict,axiom,
    ! [A4: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ A4 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
              & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_nat )
         => ( ord_less_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ ( groups73079841787564623at_rat @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8522_prod__mono__strict,axiom,
    ! [A4: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ A4 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
              & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_int )
         => ( ord_less_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ ( groups1072433553688619179nt_rat @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8523_prod__mono__strict,axiom,
    ! [A4: set_real,F: real > rat,G: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ A4 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
              & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_real )
         => ( ord_less_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ ( groups4061424788464935467al_rat @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8524_prod__mono__strict,axiom,
    ! [A4: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ A4 )
           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
              & ( ord_less_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_complex )
         => ( ord_less_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) @ ( groups861055069439313189ex_nat @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8525_prod__mono__strict,axiom,
    ! [A4: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ A4 )
           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
              & ( ord_less_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_int )
         => ( ord_less_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ ( groups1707563613775114915nt_nat @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_8526_even__prod__iff,axiom,
    ! [A4: set_nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ A4 )
     => ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups3455450783089532116nteger @ F @ A4 ) )
        = ( ? [X: nat] :
              ( ( member_nat @ X @ A4 )
              & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_8527_even__prod__iff,axiom,
    ! [A4: set_int,F: int > code_integer] :
      ( ( finite_finite_int @ A4 )
     => ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups3827104343326376752nteger @ F @ A4 ) )
        = ( ? [X: int] :
              ( ( member_int @ X @ A4 )
              & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_8528_even__prod__iff,axiom,
    ! [A4: set_complex,F: complex > code_integer] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups8682486955453173170nteger @ F @ A4 ) )
        = ( ? [X: complex] :
              ( ( member_complex @ X @ A4 )
              & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_8529_even__prod__iff,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups1707563613775114915nt_nat @ F @ A4 ) )
        = ( ? [X: int] :
              ( ( member_int @ X @ A4 )
              & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_8530_even__prod__iff,axiom,
    ! [A4: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups861055069439313189ex_nat @ F @ A4 ) )
        = ( ? [X: complex] :
              ( ( member_complex @ X @ A4 )
              & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_8531_even__prod__iff,axiom,
    ! [A4: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups858564598930262913ex_int @ F @ A4 ) )
        = ( ? [X: complex] :
              ( ( member_complex @ X @ A4 )
              & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_8532_even__prod__iff,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups708209901874060359at_nat @ F @ A4 ) )
        = ( ? [X: nat] :
              ( ( member_nat @ X @ A4 )
              & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_8533_even__prod__iff,axiom,
    ! [A4: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups705719431365010083at_int @ F @ A4 ) )
        = ( ? [X: nat] :
              ( ( member_nat @ X @ A4 )
              & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_8534_even__prod__iff,axiom,
    ! [A4: set_int,F: int > int] :
      ( ( finite_finite_int @ A4 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups1705073143266064639nt_int @ F @ A4 ) )
        = ( ? [X: int] :
              ( ( member_int @ X @ A4 )
              & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_8535_sum__choose__lower,axiom,
    ! [R: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( binomial @ ( plus_plus_nat @ R @ K2 ) @ K2 )
        @ ( set_ord_atMost_nat @ N ) )
      = ( binomial @ ( suc @ ( plus_plus_nat @ R @ N ) ) @ N ) ) ).

% sum_choose_lower
thf(fact_8536_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_8537_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_8538_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: nat > real,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
        = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_8539_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: nat > rat,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
        = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_8540_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: nat > nat,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_8541_prod_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: nat > int,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_8542_norm__prod__diff,axiom,
    ! [I5: set_real,Z: real > real,W: real > real] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ I5 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I3 ) ) @ one_one_real ) )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ I5 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I3 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups1681761925125756287l_real @ Z @ I5 ) @ ( groups1681761925125756287l_real @ W @ I5 ) ) )
          @ ( groups8097168146408367636l_real
            @ ^ [I4: real] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I4 ) @ ( W @ I4 ) ) )
            @ I5 ) ) ) ) ).

% norm_prod_diff
thf(fact_8543_norm__prod__diff,axiom,
    ! [I5: set_int,Z: int > real,W: int > real] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ I5 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I3 ) ) @ one_one_real ) )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ I5 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I3 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups2316167850115554303t_real @ Z @ I5 ) @ ( groups2316167850115554303t_real @ W @ I5 ) ) )
          @ ( groups8778361861064173332t_real
            @ ^ [I4: int] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I4 ) @ ( W @ I4 ) ) )
            @ I5 ) ) ) ) ).

% norm_prod_diff
thf(fact_8544_norm__prod__diff,axiom,
    ! [I5: set_complex,Z: complex > real,W: complex > real] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ I5 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I3 ) ) @ one_one_real ) )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ I5 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I3 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups766887009212190081x_real @ Z @ I5 ) @ ( groups766887009212190081x_real @ W @ I5 ) ) )
          @ ( groups5808333547571424918x_real
            @ ^ [I4: complex] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I4 ) @ ( W @ I4 ) ) )
            @ I5 ) ) ) ) ).

% norm_prod_diff
thf(fact_8545_norm__prod__diff,axiom,
    ! [I5: set_real,Z: real > complex,W: real > complex] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ I5 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I3 ) ) @ one_one_real ) )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ I5 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I3 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups713298508707869441omplex @ Z @ I5 ) @ ( groups713298508707869441omplex @ W @ I5 ) ) )
          @ ( groups8097168146408367636l_real
            @ ^ [I4: real] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I4 ) @ ( W @ I4 ) ) )
            @ I5 ) ) ) ) ).

% norm_prod_diff
thf(fact_8546_norm__prod__diff,axiom,
    ! [I5: set_int,Z: int > complex,W: int > complex] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ I5 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I3 ) ) @ one_one_real ) )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ I5 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I3 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups7440179247065528705omplex @ Z @ I5 ) @ ( groups7440179247065528705omplex @ W @ I5 ) ) )
          @ ( groups8778361861064173332t_real
            @ ^ [I4: int] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I4 ) @ ( W @ I4 ) ) )
            @ I5 ) ) ) ) ).

% norm_prod_diff
thf(fact_8547_norm__prod__diff,axiom,
    ! [I5: set_complex,Z: complex > complex,W: complex > complex] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ I5 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I3 ) ) @ one_one_real ) )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ I5 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I3 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups3708469109370488835omplex @ Z @ I5 ) @ ( groups3708469109370488835omplex @ W @ I5 ) ) )
          @ ( groups5808333547571424918x_real
            @ ^ [I4: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I4 ) @ ( W @ I4 ) ) )
            @ I5 ) ) ) ) ).

% norm_prod_diff
thf(fact_8548_norm__prod__diff,axiom,
    ! [I5: set_nat,Z: nat > real,W: nat > real] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ I5 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I3 ) ) @ one_one_real ) )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ I5 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I3 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups129246275422532515t_real @ Z @ I5 ) @ ( groups129246275422532515t_real @ W @ I5 ) ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I4 ) @ ( W @ I4 ) ) )
            @ I5 ) ) ) ) ).

% norm_prod_diff
thf(fact_8549_norm__prod__diff,axiom,
    ! [I5: set_nat,Z: nat > complex,W: nat > complex] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ I5 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I3 ) ) @ one_one_real ) )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ I5 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I3 ) ) @ one_one_real ) )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups6464643781859351333omplex @ Z @ I5 ) @ ( groups6464643781859351333omplex @ W @ I5 ) ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I4 ) @ ( W @ I4 ) ) )
            @ I5 ) ) ) ) ).

% norm_prod_diff
thf(fact_8550_polyfun__eq__0,axiom,
    ! [C: nat > complex,N: nat] :
      ( ( ! [X: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X @ I4 ) )
              @ ( set_ord_atMost_nat @ N ) )
            = zero_zero_complex ) )
      = ( ! [I4: nat] :
            ( ( ord_less_eq_nat @ I4 @ N )
           => ( ( C @ I4 )
              = zero_zero_complex ) ) ) ) ).

% polyfun_eq_0
thf(fact_8551_polyfun__eq__0,axiom,
    ! [C: nat > real,N: nat] :
      ( ( ! [X: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X @ I4 ) )
              @ ( set_ord_atMost_nat @ N ) )
            = zero_zero_real ) )
      = ( ! [I4: nat] :
            ( ( ord_less_eq_nat @ I4 @ N )
           => ( ( C @ I4 )
              = zero_zero_real ) ) ) ) ).

% polyfun_eq_0
thf(fact_8552_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: nat > complex,N: nat,K: nat] :
      ( ! [W2: complex] :
          ( ( groups2073611262835488442omplex
            @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ W2 @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) )
          = zero_zero_complex )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( C @ K )
          = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
thf(fact_8553_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: nat > real,N: nat,K: nat] :
      ( ! [W2: real] :
          ( ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ W2 @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) )
          = zero_zero_real )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( C @ K )
          = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
thf(fact_8554_sum_OatMost__shift,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.atMost_shift
thf(fact_8555_sum_OatMost__shift,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.atMost_shift
thf(fact_8556_sum_OatMost__shift,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.atMost_shift
thf(fact_8557_sum_OatMost__shift,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.atMost_shift
thf(fact_8558_sum__up__index__split,axiom,
    ! [F: nat > rat,M: nat,N: nat] :
      ( ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N ) ) ) ) ) ).

% sum_up_index_split
thf(fact_8559_sum__up__index__split,axiom,
    ! [F: nat > int,M: nat,N: nat] :
      ( ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N ) ) ) ) ) ).

% sum_up_index_split
thf(fact_8560_sum__up__index__split,axiom,
    ! [F: nat > nat,M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N ) ) ) ) ) ).

% sum_up_index_split
thf(fact_8561_sum__up__index__split,axiom,
    ! [F: nat > real,M: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N ) ) ) ) ) ).

% sum_up_index_split
thf(fact_8562_sum_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > nat,N: nat] :
      ( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K2: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K2 @ I4 ) )
            @ ( set_ord_atMost_nat @ K2 ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% sum.triangle_reindex_eq
thf(fact_8563_sum_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > real,N: nat] :
      ( ( groups4567486121110086003t_real @ ( produc1703576794950452218t_real @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [K2: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K2 @ I4 ) )
            @ ( set_ord_atMost_nat @ K2 ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% sum.triangle_reindex_eq
thf(fact_8564_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
            @ ^ [X: nat] : X
            @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).

% fact_eq_fact_times
thf(fact_8565_prod__mono2,axiom,
    ! [B5: set_real,A4: set_real,F: real > real] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A4 @ B5 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ A4 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
           => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ ( groups1681761925125756287l_real @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_8566_prod__mono2,axiom,
    ! [B5: set_nat,A4: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( ! [B3: nat] :
              ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ A4 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
           => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A4 ) @ ( groups129246275422532515t_real @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_8567_prod__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ A4 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
           => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A4 ) @ ( groups766887009212190081x_real @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_8568_prod__mono2,axiom,
    ! [B5: set_real,A4: set_real,F: real > rat] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A4 @ B5 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ A4 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ ( groups4061424788464935467al_rat @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_8569_prod__mono2,axiom,
    ! [B5: set_nat,A4: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( ! [B3: nat] :
              ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ A4 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ ( groups73079841787564623at_rat @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_8570_prod__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ A4 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) @ ( groups225925009352817453ex_rat @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_8571_prod__mono2,axiom,
    ! [B5: set_real,A4: set_real,F: real > int] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A4 @ B5 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F @ B3 ) ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ A4 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F @ A3 ) ) )
           => ( ord_less_eq_int @ ( groups4694064378042380927al_int @ F @ A4 ) @ ( groups4694064378042380927al_int @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_8572_prod__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F @ B3 ) ) )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ A4 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F @ A3 ) ) )
           => ( ord_less_eq_int @ ( groups858564598930262913ex_int @ F @ A4 ) @ ( groups858564598930262913ex_int @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_8573_prod__mono2,axiom,
    ! [B5: set_int,A4: set_int,F: int > real] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( ! [B3: int] :
              ( ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A4 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ A4 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
           => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ ( groups2316167850115554303t_real @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_8574_prod__mono2,axiom,
    ! [B5: set_int,A4: set_int,F: int > rat] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( ! [B3: int] :
              ( ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A4 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ A4 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ ( groups1072433553688619179nt_rat @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_8575_sum__choose__diagonal,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3542108847815614940at_nat
          @ ^ [K2: nat] : ( binomial @ ( minus_minus_nat @ N @ K2 ) @ ( minus_minus_nat @ M @ K2 ) )
          @ ( set_ord_atMost_nat @ M ) )
        = ( binomial @ ( suc @ N ) @ M ) ) ) ).

% sum_choose_diagonal
thf(fact_8576_vandermonde,axiom,
    ! [M: nat,N: nat,R: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( times_times_nat @ ( binomial @ M @ K2 ) @ ( binomial @ N @ ( minus_minus_nat @ R @ K2 ) ) )
        @ ( set_ord_atMost_nat @ R ) )
      = ( binomial @ ( plus_plus_nat @ M @ N ) @ R ) ) ).

% vandermonde
thf(fact_8577_sum__gp__basic,axiom,
    ! [X3: complex,N: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X3 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_atMost_nat @ N ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ ( suc @ N ) ) ) ) ).

% sum_gp_basic
thf(fact_8578_sum__gp__basic,axiom,
    ! [X3: rat,N: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X3 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_atMost_nat @ N ) ) )
      = ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ ( suc @ N ) ) ) ) ).

% sum_gp_basic
thf(fact_8579_sum__gp__basic,axiom,
    ! [X3: int,N: nat] :
      ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X3 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_ord_atMost_nat @ N ) ) )
      = ( minus_minus_int @ one_one_int @ ( power_power_int @ X3 @ ( suc @ N ) ) ) ) ).

% sum_gp_basic
thf(fact_8580_sum__gp__basic,axiom,
    ! [X3: real,N: nat] :
      ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X3 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_atMost_nat @ N ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( suc @ N ) ) ) ) ).

% sum_gp_basic
thf(fact_8581_polyfun__finite__roots,axiom,
    ! [C: nat > complex,N: nat] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X: complex] :
              ( ( groups2073611262835488442omplex
                @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X @ I4 ) )
                @ ( set_ord_atMost_nat @ N ) )
              = zero_zero_complex ) ) )
      = ( ? [I4: nat] :
            ( ( ord_less_eq_nat @ I4 @ N )
            & ( ( C @ I4 )
             != zero_zero_complex ) ) ) ) ).

% polyfun_finite_roots
thf(fact_8582_polyfun__finite__roots,axiom,
    ! [C: nat > real,N: nat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( groups6591440286371151544t_real
                @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X @ I4 ) )
                @ ( set_ord_atMost_nat @ N ) )
              = zero_zero_real ) ) )
      = ( ? [I4: nat] :
            ( ( ord_less_eq_nat @ I4 @ N )
            & ( ( C @ I4 )
             != zero_zero_real ) ) ) ) ).

% polyfun_finite_roots
thf(fact_8583_polyfun__roots__finite,axiom,
    ! [C: nat > complex,K: nat,N: nat] :
      ( ( ( C @ K )
       != zero_zero_complex )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( groups2073611262835488442omplex
                  @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ Z2 @ I4 ) )
                  @ ( set_ord_atMost_nat @ N ) )
                = zero_zero_complex ) ) ) ) ) ).

% polyfun_roots_finite
thf(fact_8584_polyfun__roots__finite,axiom,
    ! [C: nat > real,K: nat,N: nat] :
      ( ( ( C @ K )
       != zero_zero_real )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [Z2: real] :
                ( ( groups6591440286371151544t_real
                  @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ Z2 @ I4 ) )
                  @ ( set_ord_atMost_nat @ N ) )
                = zero_zero_real ) ) ) ) ) ).

% polyfun_roots_finite
thf(fact_8585_pochhammer__Suc__prod,axiom,
    ! [A: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
      = ( groups73079841787564623at_rat
        @ ^ [I4: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod
thf(fact_8586_pochhammer__Suc__prod,axiom,
    ! [A: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
      = ( groups129246275422532515t_real
        @ ^ [I4: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod
thf(fact_8587_pochhammer__Suc__prod,axiom,
    ! [A: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
      = ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod
thf(fact_8588_pochhammer__Suc__prod,axiom,
    ! [A: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
      = ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod
thf(fact_8589_polyfun__linear__factor__root,axiom,
    ! [C: nat > complex,A: complex,N: nat] :
      ( ( ( groups2073611262835488442omplex
          @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ A @ I4 ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_complex )
     => ~ ! [B3: nat > complex] :
            ~ ! [Z3: complex] :
                ( ( groups2073611262835488442omplex
                  @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ Z3 @ I4 ) )
                  @ ( set_ord_atMost_nat @ N ) )
                = ( times_times_complex @ ( minus_minus_complex @ Z3 @ A )
                  @ ( groups2073611262835488442omplex
                    @ ^ [I4: nat] : ( times_times_complex @ ( B3 @ I4 ) @ ( power_power_complex @ Z3 @ I4 ) )
                    @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_8590_polyfun__linear__factor__root,axiom,
    ! [C: nat > rat,A: rat,N: nat] :
      ( ( ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ A @ I4 ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_rat )
     => ~ ! [B3: nat > rat] :
            ~ ! [Z3: rat] :
                ( ( groups2906978787729119204at_rat
                  @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ Z3 @ I4 ) )
                  @ ( set_ord_atMost_nat @ N ) )
                = ( times_times_rat @ ( minus_minus_rat @ Z3 @ A )
                  @ ( groups2906978787729119204at_rat
                    @ ^ [I4: nat] : ( times_times_rat @ ( B3 @ I4 ) @ ( power_power_rat @ Z3 @ I4 ) )
                    @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_8591_polyfun__linear__factor__root,axiom,
    ! [C: nat > int,A: int,N: nat] :
      ( ( ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( times_times_int @ ( C @ I4 ) @ ( power_power_int @ A @ I4 ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_int )
     => ~ ! [B3: nat > int] :
            ~ ! [Z3: int] :
                ( ( groups3539618377306564664at_int
                  @ ^ [I4: nat] : ( times_times_int @ ( C @ I4 ) @ ( power_power_int @ Z3 @ I4 ) )
                  @ ( set_ord_atMost_nat @ N ) )
                = ( times_times_int @ ( minus_minus_int @ Z3 @ A )
                  @ ( groups3539618377306564664at_int
                    @ ^ [I4: nat] : ( times_times_int @ ( B3 @ I4 ) @ ( power_power_int @ Z3 @ I4 ) )
                    @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_8592_polyfun__linear__factor__root,axiom,
    ! [C: nat > real,A: real,N: nat] :
      ( ( ( groups6591440286371151544t_real
          @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ A @ I4 ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_real )
     => ~ ! [B3: nat > real] :
            ~ ! [Z3: real] :
                ( ( groups6591440286371151544t_real
                  @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ Z3 @ I4 ) )
                  @ ( set_ord_atMost_nat @ N ) )
                = ( times_times_real @ ( minus_minus_real @ Z3 @ A )
                  @ ( groups6591440286371151544t_real
                    @ ^ [I4: nat] : ( times_times_real @ ( B3 @ I4 ) @ ( power_power_real @ Z3 @ I4 ) )
                    @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_8593_polyfun__linear__factor,axiom,
    ! [C: nat > complex,N: nat,A: complex] :
    ? [B3: nat > complex] :
    ! [Z3: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ Z3 @ I4 ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_complex
        @ ( times_times_complex @ ( minus_minus_complex @ Z3 @ A )
          @ ( groups2073611262835488442omplex
            @ ^ [I4: nat] : ( times_times_complex @ ( B3 @ I4 ) @ ( power_power_complex @ Z3 @ I4 ) )
            @ ( set_ord_lessThan_nat @ N ) ) )
        @ ( groups2073611262835488442omplex
          @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ A @ I4 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% polyfun_linear_factor
thf(fact_8594_polyfun__linear__factor,axiom,
    ! [C: nat > rat,N: nat,A: rat] :
    ? [B3: nat > rat] :
    ! [Z3: rat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ Z3 @ I4 ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_rat
        @ ( times_times_rat @ ( minus_minus_rat @ Z3 @ A )
          @ ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( times_times_rat @ ( B3 @ I4 ) @ ( power_power_rat @ Z3 @ I4 ) )
            @ ( set_ord_lessThan_nat @ N ) ) )
        @ ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ A @ I4 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% polyfun_linear_factor
thf(fact_8595_polyfun__linear__factor,axiom,
    ! [C: nat > int,N: nat,A: int] :
    ? [B3: nat > int] :
    ! [Z3: int] :
      ( ( groups3539618377306564664at_int
        @ ^ [I4: nat] : ( times_times_int @ ( C @ I4 ) @ ( power_power_int @ Z3 @ I4 ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_int
        @ ( times_times_int @ ( minus_minus_int @ Z3 @ A )
          @ ( groups3539618377306564664at_int
            @ ^ [I4: nat] : ( times_times_int @ ( B3 @ I4 ) @ ( power_power_int @ Z3 @ I4 ) )
            @ ( set_ord_lessThan_nat @ N ) ) )
        @ ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( times_times_int @ ( C @ I4 ) @ ( power_power_int @ A @ I4 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% polyfun_linear_factor
thf(fact_8596_polyfun__linear__factor,axiom,
    ! [C: nat > real,N: nat,A: real] :
    ? [B3: nat > real] :
    ! [Z3: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ Z3 @ I4 ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_real
        @ ( times_times_real @ ( minus_minus_real @ Z3 @ A )
          @ ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( times_times_real @ ( B3 @ I4 ) @ ( power_power_real @ Z3 @ I4 ) )
            @ ( set_ord_lessThan_nat @ N ) ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ A @ I4 ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% polyfun_linear_factor
thf(fact_8597_pochhammer__prod__rev,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A5: rat,N2: nat] :
          ( groups73079841787564623at_rat
          @ ^ [I4: nat] : ( plus_plus_rat @ A5 @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N2 @ I4 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_8598_pochhammer__prod__rev,axiom,
    ( comm_s7457072308508201937r_real
    = ( ^ [A5: real,N2: nat] :
          ( groups129246275422532515t_real
          @ ^ [I4: nat] : ( plus_plus_real @ A5 @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ I4 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_8599_pochhammer__prod__rev,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A5: nat,N2: nat] :
          ( groups708209901874060359at_nat
          @ ^ [I4: nat] : ( plus_plus_nat @ A5 @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N2 @ I4 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_8600_pochhammer__prod__rev,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A5: int,N2: nat] :
          ( groups705719431365010083at_int
          @ ^ [I4: nat] : ( plus_plus_int @ A5 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ I4 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_8601_sum__power__shift,axiom,
    ! [M: nat,N: nat,X3: complex] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( times_times_complex @ ( power_power_complex @ X3 @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_8602_sum__power__shift,axiom,
    ! [M: nat,N: nat,X3: rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( times_times_rat @ ( power_power_rat @ X3 @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_8603_sum__power__shift,axiom,
    ! [M: nat,N: nat,X3: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( times_times_int @ ( power_power_int @ X3 @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_8604_sum__power__shift,axiom,
    ! [M: nat,N: nat,X3: real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( times_times_real @ ( power_power_real @ X3 @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_8605_sum_Otriangle__reindex,axiom,
    ! [G: nat > nat > nat,N: nat] :
      ( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K2: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K2 @ I4 ) )
            @ ( set_ord_atMost_nat @ K2 ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.triangle_reindex
thf(fact_8606_sum_Otriangle__reindex,axiom,
    ! [G: nat > nat > real,N: nat] :
      ( ( groups4567486121110086003t_real @ ( produc1703576794950452218t_real @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [K2: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K2 @ I4 ) )
            @ ( set_ord_atMost_nat @ K2 ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.triangle_reindex
thf(fact_8607_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
          @ ^ [X: nat] : X
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).

% fact_div_fact
thf(fact_8608_summable__Cauchy__product,axiom,
    ! [A: nat > complex,B2: nat > complex] :
      ( ( summable_real
        @ ^ [K2: nat] : ( real_V1022390504157884413omplex @ ( A @ K2 ) ) )
     => ( ( summable_real
          @ ^ [K2: nat] : ( real_V1022390504157884413omplex @ ( B2 @ K2 ) ) )
       => ( summable_complex
          @ ^ [K2: nat] :
              ( groups2073611262835488442omplex
              @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( B2 @ ( minus_minus_nat @ K2 @ I4 ) ) )
              @ ( set_ord_atMost_nat @ K2 ) ) ) ) ) ).

% summable_Cauchy_product
thf(fact_8609_summable__Cauchy__product,axiom,
    ! [A: nat > real,B2: nat > real] :
      ( ( summable_real
        @ ^ [K2: nat] : ( real_V7735802525324610683m_real @ ( A @ K2 ) ) )
     => ( ( summable_real
          @ ^ [K2: nat] : ( real_V7735802525324610683m_real @ ( B2 @ K2 ) ) )
       => ( summable_real
          @ ^ [K2: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( B2 @ ( minus_minus_nat @ K2 @ I4 ) ) )
              @ ( set_ord_atMost_nat @ K2 ) ) ) ) ) ).

% summable_Cauchy_product
thf(fact_8610_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_8611_Cauchy__product,axiom,
    ! [A: nat > complex,B2: nat > complex] :
      ( ( summable_real
        @ ^ [K2: nat] : ( real_V1022390504157884413omplex @ ( A @ K2 ) ) )
     => ( ( summable_real
          @ ^ [K2: nat] : ( real_V1022390504157884413omplex @ ( B2 @ K2 ) ) )
       => ( ( times_times_complex @ ( suminf_complex @ A ) @ ( suminf_complex @ B2 ) )
          = ( suminf_complex
            @ ^ [K2: nat] :
                ( groups2073611262835488442omplex
                @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( B2 @ ( minus_minus_nat @ K2 @ I4 ) ) )
                @ ( set_ord_atMost_nat @ K2 ) ) ) ) ) ) ).

% Cauchy_product
thf(fact_8612_Cauchy__product,axiom,
    ! [A: nat > real,B2: nat > real] :
      ( ( summable_real
        @ ^ [K2: nat] : ( real_V7735802525324610683m_real @ ( A @ K2 ) ) )
     => ( ( summable_real
          @ ^ [K2: nat] : ( real_V7735802525324610683m_real @ ( B2 @ K2 ) ) )
       => ( ( times_times_real @ ( suminf_real @ A ) @ ( suminf_real @ B2 ) )
          = ( suminf_real
            @ ^ [K2: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( B2 @ ( minus_minus_nat @ K2 @ I4 ) ) )
                @ ( set_ord_atMost_nat @ K2 ) ) ) ) ) ) ).

% Cauchy_product
thf(fact_8613_binomial,axiom,
    ! [A: nat,B2: nat,N: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B2 ) @ N )
      = ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K2 ) ) @ ( power_power_nat @ A @ K2 ) ) @ ( power_power_nat @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% binomial
thf(fact_8614_prod_Oin__pairs,axiom,
    ! [G: nat > real,M: nat,N: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups129246275422532515t_real
        @ ^ [I4: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% prod.in_pairs
thf(fact_8615_prod_Oin__pairs,axiom,
    ! [G: nat > rat,M: nat,N: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [I4: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% prod.in_pairs
thf(fact_8616_prod_Oin__pairs,axiom,
    ! [G: nat > nat,M: nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% prod.in_pairs
thf(fact_8617_prod_Oin__pairs,axiom,
    ! [G: nat > int,M: nat,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% prod.in_pairs
thf(fact_8618_sum_Oin__pairs__0,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I4: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% sum.in_pairs_0
thf(fact_8619_sum_Oin__pairs__0,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups3539618377306564664at_int
        @ ^ [I4: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% sum.in_pairs_0
thf(fact_8620_sum_Oin__pairs__0,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% sum.in_pairs_0
thf(fact_8621_sum_Oin__pairs__0,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% sum.in_pairs_0
thf(fact_8622_polynomial__product,axiom,
    ! [M: nat,A: nat > complex,N: nat,B2: nat > complex,X3: complex] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ M @ I3 )
         => ( ( A @ I3 )
            = zero_zero_complex ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N @ J2 )
           => ( ( B2 @ J2 )
              = zero_zero_complex ) )
       => ( ( times_times_complex
            @ ( groups2073611262835488442omplex
              @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ X3 @ I4 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups2073611262835488442omplex
              @ ^ [J3: nat] : ( times_times_complex @ ( B2 @ J3 ) @ ( power_power_complex @ X3 @ J3 ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups2073611262835488442omplex
            @ ^ [R5: nat] :
                ( times_times_complex
                @ ( groups2073611262835488442omplex
                  @ ^ [K2: nat] : ( times_times_complex @ ( A @ K2 ) @ ( B2 @ ( minus_minus_nat @ R5 @ K2 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_complex @ X3 @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).

% polynomial_product
thf(fact_8623_polynomial__product,axiom,
    ! [M: nat,A: nat > rat,N: nat,B2: nat > rat,X3: rat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ M @ I3 )
         => ( ( A @ I3 )
            = zero_zero_rat ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N @ J2 )
           => ( ( B2 @ J2 )
              = zero_zero_rat ) )
       => ( ( times_times_rat
            @ ( groups2906978787729119204at_rat
              @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ X3 @ I4 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups2906978787729119204at_rat
              @ ^ [J3: nat] : ( times_times_rat @ ( B2 @ J3 ) @ ( power_power_rat @ X3 @ J3 ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups2906978787729119204at_rat
            @ ^ [R5: nat] :
                ( times_times_rat
                @ ( groups2906978787729119204at_rat
                  @ ^ [K2: nat] : ( times_times_rat @ ( A @ K2 ) @ ( B2 @ ( minus_minus_nat @ R5 @ K2 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_rat @ X3 @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).

% polynomial_product
thf(fact_8624_polynomial__product,axiom,
    ! [M: nat,A: nat > int,N: nat,B2: nat > int,X3: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ M @ I3 )
         => ( ( A @ I3 )
            = zero_zero_int ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N @ J2 )
           => ( ( B2 @ J2 )
              = zero_zero_int ) )
       => ( ( times_times_int
            @ ( groups3539618377306564664at_int
              @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ X3 @ I4 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups3539618377306564664at_int
              @ ^ [J3: nat] : ( times_times_int @ ( B2 @ J3 ) @ ( power_power_int @ X3 @ J3 ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups3539618377306564664at_int
            @ ^ [R5: nat] :
                ( times_times_int
                @ ( groups3539618377306564664at_int
                  @ ^ [K2: nat] : ( times_times_int @ ( A @ K2 ) @ ( B2 @ ( minus_minus_nat @ R5 @ K2 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_int @ X3 @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).

% polynomial_product
thf(fact_8625_polynomial__product,axiom,
    ! [M: nat,A: nat > real,N: nat,B2: nat > real,X3: real] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ M @ I3 )
         => ( ( A @ I3 )
            = zero_zero_real ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N @ J2 )
           => ( ( B2 @ J2 )
              = zero_zero_real ) )
       => ( ( times_times_real
            @ ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ X3 @ I4 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups6591440286371151544t_real
              @ ^ [J3: nat] : ( times_times_real @ ( B2 @ J3 ) @ ( power_power_real @ X3 @ J3 ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups6591440286371151544t_real
            @ ^ [R5: nat] :
                ( times_times_real
                @ ( groups6591440286371151544t_real
                  @ ^ [K2: nat] : ( times_times_real @ ( A @ K2 ) @ ( B2 @ ( minus_minus_nat @ R5 @ K2 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_real @ X3 @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).

% polynomial_product
thf(fact_8626_pochhammer__Suc__prod__rev,axiom,
    ! [A: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
      = ( groups73079841787564623at_rat
        @ ^ [I4: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N @ I4 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_8627_pochhammer__Suc__prod__rev,axiom,
    ! [A: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
      = ( groups129246275422532515t_real
        @ ^ [I4: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ I4 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_8628_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
      = ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N @ I4 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_8629_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
      = ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ I4 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_8630_polyfun__eq__const,axiom,
    ! [C: nat > complex,N: nat,K: complex] :
      ( ( ! [X: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X @ I4 ) )
              @ ( set_ord_atMost_nat @ N ) )
            = K ) )
      = ( ( ( C @ zero_zero_nat )
          = K )
        & ! [X: nat] :
            ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) )
           => ( ( C @ X )
              = zero_zero_complex ) ) ) ) ).

% polyfun_eq_const
thf(fact_8631_polyfun__eq__const,axiom,
    ! [C: nat > real,N: nat,K: real] :
      ( ( ! [X: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X @ I4 ) )
              @ ( set_ord_atMost_nat @ N ) )
            = K ) )
      = ( ( ( C @ zero_zero_nat )
          = K )
        & ! [X: nat] :
            ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) )
           => ( ( C @ X )
              = zero_zero_real ) ) ) ) ).

% polyfun_eq_const
thf(fact_8632_binomial__ring,axiom,
    ! [A: complex,B2: complex,N: nat] :
      ( ( power_power_complex @ ( plus_plus_complex @ A @ B2 ) @ N )
      = ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K2 ) ) @ ( power_power_complex @ A @ K2 ) ) @ ( power_power_complex @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% binomial_ring
thf(fact_8633_binomial__ring,axiom,
    ! [A: rat,B2: rat,N: nat] :
      ( ( power_power_rat @ ( plus_plus_rat @ A @ B2 ) @ N )
      = ( groups2906978787729119204at_rat
        @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K2 ) ) @ ( power_power_rat @ A @ K2 ) ) @ ( power_power_rat @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% binomial_ring
thf(fact_8634_binomial__ring,axiom,
    ! [A: int,B2: int,N: nat] :
      ( ( power_power_int @ ( plus_plus_int @ A @ B2 ) @ N )
      = ( groups3539618377306564664at_int
        @ ^ [K2: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N @ K2 ) ) @ ( power_power_int @ A @ K2 ) ) @ ( power_power_int @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% binomial_ring
thf(fact_8635_binomial__ring,axiom,
    ! [A: nat,B2: nat,N: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B2 ) @ N )
      = ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K2 ) ) @ ( power_power_nat @ A @ K2 ) ) @ ( power_power_nat @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% binomial_ring
thf(fact_8636_binomial__ring,axiom,
    ! [A: real,B2: real,N: nat] :
      ( ( power_power_real @ ( plus_plus_real @ A @ B2 ) @ N )
      = ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K2 ) ) @ ( power_power_real @ A @ K2 ) ) @ ( power_power_real @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% binomial_ring
thf(fact_8637_pochhammer__binomial__sum,axiom,
    ! [A: rat,B2: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ B2 ) @ N )
      = ( groups2906978787729119204at_rat
        @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K2 ) ) @ ( comm_s4028243227959126397er_rat @ A @ K2 ) ) @ ( comm_s4028243227959126397er_rat @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% pochhammer_binomial_sum
thf(fact_8638_pochhammer__binomial__sum,axiom,
    ! [A: int,B2: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ B2 ) @ N )
      = ( groups3539618377306564664at_int
        @ ^ [K2: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N @ K2 ) ) @ ( comm_s4660882817536571857er_int @ A @ K2 ) ) @ ( comm_s4660882817536571857er_int @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% pochhammer_binomial_sum
thf(fact_8639_pochhammer__binomial__sum,axiom,
    ! [A: real,B2: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ B2 ) @ N )
      = ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K2 ) ) @ ( comm_s7457072308508201937r_real @ A @ K2 ) ) @ ( comm_s7457072308508201937r_real @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% pochhammer_binomial_sum
thf(fact_8640_polynomial__product__nat,axiom,
    ! [M: nat,A: nat > nat,N: nat,B2: nat > nat,X3: nat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ M @ I3 )
         => ( ( A @ 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 @ ( A @ I4 ) @ ( power_power_nat @ X3 @ I4 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups3542108847815614940at_nat
              @ ^ [J3: nat] : ( times_times_nat @ ( B2 @ J3 ) @ ( power_power_nat @ X3 @ J3 ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups3542108847815614940at_nat
            @ ^ [R5: nat] :
                ( times_times_nat
                @ ( groups3542108847815614940at_nat
                  @ ^ [K2: nat] : ( times_times_nat @ ( A @ K2 ) @ ( B2 @ ( minus_minus_nat @ R5 @ K2 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_nat @ X3 @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).

% polynomial_product_nat
thf(fact_8641_choose__square__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( power_power_nat @ ( binomial @ N @ K2 ) @ ( 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_8642_Cauchy__product__sums,axiom,
    ! [A: nat > complex,B2: nat > complex] :
      ( ( summable_real
        @ ^ [K2: nat] : ( real_V1022390504157884413omplex @ ( A @ K2 ) ) )
     => ( ( summable_real
          @ ^ [K2: nat] : ( real_V1022390504157884413omplex @ ( B2 @ K2 ) ) )
       => ( sums_complex
          @ ^ [K2: nat] :
              ( groups2073611262835488442omplex
              @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( B2 @ ( minus_minus_nat @ K2 @ I4 ) ) )
              @ ( set_ord_atMost_nat @ K2 ) )
          @ ( times_times_complex @ ( suminf_complex @ A ) @ ( suminf_complex @ B2 ) ) ) ) ) ).

% Cauchy_product_sums
thf(fact_8643_Cauchy__product__sums,axiom,
    ! [A: nat > real,B2: nat > real] :
      ( ( summable_real
        @ ^ [K2: nat] : ( real_V7735802525324610683m_real @ ( A @ K2 ) ) )
     => ( ( summable_real
          @ ^ [K2: nat] : ( real_V7735802525324610683m_real @ ( B2 @ K2 ) ) )
       => ( sums_real
          @ ^ [K2: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( B2 @ ( minus_minus_nat @ K2 @ I4 ) ) )
              @ ( set_ord_atMost_nat @ K2 ) )
          @ ( times_times_real @ ( suminf_real @ A ) @ ( suminf_real @ B2 ) ) ) ) ) ).

% Cauchy_product_sums
thf(fact_8644_sum_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > complex,H2: nat > complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups2073611262835488442omplex
            @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_complex @ ( J3 = K ) @ zero_zero_complex @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups2073611262835488442omplex
            @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_8645_sum_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > rat,H2: nat > rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups2906978787729119204at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K ) @ zero_zero_rat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups2906978787729119204at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_8646_sum_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > int,H2: nat > int] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups3539618377306564664at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K ) @ zero_zero_int @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups3539618377306564664at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_8647_sum_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > nat,H2: nat > nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K ) @ zero_zero_nat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_8648_sum_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > real,H2: nat > real] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups6591440286371151544t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K ) @ zero_zero_real @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups6591440286371151544t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_8649_root__polyfun,axiom,
    ! [N: nat,Z: int,A: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( ( power_power_int @ Z @ N )
          = A )
        = ( ( groups3539618377306564664at_int
            @ ^ [I4: nat] : ( times_times_int @ ( if_int @ ( I4 = zero_zero_nat ) @ ( uminus_uminus_int @ A ) @ ( if_int @ ( I4 = N ) @ one_one_int @ zero_zero_int ) ) @ ( power_power_int @ Z @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) )
          = zero_zero_int ) ) ) ).

% root_polyfun
thf(fact_8650_root__polyfun,axiom,
    ! [N: nat,Z: complex,A: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( ( power_power_complex @ Z @ N )
          = A )
        = ( ( groups2073611262835488442omplex
            @ ^ [I4: nat] : ( times_times_complex @ ( if_complex @ ( I4 = zero_zero_nat ) @ ( uminus1482373934393186551omplex @ A ) @ ( if_complex @ ( I4 = N ) @ one_one_complex @ zero_zero_complex ) ) @ ( power_power_complex @ Z @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) )
          = zero_zero_complex ) ) ) ).

% root_polyfun
thf(fact_8651_root__polyfun,axiom,
    ! [N: nat,Z: code_integer,A: code_integer] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( ( power_8256067586552552935nteger @ Z @ N )
          = A )
        = ( ( groups7501900531339628137nteger
            @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( if_Code_integer @ ( I4 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ A ) @ ( if_Code_integer @ ( I4 = N ) @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) @ ( power_8256067586552552935nteger @ Z @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) )
          = zero_z3403309356797280102nteger ) ) ) ).

% root_polyfun
thf(fact_8652_root__polyfun,axiom,
    ! [N: nat,Z: rat,A: rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( ( power_power_rat @ Z @ N )
          = A )
        = ( ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( times_times_rat @ ( if_rat @ ( I4 = zero_zero_nat ) @ ( uminus_uminus_rat @ A ) @ ( if_rat @ ( I4 = N ) @ one_one_rat @ zero_zero_rat ) ) @ ( power_power_rat @ Z @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) )
          = zero_zero_rat ) ) ) ).

% root_polyfun
thf(fact_8653_root__polyfun,axiom,
    ! [N: nat,Z: real,A: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( ( power_power_real @ Z @ N )
          = A )
        = ( ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( times_times_real @ ( if_real @ ( I4 = zero_zero_nat ) @ ( uminus_uminus_real @ A ) @ ( if_real @ ( I4 = N ) @ one_one_real @ zero_zero_real ) ) @ ( power_power_real @ Z @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) )
          = zero_zero_real ) ) ) ).

% root_polyfun
thf(fact_8654_sum__gp0,axiom,
    ! [X3: complex,N: nat] :
      ( ( ( X3 = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_atMost_nat @ N ) )
          = ( semiri8010041392384452111omplex @ ( plus_plus_nat @ N @ one_one_nat ) ) ) )
      & ( ( X3 != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_atMost_nat @ N ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X3 ) ) ) ) ) ).

% sum_gp0
thf(fact_8655_sum__gp0,axiom,
    ! [X3: rat,N: nat] :
      ( ( ( X3 = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_atMost_nat @ N ) )
          = ( semiri681578069525770553at_rat @ ( plus_plus_nat @ N @ one_one_nat ) ) ) )
      & ( ( X3 != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_atMost_nat @ N ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X3 ) ) ) ) ) ).

% sum_gp0
thf(fact_8656_sum__gp0,axiom,
    ! [X3: real,N: nat] :
      ( ( ( X3 = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_atMost_nat @ N ) )
          = ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N @ one_one_nat ) ) ) )
      & ( ( X3 != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_atMost_nat @ N ) )
          = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ).

% sum_gp0
thf(fact_8657_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( ( groups2073611262835488442omplex
          @ ^ [I4: nat] : ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I4 ) @ ( semiri8010041392384452111omplex @ I4 ) ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_complex ) ) ).

% choose_alternating_linear_sum
thf(fact_8658_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( ( groups7501900531339628137nteger
          @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I4 ) @ ( semiri4939895301339042750nteger @ I4 ) ) @ ( semiri4939895301339042750nteger @ ( binomial @ N @ I4 ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_linear_sum
thf(fact_8659_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I4 ) @ ( semiri681578069525770553at_rat @ I4 ) ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_rat ) ) ).

% choose_alternating_linear_sum
thf(fact_8660_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( times_times_int @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I4 ) @ ( semiri1314217659103216013at_int @ I4 ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_int ) ) ).

% choose_alternating_linear_sum
thf(fact_8661_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( ( groups6591440286371151544t_real
          @ ^ [I4: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( semiri5074537144036343181t_real @ I4 ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_real ) ) ).

% choose_alternating_linear_sum
thf(fact_8662_polyfun__diff__alt,axiom,
    ! [N: nat,A: nat > complex,X3: complex,Y3: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( minus_minus_complex
          @ ( groups2073611262835488442omplex
            @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ X3 @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ Y3 @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_complex @ ( minus_minus_complex @ X3 @ Y3 )
          @ ( groups2073611262835488442omplex
            @ ^ [J3: nat] :
                ( groups2073611262835488442omplex
                @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K2 ) @ one_one_nat ) ) @ ( power_power_complex @ Y3 @ K2 ) ) @ ( power_power_complex @ X3 @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_8663_polyfun__diff__alt,axiom,
    ! [N: nat,A: nat > rat,X3: rat,Y3: rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( minus_minus_rat
          @ ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ X3 @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ Y3 @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_rat @ ( minus_minus_rat @ X3 @ Y3 )
          @ ( groups2906978787729119204at_rat
            @ ^ [J3: nat] :
                ( groups2906978787729119204at_rat
                @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K2 ) @ one_one_nat ) ) @ ( power_power_rat @ Y3 @ K2 ) ) @ ( power_power_rat @ X3 @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_8664_polyfun__diff__alt,axiom,
    ! [N: nat,A: nat > int,X3: int,Y3: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( minus_minus_int
          @ ( groups3539618377306564664at_int
            @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ X3 @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ Y3 @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_int @ ( minus_minus_int @ X3 @ Y3 )
          @ ( groups3539618377306564664at_int
            @ ^ [J3: nat] :
                ( groups3539618377306564664at_int
                @ ^ [K2: nat] : ( times_times_int @ ( times_times_int @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K2 ) @ one_one_nat ) ) @ ( power_power_int @ Y3 @ K2 ) ) @ ( power_power_int @ X3 @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_8665_polyfun__diff__alt,axiom,
    ! [N: nat,A: nat > real,X3: real,Y3: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( minus_minus_real
          @ ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ X3 @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ Y3 @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_real @ ( minus_minus_real @ X3 @ Y3 )
          @ ( groups6591440286371151544t_real
            @ ^ [J3: nat] :
                ( groups6591440286371151544t_real
                @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K2 ) @ one_one_nat ) ) @ ( power_power_real @ Y3 @ K2 ) ) @ ( power_power_real @ X3 @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_8666_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_8667_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_8668_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( groups2073611262835488442omplex
          @ ^ [I4: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I4 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_complex ) ) ).

% choose_alternating_sum
thf(fact_8669_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( groups7501900531339628137nteger
          @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I4 ) @ ( semiri4939895301339042750nteger @ ( binomial @ N @ I4 ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_sum
thf(fact_8670_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I4 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_rat ) ) ).

% choose_alternating_sum
thf(fact_8671_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I4 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_int ) ) ).

% choose_alternating_sum
thf(fact_8672_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( groups6591440286371151544t_real
          @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_real ) ) ).

% choose_alternating_sum
thf(fact_8673_polyfun__extremal__lemma,axiom,
    ! [E: real,C: nat > complex,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ E )
     => ? [M8: real] :
        ! [Z3: complex] :
          ( ( ord_less_eq_real @ M8 @ ( real_V1022390504157884413omplex @ Z3 ) )
         => ( ord_less_eq_real
            @ ( real_V1022390504157884413omplex
              @ ( groups2073611262835488442omplex
                @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ Z3 @ I4 ) )
                @ ( set_ord_atMost_nat @ N ) ) )
            @ ( times_times_real @ E @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( suc @ N ) ) ) ) ) ) ).

% polyfun_extremal_lemma
thf(fact_8674_polyfun__extremal__lemma,axiom,
    ! [E: real,C: nat > real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ E )
     => ? [M8: real] :
        ! [Z3: real] :
          ( ( ord_less_eq_real @ M8 @ ( real_V7735802525324610683m_real @ Z3 ) )
         => ( ord_less_eq_real
            @ ( real_V7735802525324610683m_real
              @ ( groups6591440286371151544t_real
                @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ Z3 @ I4 ) )
                @ ( set_ord_atMost_nat @ N ) ) )
            @ ( times_times_real @ E @ ( power_power_real @ ( real_V7735802525324610683m_real @ Z3 ) @ ( suc @ N ) ) ) ) ) ) ).

% polyfun_extremal_lemma
thf(fact_8675_polyfun__diff,axiom,
    ! [N: nat,A: nat > complex,X3: complex,Y3: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( minus_minus_complex
          @ ( groups2073611262835488442omplex
            @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ X3 @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ Y3 @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_complex @ ( minus_minus_complex @ X3 @ Y3 )
          @ ( groups2073611262835488442omplex
            @ ^ [J3: nat] :
                ( times_times_complex
                @ ( groups2073611262835488442omplex
                  @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
                @ ( power_power_complex @ X3 @ J3 ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff
thf(fact_8676_polyfun__diff,axiom,
    ! [N: nat,A: nat > rat,X3: rat,Y3: rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( minus_minus_rat
          @ ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ X3 @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ Y3 @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_rat @ ( minus_minus_rat @ X3 @ Y3 )
          @ ( groups2906978787729119204at_rat
            @ ^ [J3: nat] :
                ( times_times_rat
                @ ( groups2906978787729119204at_rat
                  @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
                @ ( power_power_rat @ X3 @ J3 ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff
thf(fact_8677_polyfun__diff,axiom,
    ! [N: nat,A: nat > int,X3: int,Y3: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( minus_minus_int
          @ ( groups3539618377306564664at_int
            @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ X3 @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ Y3 @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_int @ ( minus_minus_int @ X3 @ Y3 )
          @ ( groups3539618377306564664at_int
            @ ^ [J3: nat] :
                ( times_times_int
                @ ( groups3539618377306564664at_int
                  @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ Y3 @ ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
                @ ( power_power_int @ X3 @ J3 ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff
thf(fact_8678_polyfun__diff,axiom,
    ! [N: nat,A: nat > real,X3: real,Y3: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( minus_minus_real
          @ ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ X3 @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ Y3 @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_real @ ( minus_minus_real @ X3 @ Y3 )
          @ ( groups6591440286371151544t_real
            @ ^ [J3: nat] :
                ( times_times_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
                @ ( power_power_real @ X3 @ J3 ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff
thf(fact_8679_gbinomial__partial__row__sum,axiom,
    ! [A: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K2 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ one_one_complex ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_8680_gbinomial__partial__row__sum,axiom,
    ! [A: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K2: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K2 ) @ ( minus_minus_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ one_one_rat ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_8681_gbinomial__partial__row__sum,axiom,
    ! [A: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( gbinomial_real @ A @ K2 ) @ ( minus_minus_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ one_one_real ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_8682_gbinomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups2073611262835488442omplex @ ( gbinomial_complex @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M ) ) @ one_one_complex ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% gbinomial_r_part_sum
thf(fact_8683_gbinomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups2906978787729119204at_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M ) ) @ one_one_rat ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% gbinomial_r_part_sum
thf(fact_8684_gbinomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups6591440286371151544t_real @ ( gbinomial_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ one_one_real ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% gbinomial_r_part_sum
thf(fact_8685_cos__arcsin,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_eq_real @ X3 @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X3 ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_arcsin
thf(fact_8686_sin__arccos__abs,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
     => ( ( sin_real @ ( arccos @ Y3 ) )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_arccos_abs
thf(fact_8687_sin__arccos,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_eq_real @ X3 @ one_one_real )
       => ( ( sin_real @ ( arccos @ X3 ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_arccos
thf(fact_8688_of__nat__id,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N2: nat] : N2 ) ) ).

% of_nat_id
thf(fact_8689_prod__pos__nat__iff,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) )
        = ( ! [X: int] :
              ( ( member_int @ X @ A4 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_8690_prod__pos__nat__iff,axiom,
    ! [A4: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) )
        = ( ! [X: complex] :
              ( ( member_complex @ X @ A4 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_8691_prod__pos__nat__iff,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A4 ) )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ A4 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ X ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_8692_cos__arccos,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ( cos_real @ ( arccos @ Y3 ) )
          = Y3 ) ) ) ).

% cos_arccos
thf(fact_8693_sin__arcsin,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ( sin_real @ ( arcsin @ Y3 ) )
          = Y3 ) ) ) ).

% sin_arcsin
thf(fact_8694_arccos__0,axiom,
    ( ( arccos @ zero_zero_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arccos_0
thf(fact_8695_arcsin__1,axiom,
    ( ( arcsin @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arcsin_1
thf(fact_8696_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_8697_gbinomial__Suc__Suc,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
      = ( plus_plus_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_8698_gbinomial__Suc__Suc,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
      = ( plus_plus_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_8699_gbinomial__Suc__Suc,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
      = ( plus_plus_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_8700_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K )
        = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% gbinomial_of_nat_symmetric
thf(fact_8701_arccos__le__arccos,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_eq_real @ X3 @ Y3 )
       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
         => ( ord_less_eq_real @ ( arccos @ Y3 ) @ ( arccos @ X3 ) ) ) ) ) ).

% arccos_le_arccos
thf(fact_8702_arccos__eq__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
        & ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real ) )
     => ( ( ( arccos @ X3 )
          = ( arccos @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% arccos_eq_iff
thf(fact_8703_arccos__le__mono,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arccos @ X3 ) @ ( arccos @ Y3 ) )
          = ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ).

% arccos_le_mono
thf(fact_8704_arcsin__minus,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_eq_real @ X3 @ one_one_real )
       => ( ( arcsin @ ( uminus_uminus_real @ X3 ) )
          = ( uminus_uminus_real @ ( arcsin @ X3 ) ) ) ) ) ).

% arcsin_minus
thf(fact_8705_arcsin__le__arcsin,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_eq_real @ X3 @ Y3 )
       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
         => ( ord_less_eq_real @ ( arcsin @ X3 ) @ ( arcsin @ Y3 ) ) ) ) ) ).

% arcsin_le_arcsin
thf(fact_8706_arcsin__eq__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
       => ( ( ( arcsin @ X3 )
            = ( arcsin @ Y3 ) )
          = ( X3 = Y3 ) ) ) ) ).

% arcsin_eq_iff
thf(fact_8707_arcsin__le__mono,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arcsin @ X3 ) @ ( arcsin @ Y3 ) )
          = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ).

% arcsin_le_mono
thf(fact_8708_gbinomial__addition__formula,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ A @ ( suc @ K ) )
      = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).

% gbinomial_addition_formula
thf(fact_8709_gbinomial__addition__formula,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ A @ ( suc @ K ) )
      = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( suc @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).

% gbinomial_addition_formula
thf(fact_8710_gbinomial__addition__formula,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ A @ ( suc @ K ) )
      = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).

% gbinomial_addition_formula
thf(fact_8711_gbinomial__absorb__comp,axiom,
    ! [A: complex,K: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ A @ K ) )
      = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).

% gbinomial_absorb_comp
thf(fact_8712_gbinomial__absorb__comp,axiom,
    ! [A: rat,K: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ A @ K ) )
      = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).

% gbinomial_absorb_comp
thf(fact_8713_gbinomial__absorb__comp,axiom,
    ! [A: real,K: nat] :
      ( ( times_times_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ A @ K ) )
      = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).

% gbinomial_absorb_comp
thf(fact_8714_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A )
     => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A @ K ) ) ) ).

% gbinomial_ge_n_over_k_pow_k
thf(fact_8715_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: rat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A )
     => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A @ K ) ) ) ).

% gbinomial_ge_n_over_k_pow_k
thf(fact_8716_gbinomial__mult__1,axiom,
    ! [A: rat,K: nat] :
      ( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_8717_gbinomial__mult__1,axiom,
    ! [A: real,K: nat] :
      ( ( times_times_real @ A @ ( gbinomial_real @ A @ K ) )
      = ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_8718_gbinomial__mult__1_H,axiom,
    ! [A: rat,K: nat] :
      ( ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ A )
      = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_8719_gbinomial__mult__1_H,axiom,
    ! [A: real,K: nat] :
      ( ( times_times_real @ ( gbinomial_real @ A @ K ) @ A )
      = ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_8720_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X: int] : X
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_8721_arccos__lbound,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) ) ) ) ).

% arccos_lbound
thf(fact_8722_arccos__less__arccos,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_real @ X3 @ Y3 )
       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
         => ( ord_less_real @ ( arccos @ Y3 ) @ ( arccos @ X3 ) ) ) ) ) ).

% arccos_less_arccos
thf(fact_8723_arccos__less__mono,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
       => ( ( ord_less_real @ ( arccos @ X3 ) @ ( arccos @ Y3 ) )
          = ( ord_less_real @ Y3 @ X3 ) ) ) ) ).

% arccos_less_mono
thf(fact_8724_arccos__ubound,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi ) ) ) ).

% arccos_ubound
thf(fact_8725_arccos__cos,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ X3 @ pi )
       => ( ( arccos @ ( cos_real @ X3 ) )
          = X3 ) ) ) ).

% arccos_cos
thf(fact_8726_arcsin__less__arcsin,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_real @ X3 @ Y3 )
       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
         => ( ord_less_real @ ( arcsin @ X3 ) @ ( arcsin @ Y3 ) ) ) ) ) ).

% arcsin_less_arcsin
thf(fact_8727_arcsin__less__mono,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
       => ( ( ord_less_real @ ( arcsin @ X3 ) @ ( arcsin @ Y3 ) )
          = ( ord_less_real @ X3 @ Y3 ) ) ) ) ).

% arcsin_less_mono
thf(fact_8728_cos__arccos__abs,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
     => ( ( cos_real @ ( arccos @ Y3 ) )
        = Y3 ) ) ).

% cos_arccos_abs
thf(fact_8729_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
     => ( ( arccos @ ( cos_real @ Theta ) )
        = ( abs_abs_real @ Theta ) ) ) ).

% arccos_cos_eq_abs
thf(fact_8730_Suc__times__gbinomial,axiom,
    ! [K: nat,A: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) )
      = ( times_times_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( gbinomial_complex @ A @ K ) ) ) ).

% Suc_times_gbinomial
thf(fact_8731_Suc__times__gbinomial,axiom,
    ! [K: nat,A: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) )
      = ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K ) ) ) ).

% Suc_times_gbinomial
thf(fact_8732_Suc__times__gbinomial,axiom,
    ! [K: nat,A: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) ) )
      = ( times_times_real @ ( plus_plus_real @ A @ one_one_real ) @ ( gbinomial_real @ A @ K ) ) ) ).

% Suc_times_gbinomial
thf(fact_8733_gbinomial__absorption,axiom,
    ! [K: nat,A: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) )
      = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).

% gbinomial_absorption
thf(fact_8734_gbinomial__absorption,axiom,
    ! [K: nat,A: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) )
      = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).

% gbinomial_absorption
thf(fact_8735_gbinomial__absorption,axiom,
    ! [K: nat,A: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) )
      = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).

% gbinomial_absorption
thf(fact_8736_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: complex] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( times_times_complex @ ( gbinomial_complex @ A @ M ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M ) @ K ) )
        = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_8737_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: rat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( times_times_rat @ ( gbinomial_rat @ A @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
        = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_8738_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: real] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( times_times_real @ ( gbinomial_real @ A @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
        = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_8739_ln__prod,axiom,
    ! [I5: set_real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ I5 )
           => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ln_ln_real @ ( groups1681761925125756287l_real @ F @ I5 ) )
          = ( groups8097168146408367636l_real
            @ ^ [X: real] : ( ln_ln_real @ ( F @ X ) )
            @ I5 ) ) ) ) ).

% ln_prod
thf(fact_8740_ln__prod,axiom,
    ! [I5: set_int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ I5 )
           => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ln_ln_real @ ( groups2316167850115554303t_real @ F @ I5 ) )
          = ( groups8778361861064173332t_real
            @ ^ [X: int] : ( ln_ln_real @ ( F @ X ) )
            @ I5 ) ) ) ) ).

% ln_prod
thf(fact_8741_ln__prod,axiom,
    ! [I5: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ I5 )
           => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ln_ln_real @ ( groups766887009212190081x_real @ F @ I5 ) )
          = ( groups5808333547571424918x_real
            @ ^ [X: complex] : ( ln_ln_real @ ( F @ X ) )
            @ I5 ) ) ) ) ).

% ln_prod
thf(fact_8742_ln__prod,axiom,
    ! [I5: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ I5 )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ I5 )
           => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ln_ln_real @ ( groups129246275422532515t_real @ F @ I5 ) )
          = ( groups6591440286371151544t_real
            @ ^ [X: nat] : ( ln_ln_real @ ( F @ X ) )
            @ I5 ) ) ) ) ).

% ln_prod
thf(fact_8743_gbinomial__parallel__sum,axiom,
    ! [A: complex,N: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( gbinomial_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K2 ) ) @ K2 )
        @ ( set_ord_atMost_nat @ N ) )
      = ( gbinomial_complex @ ( plus_plus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ N ) ) ).

% gbinomial_parallel_sum
thf(fact_8744_gbinomial__parallel__sum,axiom,
    ! [A: rat,N: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K2: nat] : ( gbinomial_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K2 ) ) @ K2 )
        @ ( set_ord_atMost_nat @ N ) )
      = ( gbinomial_rat @ ( plus_plus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) @ one_one_rat ) @ N ) ) ).

% gbinomial_parallel_sum
thf(fact_8745_gbinomial__parallel__sum,axiom,
    ! [A: real,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( gbinomial_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K2 ) ) @ K2 )
        @ ( set_ord_atMost_nat @ N ) )
      = ( gbinomial_real @ ( plus_plus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) @ one_one_real ) @ N ) ) ).

% gbinomial_parallel_sum
thf(fact_8746_arccos__lt__bounded,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_real @ Y3 @ one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y3 ) )
          & ( ord_less_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).

% arccos_lt_bounded
thf(fact_8747_arccos__bounded,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) )
          & ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).

% arccos_bounded
thf(fact_8748_gbinomial__rec,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
      = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) ) ) ).

% gbinomial_rec
thf(fact_8749_gbinomial__rec,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
      = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) ) ) ).

% gbinomial_rec
thf(fact_8750_gbinomial__rec,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
      = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) ) ).

% gbinomial_rec
thf(fact_8751_gbinomial__factors,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) @ ( gbinomial_complex @ A @ K ) ) ) ).

% gbinomial_factors
thf(fact_8752_gbinomial__factors,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) @ ( gbinomial_rat @ A @ K ) ) ) ).

% gbinomial_factors
thf(fact_8753_gbinomial__factors,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
      = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) @ ( gbinomial_real @ A @ K ) ) ) ).

% gbinomial_factors
thf(fact_8754_gbinomial__index__swap,axiom,
    ! [K: nat,N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ K ) )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ N ) ) ) ).

% gbinomial_index_swap
thf(fact_8755_gbinomial__index__swap,axiom,
    ! [K: nat,N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ one_one_rat ) @ K ) )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ N ) ) ) ).

% gbinomial_index_swap
thf(fact_8756_gbinomial__index__swap,axiom,
    ! [K: nat,N: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ one_one_real ) @ K ) )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ N ) ) ) ).

% gbinomial_index_swap
thf(fact_8757_gbinomial__negated__upper,axiom,
    ( gbinomial_complex
    = ( ^ [A5: complex,K2: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ K2 ) @ A5 ) @ one_one_complex ) @ K2 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_8758_gbinomial__negated__upper,axiom,
    ( gbinomial_rat
    = ( ^ [A5: rat,K2: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ K2 ) @ A5 ) @ one_one_rat ) @ K2 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_8759_gbinomial__negated__upper,axiom,
    ( gbinomial_real
    = ( ^ [A5: real,K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( gbinomial_real @ ( minus_minus_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ K2 ) @ A5 ) @ one_one_real ) @ K2 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_8760_sin__arccos__nonzero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_real @ X3 @ one_one_real )
       => ( ( sin_real @ ( arccos @ X3 ) )
         != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
thf(fact_8761_arccos__cos2,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ X3 @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X3 )
       => ( ( arccos @ ( cos_real @ X3 ) )
          = ( uminus_uminus_real @ X3 ) ) ) ) ).

% arccos_cos2
thf(fact_8762_arccos__minus,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_eq_real @ X3 @ one_one_real )
       => ( ( arccos @ ( uminus_uminus_real @ X3 ) )
          = ( minus_minus_real @ pi @ ( arccos @ X3 ) ) ) ) ) ).

% arccos_minus
thf(fact_8763_cos__arcsin__nonzero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_real @ X3 @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X3 ) )
         != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
thf(fact_8764_prod_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > nat,N: nat] :
      ( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [K2: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K2 @ I4 ) )
            @ ( set_ord_atMost_nat @ K2 ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% prod.triangle_reindex_eq
thf(fact_8765_prod_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > int,N: nat] :
      ( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [K2: nat] :
            ( groups705719431365010083at_int
            @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K2 @ I4 ) )
            @ ( set_ord_atMost_nat @ K2 ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% prod.triangle_reindex_eq
thf(fact_8766_gbinomial__minus,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).

% gbinomial_minus
thf(fact_8767_gbinomial__minus,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).

% gbinomial_minus
thf(fact_8768_gbinomial__minus,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).

% gbinomial_minus
thf(fact_8769_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_complex @ A @ K )
        = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_8770_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_real @ A @ K )
        = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_8771_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_rat @ A @ K )
        = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_8772_arccos,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) )
          & ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi )
          & ( ( cos_real @ ( arccos @ Y3 ) )
            = Y3 ) ) ) ) ).

% arccos
thf(fact_8773_prod_Otriangle__reindex,axiom,
    ! [G: nat > nat > nat,N: nat] :
      ( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [K2: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K2 @ I4 ) )
            @ ( set_ord_atMost_nat @ K2 ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% prod.triangle_reindex
thf(fact_8774_prod_Otriangle__reindex,axiom,
    ! [G: nat > nat > int,N: nat] :
      ( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [K2: nat] :
            ( groups705719431365010083at_int
            @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K2 @ I4 ) )
            @ ( set_ord_atMost_nat @ K2 ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% prod.triangle_reindex
thf(fact_8775_gbinomial__pochhammer,axiom,
    ( gbinomial_complex
    = ( ^ [A5: complex,K2: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ A5 ) @ K2 ) ) @ ( semiri5044797733671781792omplex @ K2 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_8776_gbinomial__pochhammer,axiom,
    ( gbinomial_rat
    = ( ^ [A5: rat,K2: nat] : ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ A5 ) @ K2 ) ) @ ( semiri773545260158071498ct_rat @ K2 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_8777_gbinomial__pochhammer,axiom,
    ( gbinomial_real
    = ( ^ [A5: real,K2: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ A5 ) @ K2 ) ) @ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_8778_gbinomial__pochhammer_H,axiom,
    ( gbinomial_complex
    = ( ^ [A5: complex,K2: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A5 @ ( semiri8010041392384452111omplex @ K2 ) ) @ one_one_complex ) @ K2 ) @ ( semiri5044797733671781792omplex @ K2 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_8779_gbinomial__pochhammer_H,axiom,
    ( gbinomial_rat
    = ( ^ [A5: rat,K2: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A5 @ ( semiri681578069525770553at_rat @ K2 ) ) @ one_one_rat ) @ K2 ) @ ( semiri773545260158071498ct_rat @ K2 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_8780_gbinomial__pochhammer_H,axiom,
    ( gbinomial_real
    = ( ^ [A5: real,K2: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A5 @ ( semiri5074537144036343181t_real @ K2 ) ) @ one_one_real ) @ K2 ) @ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_8781_arccos__minus__abs,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( ( arccos @ ( uminus_uminus_real @ X3 ) )
        = ( minus_minus_real @ pi @ ( arccos @ X3 ) ) ) ) ).

% arccos_minus_abs
thf(fact_8782_gbinomial__sum__lower__neg,axiom,
    ! [A: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ M ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ M ) ) ) ).

% gbinomial_sum_lower_neg
thf(fact_8783_gbinomial__sum__lower__neg,axiom,
    ! [A: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K2: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ M ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ M ) ) ) ).

% gbinomial_sum_lower_neg
thf(fact_8784_gbinomial__sum__lower__neg,axiom,
    ! [A: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( gbinomial_real @ A @ K2 ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ M ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ M ) ) ) ).

% gbinomial_sum_lower_neg
thf(fact_8785_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: complex,X3: complex,Y3: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ A ) @ K2 ) @ ( power_power_complex @ X3 @ K2 ) ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ M @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ X3 ) @ K2 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X3 @ Y3 ) @ ( minus_minus_nat @ M @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_8786_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: rat,X3: rat,Y3: rat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ A ) @ K2 ) @ ( power_power_rat @ X3 @ K2 ) ) @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ M @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups2906978787729119204at_rat
        @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ X3 ) @ K2 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( minus_minus_nat @ M @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_8787_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: real,X3: real,Y3: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ A ) @ K2 ) @ ( power_power_real @ X3 @ K2 ) ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ M @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K2 ) @ ( power_power_real @ ( uminus_uminus_real @ X3 ) @ K2 ) ) @ ( power_power_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( minus_minus_nat @ M @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_8788_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [J3: nat] : ( gbinomial_complex @ ( semiri8010041392384452111omplex @ J3 ) @ K )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_8789_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [J3: nat] : ( gbinomial_rat @ ( semiri681578069525770553at_rat @ J3 ) @ K )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_8790_gbinomial__sum__up__index,axiom,
    ! [K: nat,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [J3: nat] : ( gbinomial_real @ ( semiri5074537144036343181t_real @ J3 ) @ K )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_8791_gbinomial__Suc,axiom,
    ! [A: complex,K: nat] :
      ( ( gbinomial_complex @ A @ ( suc @ K ) )
      = ( divide1717551699836669952omplex
        @ ( groups6464643781859351333omplex
          @ ^ [I4: nat] : ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri5044797733671781792omplex @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_8792_gbinomial__Suc,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ A @ ( suc @ K ) )
      = ( divide_divide_rat
        @ ( groups73079841787564623at_rat
          @ ^ [I4: nat] : ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri773545260158071498ct_rat @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_8793_gbinomial__Suc,axiom,
    ! [A: real,K: nat] :
      ( ( gbinomial_real @ A @ ( suc @ K ) )
      = ( divide_divide_real
        @ ( groups129246275422532515t_real
          @ ^ [I4: nat] : ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri2265585572941072030t_real @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_8794_gbinomial__Suc,axiom,
    ! [A: nat,K: nat] :
      ( ( gbinomial_nat @ A @ ( suc @ K ) )
      = ( divide_divide_nat
        @ ( groups708209901874060359at_nat
          @ ^ [I4: nat] : ( minus_minus_nat @ A @ ( semiri1316708129612266289at_nat @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri1408675320244567234ct_nat @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_8795_gbinomial__Suc,axiom,
    ! [A: int,K: nat] :
      ( ( gbinomial_int @ A @ ( suc @ K ) )
      = ( divide_divide_int
        @ ( groups705719431365010083at_int
          @ ^ [I4: nat] : ( minus_minus_int @ A @ ( semiri1314217659103216013at_int @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
        @ ( semiri1406184849735516958ct_int @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc
thf(fact_8796_gbinomial__absorption_H,axiom,
    ! [K: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_complex @ A @ K )
        = ( times_times_complex @ ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_8797_gbinomial__absorption_H,axiom,
    ! [K: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_rat @ A @ K )
        = ( times_times_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_8798_gbinomial__absorption_H,axiom,
    ! [K: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_real @ A @ K )
        = ( times_times_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_8799_arccos__le__pi2,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arccos_le_pi2
thf(fact_8800_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( divide1717551699836669952omplex @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ K2 ) ) @ K2 ) @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ K2 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ M ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_8801_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K2: nat] : ( divide_divide_rat @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ K2 ) ) @ K2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ K2 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ M ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_8802_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( divide_divide_real @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ K2 ) ) @ K2 ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ K2 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ M ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_8803_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: complex,X3: complex,Y3: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ A ) @ K2 ) @ ( power_power_complex @ X3 @ K2 ) ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ M @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ K2 ) @ A ) @ one_one_complex ) @ K2 ) @ ( power_power_complex @ X3 @ K2 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X3 @ Y3 ) @ ( minus_minus_nat @ M @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_8804_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: rat,X3: rat,Y3: rat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ A ) @ K2 ) @ ( power_power_rat @ X3 @ K2 ) ) @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ M @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups2906978787729119204at_rat
        @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ K2 ) @ A ) @ one_one_rat ) @ K2 ) @ ( power_power_rat @ X3 @ K2 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( minus_minus_nat @ M @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_8805_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: real,X3: real,Y3: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ A ) @ K2 ) @ ( power_power_real @ X3 @ K2 ) ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ M @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ K2 ) @ A ) @ one_one_real ) @ K2 ) @ ( power_power_real @ X3 @ K2 ) ) @ ( power_power_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( minus_minus_nat @ M @ K2 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_8806_arcsin__lt__bounded,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_real @ Y3 @ one_one_real )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
          & ( ord_less_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arcsin_lt_bounded
thf(fact_8807_arcsin__lbound,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) ) ) ) ).

% arcsin_lbound
thf(fact_8808_arcsin__ubound,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arcsin_ubound
thf(fact_8809_arcsin__bounded,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
          & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arcsin_bounded
thf(fact_8810_arcsin__sin,axiom,
    ! [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 ) ) ) )
       => ( ( arcsin @ ( sin_real @ X3 ) )
          = X3 ) ) ) ).

% arcsin_sin
thf(fact_8811_arcsin,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
          & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( sin_real @ ( arcsin @ Y3 ) )
            = Y3 ) ) ) ) ).

% arcsin
thf(fact_8812_arcsin__pi,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
          & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ pi )
          & ( ( sin_real @ ( arcsin @ Y3 ) )
            = Y3 ) ) ) ) ).

% arcsin_pi
thf(fact_8813_arcsin__le__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_eq_real @ X3 @ one_one_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 )
         => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( arcsin @ X3 ) @ Y3 )
              = ( ord_less_eq_real @ X3 @ ( sin_real @ Y3 ) ) ) ) ) ) ) ).

% arcsin_le_iff
thf(fact_8814_le__arcsin__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_eq_real @ X3 @ one_one_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 )
         => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ Y3 @ ( arcsin @ X3 ) )
              = ( ord_less_eq_real @ ( sin_real @ Y3 ) @ X3 ) ) ) ) ) ) ).

% le_arcsin_iff
thf(fact_8815_gbinomial__code,axiom,
    ( gbinomial_complex
    = ( ^ [A5: complex,K2: nat] :
          ( if_complex @ ( K2 = zero_zero_nat ) @ one_one_complex
          @ ( divide1717551699836669952omplex
            @ ( set_fo1517530859248394432omplex
              @ ^ [L2: nat] : ( times_times_complex @ ( minus_minus_complex @ A5 @ ( semiri8010041392384452111omplex @ L2 ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K2 @ one_one_nat )
              @ one_one_complex )
            @ ( semiri5044797733671781792omplex @ K2 ) ) ) ) ) ).

% gbinomial_code
thf(fact_8816_gbinomial__code,axiom,
    ( gbinomial_rat
    = ( ^ [A5: rat,K2: nat] :
          ( if_rat @ ( K2 = zero_zero_nat ) @ one_one_rat
          @ ( divide_divide_rat
            @ ( set_fo1949268297981939178at_rat
              @ ^ [L2: nat] : ( times_times_rat @ ( minus_minus_rat @ A5 @ ( semiri681578069525770553at_rat @ L2 ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K2 @ one_one_nat )
              @ one_one_rat )
            @ ( semiri773545260158071498ct_rat @ K2 ) ) ) ) ) ).

% gbinomial_code
thf(fact_8817_gbinomial__code,axiom,
    ( gbinomial_real
    = ( ^ [A5: real,K2: nat] :
          ( if_real @ ( K2 = zero_zero_nat ) @ one_one_real
          @ ( divide_divide_real
            @ ( set_fo3111899725591712190t_real
              @ ^ [L2: nat] : ( times_times_real @ ( minus_minus_real @ A5 @ ( semiri5074537144036343181t_real @ L2 ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K2 @ one_one_nat )
              @ one_one_real )
            @ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ) ).

% gbinomial_code
thf(fact_8818_gchoose__row__sum__weighted,axiom,
    ! [R: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K2: nat] : ( times_times_complex @ ( gbinomial_complex @ R @ K2 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ R @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K2 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ ( suc @ M ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ R @ ( suc @ M ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_8819_gchoose__row__sum__weighted,axiom,
    ! [R: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K2: nat] : ( times_times_rat @ ( gbinomial_rat @ R @ K2 ) @ ( minus_minus_rat @ ( divide_divide_rat @ R @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K2 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ ( suc @ M ) ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ R @ ( suc @ M ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_8820_gchoose__row__sum__weighted,axiom,
    ! [R: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( times_times_real @ ( gbinomial_real @ R @ K2 ) @ ( minus_minus_real @ ( divide_divide_real @ R @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
      = ( times_times_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ R @ ( suc @ M ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_8821_of__nat__code,axiom,
    ( semiri8010041392384452111omplex
    = ( ^ [N2: nat] :
          ( semiri2816024913162550771omplex
          @ ^ [I4: complex] : ( plus_plus_complex @ I4 @ one_one_complex )
          @ N2
          @ zero_zero_complex ) ) ) ).

% of_nat_code
thf(fact_8822_of__nat__code,axiom,
    ( semiri681578069525770553at_rat
    = ( ^ [N2: nat] :
          ( semiri7787848453975740701ux_rat
          @ ^ [I4: rat] : ( plus_plus_rat @ I4 @ one_one_rat )
          @ N2
          @ zero_zero_rat ) ) ) ).

% of_nat_code
thf(fact_8823_of__nat__code,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N2: nat] :
          ( semiri8420488043553186161ux_int
          @ ^ [I4: int] : ( plus_plus_int @ I4 @ one_one_int )
          @ N2
          @ zero_zero_int ) ) ) ).

% of_nat_code
thf(fact_8824_of__nat__code,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N2: nat] :
          ( semiri7260567687927622513x_real
          @ ^ [I4: real] : ( plus_plus_real @ I4 @ one_one_real )
          @ N2
          @ zero_zero_real ) ) ) ).

% of_nat_code
thf(fact_8825_of__nat__code,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N2: nat] :
          ( semiri8422978514062236437ux_nat
          @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ one_one_nat )
          @ N2
          @ zero_zero_nat ) ) ) ).

% of_nat_code
thf(fact_8826_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Y3: $o] :
      ( ( ( vEBT_VEBT_minNull @ X3 )
        = Y3 )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
       => ( ( ( X3
              = ( vEBT_Leaf @ $false @ $false ) )
           => ( Y3
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
         => ( ! [Uv2: $o] :
                ( ( X3
                  = ( vEBT_Leaf @ $true @ Uv2 ) )
               => ( ~ Y3
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
           => ( ! [Uu2: $o] :
                  ( ( X3
                    = ( vEBT_Leaf @ Uu2 @ $true ) )
                 => ( ~ Y3
                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X3
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
                   => ( Y3
                     => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                     => ( ~ Y3
                       => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
thf(fact_8827_sin__x__sin__y,axiom,
    ! [X3: real,Y3: real] :
      ( sums_real
      @ ^ [P5: nat] :
          ( groups6591440286371151544t_real
          @ ^ [N2: nat] :
              ( if_real
              @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 )
                & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
              @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) ) @ ( power_power_real @ X3 @ N2 ) ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ P5 @ N2 ) ) )
              @ zero_zero_real )
          @ ( set_ord_atMost_nat @ P5 ) )
      @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ Y3 ) ) ) ).

% sin_x_sin_y
thf(fact_8828_sin__x__sin__y,axiom,
    ! [X3: complex,Y3: complex] :
      ( sums_complex
      @ ^ [P5: nat] :
          ( groups2073611262835488442omplex
          @ ^ [N2: nat] :
              ( if_complex
              @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 )
                & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
              @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( uminus_uminus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) ) @ ( power_power_complex @ X3 @ N2 ) ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ P5 @ N2 ) ) )
              @ zero_zero_complex )
          @ ( set_ord_atMost_nat @ P5 ) )
      @ ( times_times_complex @ ( sin_complex @ X3 ) @ ( sin_complex @ Y3 ) ) ) ).

% sin_x_sin_y
thf(fact_8829_Maclaurin__sin__bound,axiom,
    ! [X3: real,N: nat] :
      ( ord_less_eq_real
      @ ( abs_abs_real
        @ ( minus_minus_real @ ( sin_real @ X3 )
          @ ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X3 @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) )
      @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X3 ) @ N ) ) ) ).

% Maclaurin_sin_bound
thf(fact_8830_sums__cos__x__plus__y,axiom,
    ! [X3: real,Y3: real] :
      ( sums_real
      @ ^ [P5: nat] :
          ( groups6591440286371151544t_real
          @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 ) @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ X3 @ N2 ) ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ P5 @ N2 ) ) ) @ zero_zero_real )
          @ ( set_ord_atMost_nat @ P5 ) )
      @ ( cos_real @ ( plus_plus_real @ X3 @ Y3 ) ) ) ).

% sums_cos_x_plus_y
thf(fact_8831_sums__cos__x__plus__y,axiom,
    ! [X3: complex,Y3: complex] :
      ( sums_complex
      @ ^ [P5: nat] :
          ( groups2073611262835488442omplex
          @ ^ [N2: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 ) @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_complex @ X3 @ N2 ) ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ P5 @ N2 ) ) ) @ zero_zero_complex )
          @ ( set_ord_atMost_nat @ P5 ) )
      @ ( cos_complex @ ( plus_plus_complex @ X3 @ Y3 ) ) ) ).

% sums_cos_x_plus_y
thf(fact_8832_mult__scaleR__left,axiom,
    ! [A: real,X3: real,Y3: real] :
      ( ( times_times_real @ ( real_V1485227260804924795R_real @ A @ X3 ) @ Y3 )
      = ( real_V1485227260804924795R_real @ A @ ( times_times_real @ X3 @ Y3 ) ) ) ).

% mult_scaleR_left
thf(fact_8833_mult__scaleR__left,axiom,
    ! [A: real,X3: complex,Y3: complex] :
      ( ( times_times_complex @ ( real_V2046097035970521341omplex @ A @ X3 ) @ Y3 )
      = ( real_V2046097035970521341omplex @ A @ ( times_times_complex @ X3 @ Y3 ) ) ) ).

% mult_scaleR_left
thf(fact_8834_mult__scaleR__right,axiom,
    ! [X3: real,A: real,Y3: real] :
      ( ( times_times_real @ X3 @ ( real_V1485227260804924795R_real @ A @ Y3 ) )
      = ( real_V1485227260804924795R_real @ A @ ( times_times_real @ X3 @ Y3 ) ) ) ).

% mult_scaleR_right
thf(fact_8835_mult__scaleR__right,axiom,
    ! [X3: complex,A: real,Y3: complex] :
      ( ( times_times_complex @ X3 @ ( real_V2046097035970521341omplex @ A @ Y3 ) )
      = ( real_V2046097035970521341omplex @ A @ ( times_times_complex @ X3 @ Y3 ) ) ) ).

% mult_scaleR_right
thf(fact_8836_inverse__mult__distrib,axiom,
    ! [A: real,B2: real] :
      ( ( inverse_inverse_real @ ( times_times_real @ A @ B2 ) )
      = ( times_times_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) ) ) ).

% inverse_mult_distrib
thf(fact_8837_inverse__mult__distrib,axiom,
    ! [A: complex,B2: complex] :
      ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B2 ) )
      = ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B2 ) ) ) ).

% inverse_mult_distrib
thf(fact_8838_inverse__mult__distrib,axiom,
    ! [A: rat,B2: rat] :
      ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B2 ) )
      = ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) ) ) ).

% inverse_mult_distrib
thf(fact_8839_inverse__divide,axiom,
    ! [A: real,B2: real] :
      ( ( inverse_inverse_real @ ( divide_divide_real @ A @ B2 ) )
      = ( divide_divide_real @ B2 @ A ) ) ).

% inverse_divide
thf(fact_8840_inverse__divide,axiom,
    ! [A: complex,B2: complex] :
      ( ( invers8013647133539491842omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
      = ( divide1717551699836669952omplex @ B2 @ A ) ) ).

% inverse_divide
thf(fact_8841_inverse__divide,axiom,
    ! [A: rat,B2: rat] :
      ( ( inverse_inverse_rat @ ( divide_divide_rat @ A @ B2 ) )
      = ( divide_divide_rat @ B2 @ A ) ) ).

% inverse_divide
thf(fact_8842_scaleR__scaleR,axiom,
    ! [A: real,B2: real,X3: real] :
      ( ( real_V1485227260804924795R_real @ A @ ( real_V1485227260804924795R_real @ B2 @ X3 ) )
      = ( real_V1485227260804924795R_real @ ( times_times_real @ A @ B2 ) @ X3 ) ) ).

% scaleR_scaleR
thf(fact_8843_scaleR__scaleR,axiom,
    ! [A: real,B2: real,X3: complex] :
      ( ( real_V2046097035970521341omplex @ A @ ( real_V2046097035970521341omplex @ B2 @ X3 ) )
      = ( real_V2046097035970521341omplex @ ( times_times_real @ A @ B2 ) @ X3 ) ) ).

% scaleR_scaleR
thf(fact_8844_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% inverse_nonpositive_iff_nonpositive
thf(fact_8845_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% inverse_nonpositive_iff_nonpositive
thf(fact_8846_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% inverse_nonnegative_iff_nonnegative
thf(fact_8847_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% inverse_nonnegative_iff_nonnegative
thf(fact_8848_inverse__positive__iff__positive,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% inverse_positive_iff_positive
thf(fact_8849_inverse__positive__iff__positive,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% inverse_positive_iff_positive
thf(fact_8850_inverse__negative__iff__negative,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% inverse_negative_iff_negative
thf(fact_8851_inverse__negative__iff__negative,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% inverse_negative_iff_negative
thf(fact_8852_inverse__less__iff__less__neg,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
          = ( ord_less_real @ B2 @ A ) ) ) ) ).

% inverse_less_iff_less_neg
thf(fact_8853_inverse__less__iff__less__neg,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
          = ( ord_less_rat @ B2 @ A ) ) ) ) ).

% inverse_less_iff_less_neg
thf(fact_8854_inverse__less__iff__less,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
          = ( ord_less_real @ B2 @ A ) ) ) ) ).

% inverse_less_iff_less
thf(fact_8855_inverse__less__iff__less,axiom,
    ! [A: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B2 )
       => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
          = ( ord_less_rat @ B2 @ A ) ) ) ) ).

% inverse_less_iff_less
thf(fact_8856_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_8857_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_8858_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_8859_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_8860_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_8861_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_8862_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_8863_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
      = ( numera6690914467698888265omplex @ K ) ) ).

% of_int_numeral
thf(fact_8864_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_real @ K ) ) ).

% of_int_numeral
thf(fact_8865_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_rat @ K ) ) ).

% of_int_numeral
thf(fact_8866_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% of_int_numeral
thf(fact_8867_scaleR__eq__iff,axiom,
    ! [B2: complex,U: real,A: complex] :
      ( ( ( plus_plus_complex @ B2 @ ( real_V2046097035970521341omplex @ U @ A ) )
        = ( plus_plus_complex @ A @ ( real_V2046097035970521341omplex @ U @ B2 ) ) )
      = ( ( A = B2 )
        | ( U = one_one_real ) ) ) ).

% scaleR_eq_iff
thf(fact_8868_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_8869_tan__periodic__int,axiom,
    ! [X3: real,I: int] :
      ( ( tan_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( ring_1_of_int_real @ I ) @ pi ) ) )
      = ( tan_real @ X3 ) ) ).

% tan_periodic_int
thf(fact_8870_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_8871_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_8872_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_8873_real__scaleR__def,axiom,
    real_V1485227260804924795R_real = times_times_real ).

% real_scaleR_def
thf(fact_8874_real__sqrt__inverse,axiom,
    ! [X3: real] :
      ( ( sqrt @ ( inverse_inverse_real @ X3 ) )
      = ( inverse_inverse_real @ ( sqrt @ X3 ) ) ) ).

% real_sqrt_inverse
thf(fact_8875_divide__real__def,axiom,
    ( divide_divide_real
    = ( ^ [X: real,Y: real] : ( times_times_real @ X @ ( inverse_inverse_real @ Y ) ) ) ) ).

% divide_real_def
thf(fact_8876_complex__scaleR,axiom,
    ! [R: real,A: real,B2: real] :
      ( ( real_V2046097035970521341omplex @ R @ ( complex2 @ A @ B2 ) )
      = ( complex2 @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ B2 ) ) ) ).

% complex_scaleR
thf(fact_8877_real__of__int__div4,axiom,
    ! [N: int,X3: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X3 ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X3 ) ) ) ).

% real_of_int_div4
thf(fact_8878_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_8879_forall__pos__mono__1,axiom,
    ! [P: real > $o,E: real] :
      ( ! [D3: real,E2: real] :
          ( ( ord_less_real @ D3 @ E2 )
         => ( ( P @ D3 )
           => ( P @ E2 ) ) )
     => ( ! [N3: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E )
         => ( P @ E ) ) ) ) ).

% forall_pos_mono_1
thf(fact_8880_forall__pos__mono,axiom,
    ! [P: real > $o,E: real] :
      ( ! [D3: real,E2: real] :
          ( ( ord_less_real @ D3 @ E2 )
         => ( ( P @ D3 )
           => ( P @ E2 ) ) )
     => ( ! [N3: nat] :
            ( ( N3 != zero_zero_nat )
           => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E )
         => ( P @ E ) ) ) ) ).

% forall_pos_mono
thf(fact_8881_real__arch__inverse,axiom,
    ! [E: real] :
      ( ( ord_less_real @ zero_zero_real @ E )
      = ( ? [N2: nat] :
            ( ( N2 != zero_zero_nat )
            & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) )
            & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ E ) ) ) ) ).

% real_arch_inverse
thf(fact_8882_int__le__real__less,axiom,
    ( ord_less_eq_int
    = ( ^ [N2: int,M3: int] : ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M3 ) @ one_one_real ) ) ) ) ).

% int_le_real_less
thf(fact_8883_sqrt__divide__self__eq,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( divide_divide_real @ ( sqrt @ X3 ) @ X3 )
        = ( inverse_inverse_real @ ( sqrt @ X3 ) ) ) ) ).

% sqrt_divide_self_eq
thf(fact_8884_int__less__real__le,axiom,
    ( ord_less_int
    = ( ^ [N2: int,M3: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) @ ( ring_1_of_int_real @ M3 ) ) ) ) ).

% int_less_real_le
thf(fact_8885_ln__inverse,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ln_ln_real @ ( inverse_inverse_real @ X3 ) )
        = ( uminus_uminus_real @ ( ln_ln_real @ X3 ) ) ) ) ).

% ln_inverse
thf(fact_8886_sin__zero__iff__int2,axiom,
    ! [X3: real] :
      ( ( ( sin_real @ X3 )
        = zero_zero_real )
      = ( ? [I4: int] :
            ( X3
            = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ pi ) ) ) ) ).

% sin_zero_iff_int2
thf(fact_8887_real__of__int__div__aux,axiom,
    ! [X3: int,D: int] :
      ( ( divide_divide_real @ ( ring_1_of_int_real @ X3 ) @ ( ring_1_of_int_real @ D ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X3 @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X3 @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).

% real_of_int_div_aux
thf(fact_8888_real__of__int__div2,axiom,
    ! [N: int,X3: 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 @ X3 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X3 ) ) ) ) ).

% real_of_int_div2
thf(fact_8889_real__of__int__div3,axiom,
    ! [N: int,X3: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X3 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X3 ) ) ) @ one_one_real ) ).

% real_of_int_div3
thf(fact_8890_exp__plus__inverse__exp,axiom,
    ! [X3: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X3 ) @ ( inverse_inverse_real @ ( exp_real @ X3 ) ) ) ) ).

% exp_plus_inverse_exp
thf(fact_8891_plus__inverse__ge__2,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X3 @ ( inverse_inverse_real @ X3 ) ) ) ) ).

% plus_inverse_ge_2
thf(fact_8892_real__inv__sqrt__pow2,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( inverse_inverse_real @ X3 ) ) ) ).

% real_inv_sqrt_pow2
thf(fact_8893_cos__one__2pi__int,axiom,
    ! [X3: real] :
      ( ( ( cos_real @ X3 )
        = one_one_real )
      = ( ? [X: int] :
            ( X3
            = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).

% cos_one_2pi_int
thf(fact_8894_tan__cot,axiom,
    ! [X3: real] :
      ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) )
      = ( inverse_inverse_real @ ( tan_real @ X3 ) ) ) ).

% tan_cot
thf(fact_8895_real__le__x__sinh,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ord_less_eq_real @ X3 @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X3 ) @ ( inverse_inverse_real @ ( exp_real @ X3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_x_sinh
thf(fact_8896_real__le__abs__sinh,axiom,
    ! [X3: real] : ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X3 ) @ ( inverse_inverse_real @ ( exp_real @ X3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_abs_sinh
thf(fact_8897_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K3: int] :
          ( ( arccos @ ( cos_real @ Theta ) )
         != ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K3 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).

% arccos_cos_eq_abs_2pi
thf(fact_8898_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X3 )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
       => ( ! [Uv2: $o] :
              ( ( X3
                = ( vEBT_Leaf @ $true @ Uv2 ) )
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
         => ( ! [Uu2: $o] :
                ( ( X3
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
thf(fact_8899_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X3 )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
       => ( ( ( X3
              = ( vEBT_Leaf @ $false @ $false ) )
           => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
thf(fact_8900_cos__zero__iff__int,axiom,
    ! [X3: real] :
      ( ( ( cos_real @ X3 )
        = zero_zero_real )
      = ( ? [I4: int] :
            ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I4 )
            & ( X3
              = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_iff_int
thf(fact_8901_sin__zero__iff__int,axiom,
    ! [X3: real] :
      ( ( ( sin_real @ X3 )
        = zero_zero_real )
      = ( ? [I4: int] :
            ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I4 )
            & ( X3
              = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_iff_int
thf(fact_8902_sinh__real__less__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( sinh_real @ X3 ) @ ( sinh_real @ Y3 ) )
      = ( ord_less_real @ X3 @ Y3 ) ) ).

% sinh_real_less_iff
thf(fact_8903_sinh__real__le__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X3 ) @ ( sinh_real @ Y3 ) )
      = ( ord_less_eq_real @ X3 @ Y3 ) ) ).

% sinh_real_le_iff
thf(fact_8904_sinh__real__neg__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( sinh_real @ X3 ) @ zero_zero_real )
      = ( ord_less_real @ X3 @ zero_zero_real ) ) ).

% sinh_real_neg_iff
thf(fact_8905_sinh__real__pos__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X3 ) )
      = ( ord_less_real @ zero_zero_real @ X3 ) ) ).

% sinh_real_pos_iff
thf(fact_8906_sinh__real__nonneg__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X3 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).

% sinh_real_nonneg_iff
thf(fact_8907_sinh__real__nonpos__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X3 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).

% sinh_real_nonpos_iff
thf(fact_8908_divide__complex__def,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [X: complex,Y: complex] : ( times_times_complex @ X @ ( invers8013647133539491842omplex @ Y ) ) ) ) ).

% divide_complex_def
thf(fact_8909_complex__inverse,axiom,
    ! [A: real,B2: real] :
      ( ( invers8013647133539491842omplex @ ( complex2 @ A @ B2 ) )
      = ( complex2 @ ( divide_divide_real @ A @ ( plus_plus_real @ ( power_power_real @ A @ ( 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 @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% complex_inverse
thf(fact_8910_sinh__ln__real,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( sinh_real @ ( ln_ln_real @ X3 ) )
        = ( divide_divide_real @ ( minus_minus_real @ X3 @ ( inverse_inverse_real @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% sinh_ln_real
thf(fact_8911_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_8912_cot__less__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 )
     => ( ( ord_less_real @ X3 @ zero_zero_real )
       => ( ord_less_real @ ( cot_real @ X3 ) @ zero_zero_real ) ) ) ).

% cot_less_zero
thf(fact_8913_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_8914_complex__i__mult__minus,axiom,
    ! [X3: complex] :
      ( ( times_times_complex @ imaginary_unit @ ( times_times_complex @ imaginary_unit @ X3 ) )
      = ( uminus1482373934393186551omplex @ X3 ) ) ).

% complex_i_mult_minus
thf(fact_8915_pred__numeral__simps_I2_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit0 @ K ) )
      = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).

% pred_numeral_simps(2)
thf(fact_8916_divide__i,axiom,
    ! [X3: complex] :
      ( ( divide1717551699836669952omplex @ X3 @ imaginary_unit )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X3 ) ) ).

% divide_i
thf(fact_8917_i__squared,axiom,
    ( ( times_times_complex @ imaginary_unit @ imaginary_unit )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% i_squared
thf(fact_8918_cot__npi,axiom,
    ! [N: nat] :
      ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% cot_npi
thf(fact_8919_divide__numeral__i,axiom,
    ! [Z: complex,N: num] :
      ( ( divide1717551699836669952omplex @ Z @ ( times_times_complex @ ( numera6690914467698888265omplex @ N ) @ imaginary_unit ) )
      = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) @ ( numera6690914467698888265omplex @ N ) ) ) ).

% divide_numeral_i
thf(fact_8920_cot__periodic,axiom,
    ! [X3: real] :
      ( ( cot_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( cot_real @ X3 ) ) ).

% cot_periodic
thf(fact_8921_power2__i,axiom,
    ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power2_i
thf(fact_8922_sinh__less__cosh__real,axiom,
    ! [X3: real] : ( ord_less_real @ ( sinh_real @ X3 ) @ ( cosh_real @ X3 ) ) ).

% sinh_less_cosh_real
thf(fact_8923_sinh__le__cosh__real,axiom,
    ! [X3: real] : ( ord_less_eq_real @ ( sinh_real @ X3 ) @ ( cosh_real @ X3 ) ) ).

% sinh_le_cosh_real
thf(fact_8924_semiring__norm_I26_J,axiom,
    ( ( bitM @ one )
    = one ) ).

% semiring_norm(26)
thf(fact_8925_complex__i__not__numeral,axiom,
    ! [W: num] :
      ( imaginary_unit
     != ( numera6690914467698888265omplex @ W ) ) ).

% complex_i_not_numeral
thf(fact_8926_cosh__real__pos,axiom,
    ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X3 ) ) ).

% cosh_real_pos
thf(fact_8927_cosh__real__nonneg,axiom,
    ! [X3: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X3 ) ) ).

% cosh_real_nonneg
thf(fact_8928_cosh__real__nonneg__le__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_eq_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y3 ) )
          = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ).

% cosh_real_nonneg_le_iff
thf(fact_8929_cosh__real__nonpos__le__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ X3 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
       => ( ( ord_less_eq_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y3 ) )
          = ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ).

% cosh_real_nonpos_le_iff
thf(fact_8930_cosh__real__ge__1,axiom,
    ! [X3: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X3 ) ) ).

% cosh_real_ge_1
thf(fact_8931_semiring__norm_I27_J,axiom,
    ! [N: num] :
      ( ( bitM @ ( bit0 @ N ) )
      = ( bit1 @ ( bitM @ N ) ) ) ).

% semiring_norm(27)
thf(fact_8932_semiring__norm_I28_J,axiom,
    ! [N: num] :
      ( ( bitM @ ( bit1 @ N ) )
      = ( bit1 @ ( bit0 @ N ) ) ) ).

% semiring_norm(28)
thf(fact_8933_i__times__eq__iff,axiom,
    ! [W: complex,Z: complex] :
      ( ( ( times_times_complex @ imaginary_unit @ W )
        = Z )
      = ( W
        = ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) ) ) ).

% i_times_eq_iff
thf(fact_8934_complex__i__not__neg__numeral,axiom,
    ! [W: num] :
      ( imaginary_unit
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).

% complex_i_not_neg_numeral
thf(fact_8935_cosh__real__strict__mono,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ Y3 )
       => ( ord_less_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y3 ) ) ) ) ).

% cosh_real_strict_mono
thf(fact_8936_cosh__real__nonneg__less__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y3 ) )
          = ( ord_less_real @ X3 @ Y3 ) ) ) ) ).

% cosh_real_nonneg_less_iff
thf(fact_8937_cosh__real__nonpos__less__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ X3 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
       => ( ( ord_less_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y3 ) )
          = ( ord_less_real @ Y3 @ X3 ) ) ) ) ).

% cosh_real_nonpos_less_iff
thf(fact_8938_arcosh__cosh__real,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( arcosh_real @ ( cosh_real @ X3 ) )
        = X3 ) ) ).

% arcosh_cosh_real
thf(fact_8939_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_8940_one__plus__BitM,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bitM @ N ) )
      = ( bit0 @ N ) ) ).

% one_plus_BitM
thf(fact_8941_BitM__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ ( bitM @ N ) @ one )
      = ( bit0 @ N ) ) ).

% BitM_plus_one
thf(fact_8942_Complex__mult__i,axiom,
    ! [A: real,B2: real] :
      ( ( times_times_complex @ ( complex2 @ A @ B2 ) @ imaginary_unit )
      = ( complex2 @ ( uminus_uminus_real @ B2 ) @ A ) ) ).

% Complex_mult_i
thf(fact_8943_i__mult__Complex,axiom,
    ! [A: real,B2: real] :
      ( ( times_times_complex @ imaginary_unit @ ( complex2 @ A @ B2 ) )
      = ( complex2 @ ( uminus_uminus_real @ B2 ) @ A ) ) ).

% i_mult_Complex
thf(fact_8944_cosh__ln__real,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( cosh_real @ ( ln_ln_real @ X3 ) )
        = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( inverse_inverse_real @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% cosh_ln_real
thf(fact_8945_cot__gt__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cot_real @ X3 ) ) ) ) ).

% cot_gt_zero
thf(fact_8946_tan__cot_H,axiom,
    ! [X3: real] :
      ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) )
      = ( cot_real @ X3 ) ) ).

% tan_cot'
thf(fact_8947_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_8948_Arg__ii,axiom,
    ( ( arg @ imaginary_unit )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% Arg_ii
thf(fact_8949_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_8950_log__base__10__eq1,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X3 )
        = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( ln_ln_real @ X3 ) ) ) ) ).

% log_base_10_eq1
thf(fact_8951_log__eq__one,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ A @ A )
          = one_one_real ) ) ) ).

% log_eq_one
thf(fact_8952_log__less__cancel__iff,axiom,
    ! [A: real,X3: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_real @ zero_zero_real @ Y3 )
         => ( ( ord_less_real @ ( log @ A @ X3 ) @ ( log @ A @ Y3 ) )
            = ( ord_less_real @ X3 @ Y3 ) ) ) ) ) ).

% log_less_cancel_iff
thf(fact_8953_log__less__one__cancel__iff,axiom,
    ! [A: real,X3: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_real @ ( log @ A @ X3 ) @ one_one_real )
          = ( ord_less_real @ X3 @ A ) ) ) ) ).

% log_less_one_cancel_iff
thf(fact_8954_one__less__log__cancel__iff,axiom,
    ! [A: real,X3: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_real @ one_one_real @ ( log @ A @ X3 ) )
          = ( ord_less_real @ A @ X3 ) ) ) ) ).

% one_less_log_cancel_iff
thf(fact_8955_log__less__zero__cancel__iff,axiom,
    ! [A: real,X3: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_real @ ( log @ A @ X3 ) @ zero_zero_real )
          = ( ord_less_real @ X3 @ one_one_real ) ) ) ) ).

% log_less_zero_cancel_iff
thf(fact_8956_zero__less__log__cancel__iff,axiom,
    ! [A: real,X3: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X3 ) )
          = ( ord_less_real @ one_one_real @ X3 ) ) ) ) ).

% zero_less_log_cancel_iff
thf(fact_8957_zero__le__log__cancel__iff,axiom,
    ! [A: real,X3: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X3 ) )
          = ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ) ).

% zero_le_log_cancel_iff
thf(fact_8958_log__le__zero__cancel__iff,axiom,
    ! [A: real,X3: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ ( log @ A @ X3 ) @ zero_zero_real )
          = ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ) ).

% log_le_zero_cancel_iff
thf(fact_8959_one__le__log__cancel__iff,axiom,
    ! [A: real,X3: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X3 ) )
          = ( ord_less_eq_real @ A @ X3 ) ) ) ) ).

% one_le_log_cancel_iff
thf(fact_8960_log__le__one__cancel__iff,axiom,
    ! [A: real,X3: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ ( log @ A @ X3 ) @ one_one_real )
          = ( ord_less_eq_real @ X3 @ A ) ) ) ) ).

% log_le_one_cancel_iff
thf(fact_8961_log__le__cancel__iff,axiom,
    ! [A: real,X3: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_real @ zero_zero_real @ Y3 )
         => ( ( ord_less_eq_real @ ( log @ A @ X3 ) @ ( log @ A @ Y3 ) )
            = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ) ).

% log_le_cancel_iff
thf(fact_8962_log__pow__cancel,axiom,
    ! [A: real,B2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ A @ ( power_power_real @ A @ B2 ) )
          = ( semiri5074537144036343181t_real @ B2 ) ) ) ) ).

% log_pow_cancel
thf(fact_8963_cis__pi__half,axiom,
    ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = imaginary_unit ) ).

% cis_pi_half
thf(fact_8964_cis__2pi,axiom,
    ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_complex ) ).

% cis_2pi
thf(fact_8965_cis__mult,axiom,
    ! [A: real,B2: real] :
      ( ( times_times_complex @ ( cis @ A ) @ ( cis @ B2 ) )
      = ( cis @ ( plus_plus_real @ A @ B2 ) ) ) ).

% cis_mult
thf(fact_8966_log__base__change,axiom,
    ! [A: real,B2: real,X3: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ B2 @ X3 )
          = ( divide_divide_real @ ( log @ A @ X3 ) @ ( log @ A @ B2 ) ) ) ) ) ).

% log_base_change
thf(fact_8967_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_8968_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_8969_DeMoivre,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_complex @ ( cis @ A ) @ N )
      = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).

% DeMoivre
thf(fact_8970_log__mult,axiom,
    ! [A: real,X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X3 )
         => ( ( ord_less_real @ zero_zero_real @ Y3 )
           => ( ( log @ A @ ( times_times_real @ X3 @ Y3 ) )
              = ( plus_plus_real @ ( log @ A @ X3 ) @ ( log @ A @ Y3 ) ) ) ) ) ) ) ).

% log_mult
thf(fact_8971_log__divide,axiom,
    ! [A: real,X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X3 )
         => ( ( ord_less_real @ zero_zero_real @ Y3 )
           => ( ( log @ A @ ( divide_divide_real @ X3 @ Y3 ) )
              = ( minus_minus_real @ ( log @ A @ X3 ) @ ( log @ A @ Y3 ) ) ) ) ) ) ) ).

% log_divide
thf(fact_8972_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_8973_log__base__pow,axiom,
    ! [A: real,N: nat,X3: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( log @ ( power_power_real @ A @ N ) @ X3 )
        = ( divide_divide_real @ ( log @ A @ X3 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log_base_pow
thf(fact_8974_log__nat__power,axiom,
    ! [X3: real,B2: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( log @ B2 @ ( power_power_real @ X3 @ N ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ X3 ) ) ) ) ).

% log_nat_power
thf(fact_8975_log__inverse,axiom,
    ! [A: real,X3: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X3 )
         => ( ( log @ A @ ( inverse_inverse_real @ X3 ) )
            = ( uminus_uminus_real @ ( log @ A @ X3 ) ) ) ) ) ) ).

% log_inverse
thf(fact_8976_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_8977_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_8978_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B2: real,X3: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ B2 )
         => ( ( B2 != one_one_real )
           => ( ( ord_less_real @ zero_zero_real @ X3 )
             => ( ( log @ A @ X3 )
                = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B2 ) @ ( ln_ln_real @ A ) ) @ ( log @ B2 @ X3 ) ) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
thf(fact_8979_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_8980_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_8981_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_8982_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_8983_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_8984_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_8985_log__base__10__eq2,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X3 )
        = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X3 ) ) ) ) ).

% log_base_10_eq2
thf(fact_8986_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_8987_bij__betw__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( bij_betw_nat_complex
        @ ^ [K2: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        @ ( set_ord_lessThan_nat @ N )
        @ ( collect_complex
          @ ^ [Z2: complex] :
              ( ( power_power_complex @ Z2 @ N )
              = one_one_complex ) ) ) ) ).

% bij_betw_roots_unity
thf(fact_8988_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_8989_ceiling__log2__div2,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).

% ceiling_log2_div2
thf(fact_8990_ceiling__divide__eq__div__numeral,axiom,
    ! [A: num,B2: num] :
      ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B2 ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B2 ) ) ) ) ).

% ceiling_divide_eq_div_numeral
thf(fact_8991_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B2: num] :
      ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B2 ) ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B2 ) ) ) ) ).

% ceiling_minus_divide_eq_div_numeral
thf(fact_8992_ceiling__log__eq__powr__iff,axiom,
    ! [X3: real,B2: real,K: nat] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ one_one_real @ B2 )
       => ( ( ( archim7802044766580827645g_real @ ( log @ B2 @ X3 ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
          = ( ( ord_less_real @ ( powr_real @ B2 @ ( semiri5074537144036343181t_real @ K ) ) @ X3 )
            & ( ord_less_eq_real @ X3 @ ( powr_real @ B2 @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).

% ceiling_log_eq_powr_iff
thf(fact_8993_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_8994_int__ge__less__than2__def,axiom,
    ( int_ge_less_than2
    = ( ^ [D2: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z6: int,Z2: int] :
                ( ( ord_less_eq_int @ D2 @ Z2 )
                & ( ord_less_int @ Z6 @ Z2 ) ) ) ) ) ) ).

% int_ge_less_than2_def
thf(fact_8995_int__ge__less__than__def,axiom,
    ( int_ge_less_than
    = ( ^ [D2: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z6: int,Z2: int] :
                ( ( ord_less_eq_int @ D2 @ Z6 )
                & ( ord_less_int @ Z6 @ Z2 ) ) ) ) ) ) ).

% int_ge_less_than_def
thf(fact_8996_powr__gt__zero,axiom,
    ! [X3: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X3 @ A ) )
      = ( X3 != zero_zero_real ) ) ).

% powr_gt_zero
thf(fact_8997_powr__nonneg__iff,axiom,
    ! [A: real,X3: real] :
      ( ( ord_less_eq_real @ ( powr_real @ A @ X3 ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
thf(fact_8998_powr__less__cancel__iff,axiom,
    ! [X3: real,A: real,B2: real] :
      ( ( ord_less_real @ one_one_real @ X3 )
     => ( ( ord_less_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B2 ) )
        = ( ord_less_real @ A @ B2 ) ) ) ).

% powr_less_cancel_iff
thf(fact_8999_powr__eq__one__iff,axiom,
    ! [A: real,X3: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( powr_real @ A @ X3 )
          = one_one_real )
        = ( X3 = zero_zero_real ) ) ) ).

% powr_eq_one_iff
thf(fact_9000_powr__one,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( powr_real @ X3 @ one_one_real )
        = X3 ) ) ).

% powr_one
thf(fact_9001_powr__one__gt__zero__iff,axiom,
    ! [X3: real] :
      ( ( ( powr_real @ X3 @ one_one_real )
        = X3 )
      = ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).

% powr_one_gt_zero_iff
thf(fact_9002_powr__le__cancel__iff,axiom,
    ! [X3: real,A: real,B2: real] :
      ( ( ord_less_real @ one_one_real @ X3 )
     => ( ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B2 ) )
        = ( ord_less_eq_real @ A @ B2 ) ) ) ).

% powr_le_cancel_iff
thf(fact_9003_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_9004_powr__log__cancel,axiom,
    ! [A: real,X3: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X3 )
         => ( ( powr_real @ A @ ( log @ A @ X3 ) )
            = X3 ) ) ) ) ).

% powr_log_cancel
thf(fact_9005_log__powr__cancel,axiom,
    ! [A: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ A @ ( powr_real @ A @ Y3 ) )
          = Y3 ) ) ) ).

% log_powr_cancel
thf(fact_9006_floor__divide__eq__div__numeral,axiom,
    ! [A: num,B2: num] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B2 ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B2 ) ) ) ).

% floor_divide_eq_div_numeral
thf(fact_9007_powr__numeral,axiom,
    ! [X3: real,N: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( powr_real @ X3 @ ( numeral_numeral_real @ N ) )
        = ( power_power_real @ X3 @ ( numeral_numeral_nat @ N ) ) ) ) ).

% powr_numeral
thf(fact_9008_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_9009_floor__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B2: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B2 ) ) ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B2 ) ) ) ).

% floor_minus_divide_eq_div_numeral
thf(fact_9010_square__powr__half,axiom,
    ! [X3: real] :
      ( ( powr_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( abs_abs_real @ X3 ) ) ).

% square_powr_half
thf(fact_9011_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_9012_powr__powr,axiom,
    ! [X3: real,A: real,B2: real] :
      ( ( powr_real @ ( powr_real @ X3 @ A ) @ B2 )
      = ( powr_real @ X3 @ ( times_times_real @ A @ B2 ) ) ) ).

% powr_powr
thf(fact_9013_powr__less__mono2__neg,axiom,
    ! [A: real,X3: real,Y3: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_real @ X3 @ Y3 )
         => ( ord_less_real @ ( powr_real @ Y3 @ A ) @ ( powr_real @ X3 @ A ) ) ) ) ) ).

% powr_less_mono2_neg
thf(fact_9014_powr__non__neg,axiom,
    ! [A: real,X3: real] :
      ~ ( ord_less_real @ ( powr_real @ A @ X3 ) @ zero_zero_real ) ).

% powr_non_neg
thf(fact_9015_powr__ge__pzero,axiom,
    ! [X3: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X3 @ Y3 ) ) ).

% powr_ge_pzero
thf(fact_9016_powr__mono2,axiom,
    ! [A: real,X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ X3 @ Y3 )
         => ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).

% powr_mono2
thf(fact_9017_powr__less__cancel,axiom,
    ! [X3: real,A: real,B2: real] :
      ( ( ord_less_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B2 ) )
     => ( ( ord_less_real @ one_one_real @ X3 )
       => ( ord_less_real @ A @ B2 ) ) ) ).

% powr_less_cancel
thf(fact_9018_powr__less__mono,axiom,
    ! [A: real,B2: real,X3: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ord_less_real @ one_one_real @ X3 )
       => ( ord_less_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B2 ) ) ) ) ).

% powr_less_mono
thf(fact_9019_powr__mono,axiom,
    ! [A: real,B2: real,X3: real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( ord_less_eq_real @ one_one_real @ X3 )
       => ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B2 ) ) ) ) ).

% powr_mono
thf(fact_9020_powr__less__mono2,axiom,
    ! [A: real,X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
       => ( ( ord_less_real @ X3 @ Y3 )
         => ( ord_less_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).

% powr_less_mono2
thf(fact_9021_powr__mono2_H,axiom,
    ! [A: real,X3: real,Y3: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ X3 @ Y3 )
         => ( ord_less_eq_real @ ( powr_real @ Y3 @ A ) @ ( powr_real @ X3 @ A ) ) ) ) ) ).

% powr_mono2'
thf(fact_9022_powr__inj,axiom,
    ! [A: real,X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ( powr_real @ A @ X3 )
            = ( powr_real @ A @ Y3 ) )
          = ( X3 = Y3 ) ) ) ) ).

% powr_inj
thf(fact_9023_gr__one__powr,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ X3 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ord_less_real @ one_one_real @ ( powr_real @ X3 @ Y3 ) ) ) ) ).

% gr_one_powr
thf(fact_9024_powr__le1,axiom,
    ! [A: real,X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ X3 @ one_one_real )
         => ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ one_one_real ) ) ) ) ).

% powr_le1
thf(fact_9025_powr__mono__both,axiom,
    ! [A: real,B2: real,X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ B2 )
       => ( ( ord_less_eq_real @ one_one_real @ X3 )
         => ( ( ord_less_eq_real @ X3 @ Y3 )
           => ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y3 @ B2 ) ) ) ) ) ) ).

% powr_mono_both
thf(fact_9026_ge__one__powr__ge__zero,axiom,
    ! [X3: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X3 @ A ) ) ) ) ).

% ge_one_powr_ge_zero
thf(fact_9027_powr__divide,axiom,
    ! [X3: real,Y3: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( powr_real @ ( divide_divide_real @ X3 @ Y3 ) @ A )
          = ( divide_divide_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).

% powr_divide
thf(fact_9028_powr__mult,axiom,
    ! [X3: real,Y3: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( powr_real @ ( times_times_real @ X3 @ Y3 ) @ A )
          = ( times_times_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).

% powr_mult
thf(fact_9029_inverse__powr,axiom,
    ! [Y3: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ( ( powr_real @ ( inverse_inverse_real @ Y3 ) @ A )
        = ( inverse_inverse_real @ ( powr_real @ Y3 @ A ) ) ) ) ).

% inverse_powr
thf(fact_9030_divide__powr__uminus,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( divide_divide_real @ A @ ( powr_real @ B2 @ C ) )
      = ( times_times_real @ A @ ( powr_real @ B2 @ ( uminus_uminus_real @ C ) ) ) ) ).

% divide_powr_uminus
thf(fact_9031_ln__powr,axiom,
    ! [X3: real,Y3: real] :
      ( ( X3 != zero_zero_real )
     => ( ( ln_ln_real @ ( powr_real @ X3 @ Y3 ) )
        = ( times_times_real @ Y3 @ ( ln_ln_real @ X3 ) ) ) ) ).

% ln_powr
thf(fact_9032_log__powr,axiom,
    ! [X3: real,B2: real,Y3: real] :
      ( ( X3 != zero_zero_real )
     => ( ( log @ B2 @ ( powr_real @ X3 @ Y3 ) )
        = ( times_times_real @ Y3 @ ( log @ B2 @ X3 ) ) ) ) ).

% log_powr
thf(fact_9033_floor__log__eq__powr__iff,axiom,
    ! [X3: real,B2: real,K: int] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ one_one_real @ B2 )
       => ( ( ( archim6058952711729229775r_real @ ( log @ B2 @ X3 ) )
            = K )
          = ( ( ord_less_eq_real @ ( powr_real @ B2 @ ( ring_1_of_int_real @ K ) ) @ X3 )
            & ( ord_less_real @ X3 @ ( powr_real @ B2 @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).

% floor_log_eq_powr_iff
thf(fact_9034_powr__realpow,axiom,
    ! [X3: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( powr_real @ X3 @ ( semiri5074537144036343181t_real @ N ) )
        = ( power_power_real @ X3 @ N ) ) ) ).

% powr_realpow
thf(fact_9035_powr__less__iff,axiom,
    ! [B2: real,X3: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_real @ ( powr_real @ B2 @ Y3 ) @ X3 )
          = ( ord_less_real @ Y3 @ ( log @ B2 @ X3 ) ) ) ) ) ).

% powr_less_iff
thf(fact_9036_less__powr__iff,axiom,
    ! [B2: real,X3: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_real @ X3 @ ( powr_real @ B2 @ Y3 ) )
          = ( ord_less_real @ ( log @ B2 @ X3 ) @ Y3 ) ) ) ) ).

% less_powr_iff
thf(fact_9037_log__less__iff,axiom,
    ! [B2: real,X3: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_real @ ( log @ B2 @ X3 ) @ Y3 )
          = ( ord_less_real @ X3 @ ( powr_real @ B2 @ Y3 ) ) ) ) ) ).

% log_less_iff
thf(fact_9038_less__log__iff,axiom,
    ! [B2: real,X3: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_real @ Y3 @ ( log @ B2 @ X3 ) )
          = ( ord_less_real @ ( powr_real @ B2 @ Y3 ) @ X3 ) ) ) ) ).

% less_log_iff
thf(fact_9039_floor__eq,axiom,
    ! [N: int,X3: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X3 )
     => ( ( ord_less_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X3 )
          = N ) ) ) ).

% floor_eq
thf(fact_9040_real__of__int__floor__add__one__gt,axiom,
    ! [R: real] : ( ord_less_real @ R @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_gt
thf(fact_9041_real__of__int__floor__add__one__ge,axiom,
    ! [R: real] : ( ord_less_eq_real @ R @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_ge
thf(fact_9042_real__of__int__floor__gt__diff__one,axiom,
    ! [R: real] : ( ord_less_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) ) ).

% real_of_int_floor_gt_diff_one
thf(fact_9043_real__of__int__floor__ge__diff__one,axiom,
    ! [R: real] : ( ord_less_eq_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) ) ).

% real_of_int_floor_ge_diff_one
thf(fact_9044_powr__neg__one,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( powr_real @ X3 @ ( uminus_uminus_real @ one_one_real ) )
        = ( divide_divide_real @ one_one_real @ X3 ) ) ) ).

% powr_neg_one
thf(fact_9045_powr__mult__base,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( times_times_real @ X3 @ ( powr_real @ X3 @ Y3 ) )
        = ( powr_real @ X3 @ ( plus_plus_real @ one_one_real @ Y3 ) ) ) ) ).

% powr_mult_base
thf(fact_9046_le__log__iff,axiom,
    ! [B2: real,X3: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ Y3 @ ( log @ B2 @ X3 ) )
          = ( ord_less_eq_real @ ( powr_real @ B2 @ Y3 ) @ X3 ) ) ) ) ).

% le_log_iff
thf(fact_9047_log__le__iff,axiom,
    ! [B2: real,X3: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ ( log @ B2 @ X3 ) @ Y3 )
          = ( ord_less_eq_real @ X3 @ ( powr_real @ B2 @ Y3 ) ) ) ) ) ).

% log_le_iff
thf(fact_9048_le__powr__iff,axiom,
    ! [B2: real,X3: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ X3 @ ( powr_real @ B2 @ Y3 ) )
          = ( ord_less_eq_real @ ( log @ B2 @ X3 ) @ Y3 ) ) ) ) ).

% le_powr_iff
thf(fact_9049_powr__le__iff,axiom,
    ! [B2: real,X3: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ ( powr_real @ B2 @ Y3 ) @ X3 )
          = ( ord_less_eq_real @ Y3 @ ( log @ B2 @ X3 ) ) ) ) ) ).

% powr_le_iff
thf(fact_9050_floor__eq2,axiom,
    ! [N: int,X3: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X3 )
     => ( ( ord_less_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X3 )
          = N ) ) ) ).

% floor_eq2
thf(fact_9051_floor__divide__real__eq__div,axiom,
    ! [B2: int,A: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ B2 )
     => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B2 ) ) )
        = ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B2 ) ) ) ).

% floor_divide_real_eq_div
thf(fact_9052_ln__powr__bound,axiom,
    ! [X3: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X3 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( divide_divide_real @ ( powr_real @ X3 @ A ) @ A ) ) ) ) ).

% ln_powr_bound
thf(fact_9053_ln__powr__bound2,axiom,
    ! [X3: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X3 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X3 ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X3 ) ) ) ) ).

% ln_powr_bound2
thf(fact_9054_log__add__eq__powr,axiom,
    ! [B2: real,X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ B2 )
     => ( ( B2 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X3 )
         => ( ( plus_plus_real @ ( log @ B2 @ X3 ) @ Y3 )
            = ( log @ B2 @ ( times_times_real @ X3 @ ( powr_real @ B2 @ Y3 ) ) ) ) ) ) ) ).

% log_add_eq_powr
thf(fact_9055_add__log__eq__powr,axiom,
    ! [B2: real,X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ B2 )
     => ( ( B2 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X3 )
         => ( ( plus_plus_real @ Y3 @ ( log @ B2 @ X3 ) )
            = ( log @ B2 @ ( times_times_real @ ( powr_real @ B2 @ Y3 ) @ X3 ) ) ) ) ) ) ).

% add_log_eq_powr
thf(fact_9056_minus__log__eq__powr,axiom,
    ! [B2: real,X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ B2 )
     => ( ( B2 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X3 )
         => ( ( minus_minus_real @ Y3 @ ( log @ B2 @ X3 ) )
            = ( log @ B2 @ ( divide_divide_real @ ( powr_real @ B2 @ Y3 ) @ X3 ) ) ) ) ) ) ).

% minus_log_eq_powr
thf(fact_9057_log__minus__eq__powr,axiom,
    ! [B2: real,X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ B2 )
     => ( ( B2 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X3 )
         => ( ( minus_minus_real @ ( log @ B2 @ X3 ) @ Y3 )
            = ( log @ B2 @ ( times_times_real @ X3 @ ( powr_real @ B2 @ ( uminus_uminus_real @ Y3 ) ) ) ) ) ) ) ) ).

% log_minus_eq_powr
thf(fact_9058_powr__half__sqrt,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( powr_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        = ( sqrt @ X3 ) ) ) ).

% powr_half_sqrt
thf(fact_9059_powr__neg__numeral,axiom,
    ! [X3: real,N: num] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( powr_real @ X3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
        = ( divide_divide_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ N ) ) ) ) ) ).

% powr_neg_numeral
thf(fact_9060_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_9061_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_9062_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_9063_exp__pi__i,axiom,
    ( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% exp_pi_i
thf(fact_9064_exp__pi__i_H,axiom,
    ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% exp_pi_i'
thf(fact_9065_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_9066_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_9067_complex__exp__exists,axiom,
    ! [Z: complex] :
    ? [A3: complex,R3: real] :
      ( Z
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( exp_complex @ A3 ) ) ) ).

% complex_exp_exists
thf(fact_9068_complex__of__real__mult__Complex,axiom,
    ! [R: real,X3: real,Y3: real] :
      ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ ( complex2 @ X3 @ Y3 ) )
      = ( complex2 @ ( times_times_real @ R @ X3 ) @ ( times_times_real @ R @ Y3 ) ) ) ).

% complex_of_real_mult_Complex
thf(fact_9069_Complex__mult__complex__of__real,axiom,
    ! [X3: real,Y3: real,R: real] :
      ( ( times_times_complex @ ( complex2 @ X3 @ Y3 ) @ ( real_V4546457046886955230omplex @ R ) )
      = ( complex2 @ ( times_times_real @ X3 @ R ) @ ( times_times_real @ Y3 @ R ) ) ) ).

% Complex_mult_complex_of_real
thf(fact_9070_cis__conv__exp,axiom,
    ( cis
    = ( ^ [B4: real] : ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B4 ) ) ) ) ) ).

% cis_conv_exp
thf(fact_9071_i__complex__of__real,axiom,
    ! [R: real] :
      ( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R ) )
      = ( complex2 @ zero_zero_real @ R ) ) ).

% i_complex_of_real
thf(fact_9072_complex__of__real__i,axiom,
    ! [R: real] :
      ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ imaginary_unit )
      = ( complex2 @ zero_zero_real @ R ) ) ).

% complex_of_real_i
thf(fact_9073_Complex__eq,axiom,
    ( complex2
    = ( ^ [A5: real,B4: real] : ( plus_plus_complex @ ( real_V4546457046886955230omplex @ A5 ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B4 ) ) ) ) ) ).

% Complex_eq
thf(fact_9074_complex__split__polar,axiom,
    ! [Z: complex] :
    ? [R3: real,A3: real] :
      ( Z
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A3 ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A3 ) ) ) ) ) ) ).

% complex_split_polar
thf(fact_9075_cmod__unit__one,axiom,
    ! [A: real] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) )
      = one_one_real ) ).

% cmod_unit_one
thf(fact_9076_cmod__complex__polar,axiom,
    ! [R: real,A: real] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) ) )
      = ( abs_abs_real @ R ) ) ).

% cmod_complex_polar
thf(fact_9077_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_9078_arctan__def,axiom,
    ( arctan
    = ( ^ [Y: real] :
          ( the_real
          @ ^ [X: real] :
              ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
              & ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( tan_real @ X )
                = Y ) ) ) ) ) ).

% arctan_def
thf(fact_9079_arcsin__def,axiom,
    ( arcsin
    = ( ^ [Y: real] :
          ( the_real
          @ ^ [X: real] :
              ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
              & ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( sin_real @ X )
                = Y ) ) ) ) ) ).

% arcsin_def
thf(fact_9080_even__set__encode__iff,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A4 ) )
        = ( ~ ( member_nat @ zero_zero_nat @ A4 ) ) ) ) ).

% even_set_encode_iff
thf(fact_9081_power2__csqrt,axiom,
    ! [Z: complex] :
      ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = Z ) ).

% power2_csqrt
thf(fact_9082_ln__neg__is__const,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ X3 @ zero_zero_real )
     => ( ( ln_ln_real @ X3 )
        = ( the_real
          @ ^ [X: real] : $false ) ) ) ).

% ln_neg_is_const
thf(fact_9083_of__real__sqrt,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( real_V4546457046886955230omplex @ ( sqrt @ X3 ) )
        = ( csqrt @ ( real_V4546457046886955230omplex @ X3 ) ) ) ) ).

% of_real_sqrt
thf(fact_9084_arccos__def,axiom,
    ( arccos
    = ( ^ [Y: real] :
          ( the_real
          @ ^ [X: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ X )
              & ( ord_less_eq_real @ X @ pi )
              & ( ( cos_real @ X )
                = Y ) ) ) ) ) ).

% arccos_def
thf(fact_9085_set__encode__def,axiom,
    ( nat_set_encode
    = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% set_encode_def
thf(fact_9086_pi__half,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
    = ( the_real
      @ ^ [X: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X )
          & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
          & ( ( cos_real @ X )
            = zero_zero_real ) ) ) ) ).

% pi_half
thf(fact_9087_pi__def,axiom,
    ( pi
    = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
      @ ( the_real
        @ ^ [X: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X )
            & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ X )
              = zero_zero_real ) ) ) ) ) ).

% pi_def
thf(fact_9088_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_9089_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_9090_powr__int,axiom,
    ! [X3: real,I: int] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ I ) )
            = ( power_power_real @ X3 @ ( nat2 @ I ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ I ) )
            = ( divide_divide_real @ one_one_real @ ( power_power_real @ X3 @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).

% powr_int
thf(fact_9091_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_9092_nat__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% nat_numeral
thf(fact_9093_nat__0__iff,axiom,
    ! [I: int] :
      ( ( ( nat2 @ I )
        = zero_zero_nat )
      = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).

% nat_0_iff
thf(fact_9094_nat__le__0,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ Z )
        = zero_zero_nat ) ) ).

% nat_le_0
thf(fact_9095_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_9096_nat__neg__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = zero_zero_nat ) ).

% nat_neg_numeral
thf(fact_9097_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_9098_sgn__mult__dvd__iff,axiom,
    ! [R: int,L: int,K: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R ) @ L ) @ K )
      = ( ( dvd_dvd_int @ L @ K )
        & ( ( R = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
thf(fact_9099_mult__sgn__dvd__iff,axiom,
    ! [L: int,R: int,K: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R ) ) @ K )
      = ( ( dvd_dvd_int @ L @ K )
        & ( ( R = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
thf(fact_9100_dvd__sgn__mult__iff,axiom,
    ! [L: int,R: int,K: int] :
      ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R ) @ K ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( R = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
thf(fact_9101_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R: int] :
      ( ( dvd_dvd_int @ L @ ( times_times_int @ K @ ( sgn_sgn_int @ R ) ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( R = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
thf(fact_9102_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_9103_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_9104_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X3: num,N: nat] :
      ( ( ( nat2 @ Y3 )
        = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) )
      = ( Y3
        = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).

% nat_eq_numeral_power_cancel_iff
thf(fact_9105_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X3: num,N: nat,Y3: int] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
        = ( nat2 @ Y3 ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N )
        = Y3 ) ) ).

% numeral_power_eq_nat_cancel_iff
thf(fact_9106_nat__ceiling__le__eq,axiom,
    ! [X3: real,A: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X3 ) ) @ A )
      = ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ A ) ) ) ).

% nat_ceiling_le_eq
thf(fact_9107_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_9108_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_9109_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N: nat] :
      ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).

% nat_less_numeral_power_cancel_iff
thf(fact_9110_numeral__power__less__nat__cancel__iff,axiom,
    ! [X3: num,N: nat,A: int] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) @ ( nat2 @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) @ A ) ) ).

% numeral_power_less_nat_cancel_iff
thf(fact_9111_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).

% nat_le_numeral_power_cancel_iff
thf(fact_9112_numeral__power__le__nat__cancel__iff,axiom,
    ! [X3: num,N: nat,A: int] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) @ ( nat2 @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) @ A ) ) ).

% numeral_power_le_nat_cancel_iff
thf(fact_9113_less__eq__mask,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).

% less_eq_mask
thf(fact_9114_nat__mask__eq,axiom,
    ! [N: nat] :
      ( ( nat2 @ ( bit_se2000444600071755411sk_int @ N ) )
      = ( bit_se2002935070580805687sk_nat @ N ) ) ).

% nat_mask_eq
thf(fact_9115_nat__zero__as__int,axiom,
    ( zero_zero_nat
    = ( nat2 @ zero_zero_int ) ) ).

% nat_zero_as_int
thf(fact_9116_nat__numeral__as__int,axiom,
    ( numeral_numeral_nat
    = ( ^ [I4: num] : ( nat2 @ ( numeral_numeral_int @ I4 ) ) ) ) ).

% nat_numeral_as_int
thf(fact_9117_nat__mono,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ X3 @ Y3 )
     => ( ord_less_eq_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y3 ) ) ) ).

% nat_mono
thf(fact_9118_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N3: nat,L4: int] :
          ( K
         != ( times_times_int @ ( sgn_sgn_int @ L4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% int_sgnE
thf(fact_9119_ex__nat,axiom,
    ( ( ^ [P2: nat > $o] :
        ? [X4: nat] : ( P2 @ X4 ) )
    = ( ^ [P3: nat > $o] :
        ? [X: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X )
          & ( P3 @ ( nat2 @ X ) ) ) ) ) ).

% ex_nat
thf(fact_9120_all__nat,axiom,
    ( ( ^ [P2: nat > $o] :
        ! [X4: nat] : ( P2 @ X4 ) )
    = ( ^ [P3: nat > $o] :
        ! [X: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X )
         => ( P3 @ ( nat2 @ X ) ) ) ) ) ).

% all_nat
thf(fact_9121_eq__nat__nat__iff,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
       => ( ( ( nat2 @ Z )
            = ( nat2 @ Z7 ) )
          = ( Z = Z7 ) ) ) ) ).

% eq_nat_nat_iff
thf(fact_9122_nat__one__as__int,axiom,
    ( one_one_nat
    = ( nat2 @ one_one_int ) ) ).

% nat_one_as_int
thf(fact_9123_unset__bit__nat__def,axiom,
    ( bit_se4205575877204974255it_nat
    = ( ^ [M3: nat,N2: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M3 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% unset_bit_nat_def
thf(fact_9124_mask__nonnegative__int,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).

% mask_nonnegative_int
thf(fact_9125_not__mask__negative__int,axiom,
    ! [N: nat] :
      ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).

% not_mask_negative_int
thf(fact_9126_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_9127_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_9128_nat__le__iff,axiom,
    ! [X3: int,N: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ X3 ) @ N )
      = ( ord_less_eq_int @ X3 @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% nat_le_iff
thf(fact_9129_nat__0__le,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
        = Z ) ) ).

% nat_0_le
thf(fact_9130_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_9131_nat__int__add,axiom,
    ! [A: nat,B2: nat] :
      ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) )
      = ( plus_plus_nat @ A @ B2 ) ) ).

% nat_int_add
thf(fact_9132_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_9133_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_9134_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_9135_nat__plus__as__int,axiom,
    ( plus_plus_nat
    = ( ^ [A5: nat,B4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_plus_as_int
thf(fact_9136_nat__times__as__int,axiom,
    ( times_times_nat
    = ( ^ [A5: nat,B4: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_times_as_int
thf(fact_9137_real__nat__ceiling__ge,axiom,
    ! [X3: real] : ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X3 ) ) ) ) ).

% real_nat_ceiling_ge
thf(fact_9138_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_9139_nat__minus__as__int,axiom,
    ( minus_minus_nat
    = ( ^ [A5: nat,B4: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_minus_as_int
thf(fact_9140_nat__div__as__int,axiom,
    ( divide_divide_nat
    = ( ^ [A5: nat,B4: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_div_as_int
thf(fact_9141_nat__mod__as__int,axiom,
    ( modulo_modulo_nat
    = ( ^ [A5: nat,B4: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_mod_as_int
thf(fact_9142_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_9143_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_9144_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_9145_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_9146_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_9147_split__nat,axiom,
    ! [P: nat > $o,I: int] :
      ( ( P @ ( nat2 @ I ) )
      = ( ! [N2: nat] :
            ( ( I
              = ( semiri1314217659103216013at_int @ N2 ) )
           => ( P @ N2 ) )
        & ( ( ord_less_int @ I @ zero_zero_int )
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_nat
thf(fact_9148_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_9149_nat__add__distrib,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
       => ( ( nat2 @ ( plus_plus_int @ Z @ Z7 ) )
          = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).

% nat_add_distrib
thf(fact_9150_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_9151_nat__mult__distrib,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
        = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ).

% nat_mult_distrib
thf(fact_9152_Suc__as__int,axiom,
    ( suc
    = ( ^ [A5: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ one_one_int ) ) ) ) ).

% Suc_as_int
thf(fact_9153_nat__diff__distrib_H,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( nat2 @ ( minus_minus_int @ X3 @ Y3 ) )
          = ( minus_minus_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y3 ) ) ) ) ) ).

% nat_diff_distrib'
thf(fact_9154_nat__diff__distrib,axiom,
    ! [Z7: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
     => ( ( ord_less_eq_int @ Z7 @ Z )
       => ( ( nat2 @ ( minus_minus_int @ Z @ Z7 ) )
          = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).

% nat_diff_distrib
thf(fact_9155_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_9156_nat__div__distrib_H,axiom,
    ! [Y3: int,X3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( nat2 @ ( divide_divide_int @ X3 @ Y3 ) )
        = ( divide_divide_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y3 ) ) ) ) ).

% nat_div_distrib'
thf(fact_9157_nat__div__distrib,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ( nat2 @ ( divide_divide_int @ X3 @ Y3 ) )
        = ( divide_divide_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y3 ) ) ) ) ).

% nat_div_distrib
thf(fact_9158_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( dvd_dvd_int @ L @ K )
     => ( ( divide_divide_int @ K @ L )
        = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ) ).

% div_dvd_sgn_abs
thf(fact_9159_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_9160_nat__floor__neg,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ X3 @ zero_zero_real )
     => ( ( nat2 @ ( archim6058952711729229775r_real @ X3 ) )
        = zero_zero_nat ) ) ).

% nat_floor_neg
thf(fact_9161_nat__mod__distrib,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( nat2 @ ( modulo_modulo_int @ X3 @ Y3 ) )
          = ( modulo_modulo_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y3 ) ) ) ) ) ).

% nat_mod_distrib
thf(fact_9162_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_9163_floor__eq3,axiom,
    ! [N: nat,X3: real] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X3 )
     => ( ( ord_less_real @ X3 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X3 ) )
          = N ) ) ) ).

% floor_eq3
thf(fact_9164_le__nat__floor,axiom,
    ! [X3: nat,A: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X3 ) @ A )
     => ( ord_less_eq_nat @ X3 @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).

% le_nat_floor
thf(fact_9165_divide__int__def,axiom,
    ( divide_divide_int
    = ( ^ [K2: int,L2: int] :
          ( if_int @ ( L2 = zero_zero_int ) @ zero_zero_int
          @ ( if_int
            @ ( ( sgn_sgn_int @ K2 )
              = ( sgn_sgn_int @ L2 ) )
            @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) )
            @ ( uminus_uminus_int
              @ ( semiri1314217659103216013at_int
                @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) )
                  @ ( zero_n2687167440665602831ol_nat
                    @ ~ ( dvd_dvd_int @ L2 @ K2 ) ) ) ) ) ) ) ) ) ).

% divide_int_def
thf(fact_9166_modulo__int__def,axiom,
    ( modulo_modulo_int
    = ( ^ [K2: int,L2: int] :
          ( if_int @ ( L2 = zero_zero_int ) @ K2
          @ ( if_int
            @ ( ( sgn_sgn_int @ K2 )
              = ( sgn_sgn_int @ L2 ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( 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 @ K2 ) ) )
                @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ) ) ) ).

% modulo_int_def
thf(fact_9167_nat__2,axiom,
    ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% nat_2
thf(fact_9168_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_9169_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_9170_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
        = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z7 ) ) ) ) ) ).

% nat_mult_distrib_neg
thf(fact_9171_nat__abs__int__diff,axiom,
    ! [A: nat,B2: nat] :
      ( ( ( ord_less_eq_nat @ A @ B2 )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
          = ( minus_minus_nat @ B2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ A @ B2 )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
          = ( minus_minus_nat @ A @ B2 ) ) ) ) ).

% nat_abs_int_diff
thf(fact_9172_floor__eq4,axiom,
    ! [N: nat,X3: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X3 )
     => ( ( ord_less_real @ X3 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X3 ) )
          = N ) ) ) ).

% floor_eq4
thf(fact_9173_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one )
    = zero_zero_nat ) ).

% num.size_gen(1)
thf(fact_9174_diff__nat__eq__if,axiom,
    ! [Z7: int,Z: int] :
      ( ( ( ord_less_int @ Z7 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
          = ( nat2 @ Z ) ) )
      & ( ~ ( ord_less_int @ Z7 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
          = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z7 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z7 ) ) ) ) ) ) ).

% diff_nat_eq_if
thf(fact_9175_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_9176_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_9177_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_9178_eucl__rel__int__remainderI,axiom,
    ! [R: int,L: int,K: int,Q2: int] :
      ( ( ( sgn_sgn_int @ R )
        = ( sgn_sgn_int @ L ) )
     => ( ( ord_less_int @ ( abs_abs_int @ R ) @ ( abs_abs_int @ L ) )
       => ( ( K
            = ( plus_plus_int @ ( times_times_int @ Q2 @ L ) @ R ) )
         => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R ) ) ) ) ) ).

% eucl_rel_int_remainderI
thf(fact_9179_mask__nat__def,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N2: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ).

% mask_nat_def
thf(fact_9180_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_9181_mask__int__def,axiom,
    ( bit_se2000444600071755411sk_int
    = ( ^ [N2: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ).

% mask_int_def
thf(fact_9182_eucl__rel__int_Osimps,axiom,
    ( eucl_rel_int
    = ( ^ [A1: int,A22: int,A32: product_prod_int_int] :
          ( ? [K2: int] :
              ( ( A1 = K2 )
              & ( A22 = zero_zero_int )
              & ( A32
                = ( product_Pair_int_int @ zero_zero_int @ K2 ) ) )
          | ? [L2: int,K2: int,Q4: int] :
              ( ( A1 = K2 )
              & ( A22 = L2 )
              & ( A32
                = ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
              & ( L2 != zero_zero_int )
              & ( K2
                = ( times_times_int @ Q4 @ L2 ) ) )
          | ? [R5: int,L2: int,K2: int,Q4: int] :
              ( ( A1 = K2 )
              & ( A22 = L2 )
              & ( A32
                = ( product_Pair_int_int @ Q4 @ R5 ) )
              & ( ( sgn_sgn_int @ R5 )
                = ( sgn_sgn_int @ L2 ) )
              & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L2 ) )
              & ( K2
                = ( plus_plus_int @ ( times_times_int @ Q4 @ L2 ) @ R5 ) ) ) ) ) ) ).

% eucl_rel_int.simps
thf(fact_9183_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 ) ) )
       => ( ! [Q3: int] :
              ( ( A33
                = ( product_Pair_int_int @ Q3 @ zero_zero_int ) )
             => ( ( A23 != zero_zero_int )
               => ( A12
                 != ( times_times_int @ Q3 @ A23 ) ) ) )
         => ~ ! [R3: int,Q3: int] :
                ( ( A33
                  = ( product_Pair_int_int @ Q3 @ R3 ) )
               => ( ( ( sgn_sgn_int @ R3 )
                    = ( sgn_sgn_int @ A23 ) )
                 => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A23 ) )
                   => ( A12
                     != ( plus_plus_int @ ( times_times_int @ Q3 @ A23 ) @ R3 ) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
thf(fact_9184_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_9185_powr__real__of__int,axiom,
    ! [X3: real,N: int] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
         => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ N ) )
            = ( power_power_real @ X3 @ ( nat2 @ N ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
         => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ N ) )
            = ( inverse_inverse_real @ ( power_power_real @ X3 @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).

% powr_real_of_int
thf(fact_9186_num_Osize__gen_I2_J,axiom,
    ! [X2: num] :
      ( ( size_num @ ( bit0 @ X2 ) )
      = ( plus_plus_nat @ ( size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(2)
thf(fact_9187_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_9188_and__int__unfold,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K2: int,L2: int] :
          ( if_int
          @ ( ( K2 = zero_zero_int )
            | ( L2 = zero_zero_int ) )
          @ zero_zero_int
          @ ( if_int
            @ ( K2
              = ( uminus_uminus_int @ one_one_int ) )
            @ L2
            @ ( if_int
              @ ( L2
                = ( uminus_uminus_int @ one_one_int ) )
              @ K2
              @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K2 @ ( 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 @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% and_int_unfold
thf(fact_9189_zero__le__sgn__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X3 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).

% zero_le_sgn_iff
thf(fact_9190_sgn__le__0__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( sgn_sgn_real @ X3 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).

% sgn_le_0_iff
thf(fact_9191_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_9192_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_9193_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_9194_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_9195_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_9196_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_9197_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_9198_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_9199_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_9200_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_9201_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).

% take_bit_nat_less_eq_self
thf(fact_9202_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q2 ) @ ( bit_se2925701944663578781it_nat @ N @ Q2 ) ) ) ).

% take_bit_tightened_less_eq_nat
thf(fact_9203_take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
      = ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ L ) ) ) ).

% take_bit_diff
thf(fact_9204_take__bit__mult,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
      = ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ K @ L ) ) ) ).

% take_bit_mult
thf(fact_9205_take__bit__minus,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
      = ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).

% take_bit_minus
thf(fact_9206_concat__bit__take__bit__eq,axiom,
    ! [N: nat,B2: int] :
      ( ( bit_concat_bit @ N @ ( bit_se2923211474154528505it_int @ N @ B2 ) )
      = ( bit_concat_bit @ N @ B2 ) ) ).

% concat_bit_take_bit_eq
thf(fact_9207_concat__bit__eq__iff,axiom,
    ! [N: nat,K: int,L: int,R: int,S: int] :
      ( ( ( bit_concat_bit @ N @ K @ L )
        = ( bit_concat_bit @ N @ R @ S ) )
      = ( ( ( bit_se2923211474154528505it_int @ N @ K )
          = ( bit_se2923211474154528505it_int @ N @ R ) )
        & ( L = S ) ) ) ).

% concat_bit_eq_iff
thf(fact_9208_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N: nat,K: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% take_bit_tightened_less_eq_int
thf(fact_9209_AND__lower,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X3 @ Y3 ) ) ) ).

% AND_lower
thf(fact_9210_AND__upper1,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X3 @ Y3 ) @ X3 ) ) ).

% AND_upper1
thf(fact_9211_AND__upper2,axiom,
    ! [Y3: int,X3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X3 @ Y3 ) @ Y3 ) ) ).

% AND_upper2
thf(fact_9212_AND__upper1_H,axiom,
    ! [Y3: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( ord_less_eq_int @ Y3 @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z ) ) ) ).

% AND_upper1'
thf(fact_9213_AND__upper2_H,axiom,
    ! [Y3: int,Z: int,X3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( ord_less_eq_int @ Y3 @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X3 @ Y3 ) @ Z ) ) ) ).

% AND_upper2'
thf(fact_9214_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_9215_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_9216_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_9217_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_9218_pow_Osimps_I1_J,axiom,
    ! [X3: num] :
      ( ( pow @ X3 @ one )
      = X3 ) ).

% pow.simps(1)
thf(fact_9219_AND__upper2_H_H,axiom,
    ! [Y3: int,Z: int,X3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( ord_less_int @ Y3 @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X3 @ Y3 ) @ Z ) ) ) ).

% AND_upper2''
thf(fact_9220_AND__upper1_H_H,axiom,
    ! [Y3: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( ord_less_int @ Y3 @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z ) ) ) ).

% AND_upper1''
thf(fact_9221_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_9222_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_9223_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_9224_sgn__real__def,axiom,
    ( sgn_sgn_real
    = ( ^ [A5: real] : ( if_real @ ( A5 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A5 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).

% sgn_real_def
thf(fact_9225_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_9226_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_9227_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_9228_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_9229_take__bit__nat__def,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N2: nat,M3: nat] : ( modulo_modulo_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% take_bit_nat_def
thf(fact_9230_sgn__power__injE,axiom,
    ! [A: real,N: nat,X3: real,B2: real] :
      ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
        = X3 )
     => ( ( X3
          = ( times_times_real @ ( sgn_sgn_real @ B2 ) @ ( power_power_real @ ( abs_abs_real @ B2 ) @ N ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( A = B2 ) ) ) ) ).

% sgn_power_injE
thf(fact_9231_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_9232_take__bit__int__def,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N2: nat,K2: int] : ( modulo_modulo_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% take_bit_int_def
thf(fact_9233_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_9234_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_9235_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_9236_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_9237_cis__Arg__unique,axiom,
    ! [Z: complex,X3: real] :
      ( ( ( sgn_sgn_complex @ Z )
        = ( cis @ X3 ) )
     => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X3 )
       => ( ( ord_less_eq_real @ X3 @ pi )
         => ( ( arg @ Z )
            = X3 ) ) ) ) ).

% cis_Arg_unique
thf(fact_9238_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_9239_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_9240_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_9241_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_9242_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_9243_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_9244_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_9245_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_9246_signed__take__bit__eq__take__bit__shift,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N2: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% signed_take_bit_eq_take_bit_shift
thf(fact_9247_and__int__rec,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K2: int,L2: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
              & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_int_rec
thf(fact_9248_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_9249_arctan__inverse,axiom,
    ! [X3: real] :
      ( ( X3 != zero_zero_real )
     => ( ( arctan @ ( divide_divide_real @ one_one_real @ X3 ) )
        = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X3 ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X3 ) ) ) ) ).

% arctan_inverse
thf(fact_9250_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_9251_and__int_Osimps,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K2: int,L2: int] :
          ( if_int
          @ ( ( member_int @ K2 @ ( 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 ) ) @ K2 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
          @ ( plus_plus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_int.simps
thf(fact_9252_and__int_Oelims,axiom,
    ! [X3: int,Xa2: int,Y3: int] :
      ( ( ( bit_se725231765392027082nd_int @ X3 @ Xa2 )
        = Y3 )
     => ( ( ( ( member_int @ X3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y3
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
        & ( ~ ( ( member_int @ X3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y3
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.elims
thf(fact_9253_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_9254_pred__numeral__inc,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( inc @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% pred_numeral_inc
thf(fact_9255_and__nat__numerals_I1_J,axiom,
    ! [Y3: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
      = zero_zero_nat ) ).

% and_nat_numerals(1)
thf(fact_9256_and__nat__numerals_I3_J,axiom,
    ! [X3: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% and_nat_numerals(3)
thf(fact_9257_and__nat__numerals_I4_J,axiom,
    ! [X3: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% and_nat_numerals(4)
thf(fact_9258_and__nat__numerals_I2_J,axiom,
    ! [Y3: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
      = one_one_nat ) ).

% and_nat_numerals(2)
thf(fact_9259_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_9260_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_9261_num__induct,axiom,
    ! [P: num > $o,X3: num] :
      ( ( P @ one )
     => ( ! [X5: num] :
            ( ( P @ X5 )
           => ( P @ ( inc @ X5 ) ) )
       => ( P @ X3 ) ) ) ).

% num_induct
thf(fact_9262_add__inc,axiom,
    ! [X3: num,Y3: num] :
      ( ( plus_plus_num @ X3 @ ( inc @ Y3 ) )
      = ( inc @ ( plus_plus_num @ X3 @ Y3 ) ) ) ).

% add_inc
thf(fact_9263_inc_Osimps_I1_J,axiom,
    ( ( inc @ one )
    = ( bit0 @ one ) ) ).

% inc.simps(1)
thf(fact_9264_inc_Osimps_I2_J,axiom,
    ! [X3: num] :
      ( ( inc @ ( bit0 @ X3 ) )
      = ( bit1 @ X3 ) ) ).

% inc.simps(2)
thf(fact_9265_inc_Osimps_I3_J,axiom,
    ! [X3: num] :
      ( ( inc @ ( bit1 @ X3 ) )
      = ( bit0 @ ( inc @ X3 ) ) ) ).

% inc.simps(3)
thf(fact_9266_add__One,axiom,
    ! [X3: num] :
      ( ( plus_plus_num @ X3 @ one )
      = ( inc @ X3 ) ) ).

% add_One
thf(fact_9267_inc__BitM__eq,axiom,
    ! [N: num] :
      ( ( inc @ ( bitM @ N ) )
      = ( bit0 @ N ) ) ).

% inc_BitM_eq
thf(fact_9268_BitM__inc__eq,axiom,
    ! [N: num] :
      ( ( bitM @ ( inc @ N ) )
      = ( bit1 @ N ) ) ).

% BitM_inc_eq
thf(fact_9269_and__nat__def,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M3: nat,N2: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% and_nat_def
thf(fact_9270_mult__inc,axiom,
    ! [X3: num,Y3: num] :
      ( ( times_times_num @ X3 @ ( inc @ Y3 ) )
      = ( plus_plus_num @ ( times_times_num @ X3 @ Y3 ) @ X3 ) ) ).

% mult_inc
thf(fact_9271_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_9272_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_9273_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M3: nat,N2: nat] :
          ( if_nat
          @ ( ( M3 = zero_zero_nat )
            | ( N2 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_9274_and__nat__rec,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M3: nat,N2: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
              & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_nat_rec
thf(fact_9275_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_9276_and__int_Opelims,axiom,
    ! [X3: int,Xa2: int,Y3: int] :
      ( ( ( bit_se725231765392027082nd_int @ X3 @ Xa2 )
        = Y3 )
     => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) )
       => ~ ( ( ( ( ( member_int @ X3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                  & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y3
                  = ( uminus_uminus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
              & ( ~ ( ( member_int @ X3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y3
                  = ( plus_plus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
                    @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) ) ) ) ) ).

% and_int.pelims
thf(fact_9277_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 ) )
     => ( ! [K3: int,L4: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K3 @ L4 ) )
           => ( ( ~ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( P @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P @ K3 @ L4 ) ) )
       => ( P @ A0 @ A12 ) ) ) ).

% and_int.pinduct
thf(fact_9278_Arg__def,axiom,
    ( arg
    = ( ^ [Z2: complex] :
          ( if_real @ ( Z2 = zero_zero_complex ) @ zero_zero_real
          @ ( fChoice_real
            @ ^ [A5: real] :
                ( ( ( sgn_sgn_complex @ Z2 )
                  = ( cis @ A5 ) )
                & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A5 )
                & ( ord_less_eq_real @ A5 @ pi ) ) ) ) ) ) ).

% Arg_def
thf(fact_9279_set__encode__insert,axiom,
    ! [A4: set_nat,N: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ~ ( member_nat @ N @ A4 )
       => ( ( nat_set_encode @ ( insert_nat @ N @ A4 ) )
          = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A4 ) ) ) ) ) ).

% set_encode_insert
thf(fact_9280_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_9281_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_9282_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_9283_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_9284_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_9285_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_9286_signed__take__bit__eq__take__bit__minus,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N2: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K2 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) ) ) ) ) ).

% signed_take_bit_eq_take_bit_minus
thf(fact_9287_or__int__unfold,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K2: int,L2: int] :
          ( if_int
          @ ( ( K2
              = ( uminus_uminus_int @ one_one_int ) )
            | ( L2
              = ( uminus_uminus_int @ one_one_int ) ) )
          @ ( uminus_uminus_int @ one_one_int )
          @ ( if_int @ ( K2 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K2 @ ( 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 @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% or_int_unfold
thf(fact_9288_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_9289_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_9290_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_9291_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_9292_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_9293_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_9294_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_9295_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_9296_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_9297_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_9298_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_9299_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_9300_bit__or__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ N )
      = ( ( bit_se1146084159140164899it_int @ K @ N )
        | ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).

% bit_or_int_iff
thf(fact_9301_bit__and__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ N )
      = ( ( bit_se1146084159140164899it_int @ K @ N )
        & ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).

% bit_and_int_iff
thf(fact_9302_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_9303_OR__lower,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X3 @ Y3 ) ) ) ) ).

% OR_lower
thf(fact_9304_plus__and__or,axiom,
    ! [X3: int,Y3: int] :
      ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X3 @ Y3 ) @ ( bit_se1409905431419307370or_int @ X3 @ Y3 ) )
      = ( plus_plus_int @ X3 @ Y3 ) ) ).

% plus_and_or
thf(fact_9305_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_9306_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_9307_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_9308_sin__times__pi__eq__0,axiom,
    ! [X3: real] :
      ( ( ( sin_real @ ( times_times_real @ X3 @ pi ) )
        = zero_zero_real )
      = ( member_real @ X3 @ ring_1_Ints_real ) ) ).

% sin_times_pi_eq_0
thf(fact_9309_signed__take__bit__eq__concat__bit,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N2: nat,K2: int] : ( bit_concat_bit @ N2 @ K2 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) ) ) ) ) ).

% signed_take_bit_eq_concat_bit
thf(fact_9310_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N3: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ N3 @ M2 )
             => ( ( bit_se1146084159140164899it_int @ K @ M2 )
                = ( bit_se1146084159140164899it_int @ K @ N3 ) ) )
         => ~ ( ( ord_less_nat @ zero_zero_nat @ N3 )
             => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N3 @ one_one_nat ) )
                = ( ~ ( bit_se1146084159140164899it_int @ K @ N3 ) ) ) ) ) ).

% int_bit_bound
thf(fact_9311_bit__int__def,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [K2: int,N2: nat] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% bit_int_def
thf(fact_9312_OR__upper,axiom,
    ! [X3: int,N: nat,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ( ord_less_int @ X3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
         => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X3 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% OR_upper
thf(fact_9313_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_9314_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_9315_or__int__rec,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K2: int,L2: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
              | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_int_rec
thf(fact_9316_set__bit__eq,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N2: nat,K2: int] :
          ( plus_plus_int @ K2
          @ ( times_times_int
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( bit_se1146084159140164899it_int @ K2 @ N2 ) )
            @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% set_bit_eq
thf(fact_9317_unset__bit__eq,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N2: nat,K2: int] : ( minus_minus_int @ K2 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% unset_bit_eq
thf(fact_9318_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_9319_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_9320_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_9321_or__nat__numerals_I2_J,axiom,
    ! [Y3: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).

% or_nat_numerals(2)
thf(fact_9322_or__nat__numerals_I4_J,axiom,
    ! [X3: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X3 ) ) ) ).

% or_nat_numerals(4)
thf(fact_9323_or__nat__numerals_I3_J,axiom,
    ! [X3: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X3 ) ) ) ).

% or_nat_numerals(3)
thf(fact_9324_or__nat__numerals_I1_J,axiom,
    ! [Y3: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).

% or_nat_numerals(1)
thf(fact_9325_xor__nat__numerals_I4_J,axiom,
    ! [X3: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit0 @ X3 ) ) ) ).

% xor_nat_numerals(4)
thf(fact_9326_xor__nat__numerals_I3_J,axiom,
    ! [X3: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X3 ) ) ) ).

% xor_nat_numerals(3)
thf(fact_9327_xor__nat__numerals_I2_J,axiom,
    ! [Y3: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
      = ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) ) ).

% xor_nat_numerals(2)
thf(fact_9328_xor__nat__numerals_I1_J,axiom,
    ! [Y3: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).

% xor_nat_numerals(1)
thf(fact_9329_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_9330_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_9331_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_9332_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_9333_or__not__num__neg_Osimps_I1_J,axiom,
    ( ( bit_or_not_num_neg @ one @ one )
    = one ) ).

% or_not_num_neg.simps(1)
thf(fact_9334_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_9335_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).

% not_bit_Suc_0_Suc
thf(fact_9336_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_9337_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_9338_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_9339_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_9340_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_9341_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_9342_or__not__num__neg_Osimps_I9_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit1 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).

% or_not_num_neg.simps(9)
thf(fact_9343_or__nat__def,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M3: nat,N2: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% or_nat_def
thf(fact_9344_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_9345_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_9346_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_9347_or__not__num__neg_Oelims,axiom,
    ! [X3: num,Xa2: num,Y3: num] :
      ( ( ( bit_or_not_num_neg @ X3 @ Xa2 )
        = Y3 )
     => ( ( ( X3 = one )
         => ( ( Xa2 = one )
           => ( Y3 != one ) ) )
       => ( ( ( X3 = one )
           => ! [M4: num] :
                ( ( Xa2
                  = ( bit0 @ M4 ) )
               => ( Y3
                 != ( bit1 @ M4 ) ) ) )
         => ( ( ( X3 = one )
             => ! [M4: num] :
                  ( ( Xa2
                    = ( bit1 @ M4 ) )
                 => ( Y3
                   != ( bit1 @ M4 ) ) ) )
           => ( ( ? [N3: num] :
                    ( X3
                    = ( bit0 @ N3 ) )
               => ( ( Xa2 = one )
                 => ( Y3
                   != ( bit0 @ one ) ) ) )
             => ( ! [N3: num] :
                    ( ( X3
                      = ( bit0 @ N3 ) )
                   => ! [M4: num] :
                        ( ( Xa2
                          = ( bit0 @ M4 ) )
                       => ( Y3
                         != ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) ) ) )
               => ( ! [N3: num] :
                      ( ( X3
                        = ( bit0 @ N3 ) )
                     => ! [M4: num] :
                          ( ( Xa2
                            = ( bit1 @ M4 ) )
                         => ( Y3
                           != ( bit0 @ ( bit_or_not_num_neg @ N3 @ M4 ) ) ) ) )
                 => ( ( ? [N3: num] :
                          ( X3
                          = ( bit1 @ N3 ) )
                     => ( ( Xa2 = one )
                       => ( Y3 != one ) ) )
                   => ( ! [N3: num] :
                          ( ( X3
                            = ( bit1 @ N3 ) )
                         => ! [M4: num] :
                              ( ( Xa2
                                = ( bit0 @ M4 ) )
                             => ( Y3
                               != ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) ) ) )
                     => ~ ! [N3: num] :
                            ( ( X3
                              = ( bit1 @ N3 ) )
                           => ! [M4: num] :
                                ( ( Xa2
                                  = ( bit1 @ M4 ) )
                               => ( Y3
                                 != ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
thf(fact_9348_bit__nat__def,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [M3: nat,N2: nat] :
          ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% bit_nat_def
thf(fact_9349_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_9350_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_9351_or__nat__rec,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M3: nat,N2: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
              | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_nat_rec
thf(fact_9352_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = 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 @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_9353_xor__nat__rec,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M3: nat,N2: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
             != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_nat_rec
thf(fact_9354_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_9355_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_9356_or__not__num__neg_Opelims,axiom,
    ! [X3: num,Xa2: num,Y3: num] :
      ( ( ( bit_or_not_num_neg @ X3 @ Xa2 )
        = Y3 )
     => ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X3 @ Xa2 ) )
       => ( ( ( X3 = one )
           => ( ( Xa2 = one )
             => ( ( Y3 = one )
               => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X3 = one )
             => ! [M4: num] :
                  ( ( Xa2
                    = ( bit0 @ M4 ) )
                 => ( ( Y3
                      = ( bit1 @ M4 ) )
                   => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M4 ) ) ) ) ) )
           => ( ( ( X3 = one )
               => ! [M4: num] :
                    ( ( Xa2
                      = ( bit1 @ M4 ) )
                   => ( ( Y3
                        = ( bit1 @ M4 ) )
                     => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M4 ) ) ) ) ) )
             => ( ! [N3: num] :
                    ( ( X3
                      = ( bit0 @ N3 ) )
                   => ( ( Xa2 = one )
                     => ( ( Y3
                          = ( bit0 @ one ) )
                       => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N3 ) @ one ) ) ) ) )
               => ( ! [N3: num] :
                      ( ( X3
                        = ( bit0 @ N3 ) )
                     => ! [M4: num] :
                          ( ( Xa2
                            = ( bit0 @ M4 ) )
                         => ( ( Y3
                              = ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N3 ) @ ( bit0 @ M4 ) ) ) ) ) )
                 => ( ! [N3: num] :
                        ( ( X3
                          = ( bit0 @ N3 ) )
                       => ! [M4: num] :
                            ( ( Xa2
                              = ( bit1 @ M4 ) )
                           => ( ( Y3
                                = ( bit0 @ ( bit_or_not_num_neg @ N3 @ M4 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N3 ) @ ( bit1 @ M4 ) ) ) ) ) )
                   => ( ! [N3: num] :
                          ( ( X3
                            = ( bit1 @ N3 ) )
                         => ( ( Xa2 = one )
                           => ( ( Y3 = one )
                             => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N3 ) @ one ) ) ) ) )
                     => ( ! [N3: num] :
                            ( ( X3
                              = ( bit1 @ N3 ) )
                           => ! [M4: num] :
                                ( ( Xa2
                                  = ( bit0 @ M4 ) )
                               => ( ( Y3
                                    = ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N3 ) @ ( bit0 @ M4 ) ) ) ) ) )
                       => ~ ! [N3: num] :
                              ( ( X3
                                = ( bit1 @ N3 ) )
                             => ! [M4: num] :
                                  ( ( Xa2
                                    = ( bit1 @ M4 ) )
                                 => ( ( Y3
                                      = ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N3 ) @ ( bit1 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
thf(fact_9357_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_9358_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_9359_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_9360_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_9361_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_9362_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_9363_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_9364_bit__xor__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ N )
      = ( ( bit_se1146084159140164899it_int @ K @ N )
       != ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).

% bit_xor_int_iff
thf(fact_9365_flip__bit__int__def,axiom,
    ( bit_se2159334234014336723it_int
    = ( ^ [N2: nat,K2: int] : ( bit_se6526347334894502574or_int @ K2 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).

% flip_bit_int_def
thf(fact_9366_push__bit__nat__eq,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se547839408752420682it_nat @ N @ ( nat2 @ K ) )
      = ( nat2 @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).

% push_bit_nat_eq
thf(fact_9367_XOR__lower,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X3 @ Y3 ) ) ) ) ).

% XOR_lower
thf(fact_9368_flip__bit__nat__def,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [M3: nat,N2: nat] : ( bit_se6528837805403552850or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).

% flip_bit_nat_def
thf(fact_9369_set__bit__nat__def,axiom,
    ( bit_se7882103937844011126it_nat
    = ( ^ [M3: nat,N2: nat] : ( bit_se1412395901928357646or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).

% set_bit_nat_def
thf(fact_9370_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
      = ( ( ord_less_eq_nat @ M @ N )
        & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).

% bit_push_bit_iff_int
thf(fact_9371_xor__nat__def,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M3: nat,N2: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% xor_nat_def
thf(fact_9372_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q2 ) @ N )
      = ( ( ord_less_eq_nat @ M @ N )
        & ( bit_se1148574629649215175it_nat @ Q2 @ ( minus_minus_nat @ N @ M ) ) ) ) ).

% bit_push_bit_iff_nat
thf(fact_9373_concat__bit__eq,axiom,
    ( bit_concat_bit
    = ( ^ [N2: nat,K2: int,L2: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) @ ( bit_se545348938243370406it_int @ N2 @ L2 ) ) ) ) ).

% concat_bit_eq
thf(fact_9374_concat__bit__def,axiom,
    ( bit_concat_bit
    = ( ^ [N2: nat,K2: int,L2: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) @ ( bit_se545348938243370406it_int @ N2 @ L2 ) ) ) ) ).

% concat_bit_def
thf(fact_9375_set__bit__int__def,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N2: nat,K2: int] : ( bit_se1409905431419307370or_int @ K2 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).

% set_bit_int_def
thf(fact_9376_push__bit__int__def,axiom,
    ( bit_se545348938243370406it_int
    = ( ^ [N2: nat,K2: int] : ( times_times_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% push_bit_int_def
thf(fact_9377_push__bit__nat__def,axiom,
    ( bit_se547839408752420682it_nat
    = ( ^ [N2: nat,M3: nat] : ( times_times_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% push_bit_nat_def
thf(fact_9378_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_9379_XOR__upper,axiom,
    ! [X3: int,N: nat,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ( ord_less_int @ X3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
         => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X3 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% XOR_upper
thf(fact_9380_xor__int__rec,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K2: int,L2: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) )
             != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_int_rec
thf(fact_9381_xor__int__unfold,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K2: int,L2: int] :
          ( if_int
          @ ( K2
            = ( uminus_uminus_int @ one_one_int ) )
          @ ( bit_ri7919022796975470100ot_int @ L2 )
          @ ( if_int
            @ ( L2
              = ( uminus_uminus_int @ one_one_int ) )
            @ ( bit_ri7919022796975470100ot_int @ K2 )
            @ ( if_int @ ( K2 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K2 @ ( 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 @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
thf(fact_9382_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( 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_9383_Cauchy__iff2,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X7: nat > real] :
        ! [J3: nat] :
        ? [M9: nat] :
        ! [M3: nat] :
          ( ( ord_less_eq_nat @ M9 @ M3 )
         => ! [N2: nat] :
              ( ( ord_less_eq_nat @ M9 @ N2 )
             => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X7 @ M3 ) @ ( X7 @ N2 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).

% Cauchy_iff2
thf(fact_9384_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_9385_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_9386_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_9387_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_9388_bit__not__int__iff,axiom,
    ! [K: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K ) @ N )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).

% bit_not_int_iff
thf(fact_9389_ex__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M3: nat] :
            ( ( ord_less_nat @ M3 @ N )
            & ( P @ M3 ) ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
            & ( P @ X ) ) ) ) ).

% ex_nat_less_eq
thf(fact_9390_all__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M3: nat] :
            ( ( ord_less_nat @ M3 @ N )
           => ( P @ M3 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
           => ( P @ X ) ) ) ) ).

% all_nat_less_eq
thf(fact_9391_or__int__def,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K2: int,L2: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ ( bit_ri7919022796975470100ot_int @ L2 ) ) ) ) ) ).

% or_int_def
thf(fact_9392_not__int__def,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K2: int] : ( minus_minus_int @ ( uminus_uminus_int @ K2 ) @ one_one_int ) ) ) ).

% not_int_def
thf(fact_9393_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_9394_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_9395_unset__bit__int__def,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N2: nat,K2: int] : ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ) ).

% unset_bit_int_def
thf(fact_9396_xor__int__def,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K2: int,L2: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ L2 ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ L2 ) ) ) ) ).

% xor_int_def
thf(fact_9397_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N4: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
     => ( finite_finite_nat @ N4 ) ) ).

% subset_eq_atLeast0_lessThan_finite
thf(fact_9398_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_9399_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_9400_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_9401_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_9402_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_9403_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_9404_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_9405_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_9406_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_9407_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_9408_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_9409_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_9410_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_9411_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_9412_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_9413_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_9414_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_9415_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_9416_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_9417_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_9418_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_9419_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_9420_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_9421_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_9422_not__int__rec,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K2: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% not_int_rec
thf(fact_9423_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_9424_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A: nat > nat,B2: nat > nat] :
      ( ! [I3: nat,J2: nat] :
          ( ( ord_less_eq_nat @ I3 @ J2 )
         => ( ( ord_less_nat @ J2 @ N )
           => ( ord_less_eq_nat @ ( A @ I3 ) @ ( A @ 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 @ ( A @ I4 ) @ ( B2 @ I4 ) )
              @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
          @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).

% Chebyshev_sum_upper_nat
thf(fact_9425_valid__eq,axiom,
    vEBT_VEBT_valid = vEBT_invar_vebt ).

% valid_eq
thf(fact_9426_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_invar_vebt @ T @ D )
     => ( vEBT_VEBT_valid @ T @ D ) ) ).

% valid_eq1
thf(fact_9427_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_VEBT_valid @ T @ D )
     => ( vEBT_invar_vebt @ T @ D ) ) ).

% valid_eq2
thf(fact_9428_Code__Target__Int_Opositive__def,axiom,
    code_Target_positive = numeral_numeral_int ).

% Code_Target_Int.positive_def
thf(fact_9429_divmod__step__integer__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L2: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_integer_def
thf(fact_9430_times__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(2)
thf(fact_9431_times__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(1)
thf(fact_9432_exhaustive__integer_H_Ocases,axiom,
    ! [X3: produc8763457246119570046nteger] :
      ~ ! [F2: code_integer > option6357759511663192854e_term,D3: code_integer,I3: code_integer] :
          ( X3
         != ( produc6137756002093451184nteger @ F2 @ ( produc1086072967326762835nteger @ D3 @ I3 ) ) ) ).

% exhaustive_integer'.cases
thf(fact_9433_full__exhaustive__integer_H_Ocases,axiom,
    ! [X3: produc1908205239877642774nteger] :
      ~ ! [F2: produc6241069584506657477e_term > option6357759511663192854e_term,D3: code_integer,I3: code_integer] :
          ( X3
         != ( produc8603105652947943368nteger @ F2 @ ( produc1086072967326762835nteger @ D3 @ I3 ) ) ) ).

% full_exhaustive_integer'.cases
thf(fact_9434_less__eq__integer__code_I1_J,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).

% less_eq_integer_code(1)
thf(fact_9435_divmod__integer_H__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M3: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).

% divmod_integer'_def
thf(fact_9436_sgn__integer__code,axiom,
    ( sgn_sgn_Code_integer
    = ( ^ [K2: code_integer] : ( if_Code_integer @ ( K2 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).

% sgn_integer_code
thf(fact_9437_integer__of__int__code,axiom,
    ( code_integer_of_int
    = ( ^ [K2: int] :
          ( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K2 ) ) )
          @ ( if_Code_integer @ ( K2 = zero_zero_int ) @ zero_z3403309356797280102nteger
            @ ( if_Code_integer
              @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).

% integer_of_int_code
thf(fact_9438_Code__Numeral_Opositive__def,axiom,
    code_positive = numera6620942414471956472nteger ).

% Code_Numeral.positive_def
thf(fact_9439_abs__integer__code,axiom,
    ( abs_abs_Code_integer
    = ( ^ [K2: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K2 ) @ K2 ) ) ) ).

% abs_integer_code
thf(fact_9440_less__integer_Oabs__eq,axiom,
    ! [Xa2: int,X3: int] :
      ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X3 ) )
      = ( ord_less_int @ Xa2 @ X3 ) ) ).

% less_integer.abs_eq
thf(fact_9441_less__integer__code_I1_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).

% less_integer_code(1)
thf(fact_9442_times__integer_Oabs__eq,axiom,
    ! [Xa2: int,X3: int] :
      ( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X3 ) )
      = ( code_integer_of_int @ ( times_times_int @ Xa2 @ X3 ) ) ) ).

% times_integer.abs_eq
thf(fact_9443_less__eq__integer_Oabs__eq,axiom,
    ! [Xa2: int,X3: int] :
      ( ( ord_le3102999989581377725nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X3 ) )
      = ( ord_less_eq_int @ Xa2 @ X3 ) ) ).

% less_eq_integer.abs_eq
thf(fact_9444_int__of__integer__code,axiom,
    ( code_int_of_integer
    = ( ^ [K2: code_integer] :
          ( if_int @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K2 ) ) )
          @ ( if_int @ ( K2 = 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 @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% int_of_integer_code
thf(fact_9445_bit__cut__integer__def,axiom,
    ( code_bit_cut_integer
    = ( ^ [K2: code_integer] :
          ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K2 ) ) ) ) ).

% bit_cut_integer_def
thf(fact_9446_int__of__integer__numeral,axiom,
    ! [K: num] :
      ( ( code_int_of_integer @ ( numera6620942414471956472nteger @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% int_of_integer_numeral
thf(fact_9447_times__integer_Orep__eq,axiom,
    ! [X3: code_integer,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X3 @ Xa2 ) )
      = ( times_times_int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa2 ) ) ) ).

% times_integer.rep_eq
thf(fact_9448_less__integer_Orep__eq,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).

% less_integer.rep_eq
thf(fact_9449_integer__less__iff,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [K2: code_integer,L2: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K2 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).

% integer_less_iff
thf(fact_9450_integer__less__eq__iff,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [K2: code_integer,L2: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ K2 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).

% integer_less_eq_iff
thf(fact_9451_less__eq__integer_Orep__eq,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).

% less_eq_integer.rep_eq
thf(fact_9452_divmod__integer__def,axiom,
    ( code_divmod_integer
    = ( ^ [K2: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K2 @ L2 ) @ ( modulo364778990260209775nteger @ K2 @ L2 ) ) ) ) ).

% divmod_integer_def
thf(fact_9453_num__of__integer__code,axiom,
    ( code_num_of_integer
    = ( ^ [K2: code_integer] :
          ( if_num @ ( ord_le3102999989581377725nteger @ K2 @ 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 @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% num_of_integer_code
thf(fact_9454_bit__cut__integer__code,axiom,
    ( code_bit_cut_integer
    = ( ^ [K2: code_integer] :
          ( if_Pro5737122678794959658eger_o @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
          @ ( produc9125791028180074456eger_o
            @ ^ [R5: code_integer,S6: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S6 ) ) @ ( S6 = one_one_Code_integer ) )
            @ ( code_divmod_abs @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_cut_integer_code
thf(fact_9455_nat__of__integer__code,axiom,
    ( code_nat_of_integer
    = ( ^ [K2: code_integer] :
          ( if_nat @ ( ord_le3102999989581377725nteger @ K2 @ 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 @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% nat_of_integer_code
thf(fact_9456_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_9457_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_9458_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_9459_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_9460_divmod__abs__def,axiom,
    ( code_divmod_abs
    = ( ^ [K2: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K2 ) @ ( abs_abs_Code_integer @ L2 ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K2 ) @ ( abs_abs_Code_integer @ L2 ) ) ) ) ) ).

% divmod_abs_def
thf(fact_9461_divmod__integer__code,axiom,
    ( code_divmod_integer
    = ( ^ [K2: code_integer,L2: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L2 )
            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ ( code_divmod_abs @ K2 @ L2 )
              @ ( produc6916734918728496179nteger
                @ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L2 @ S6 ) ) )
                @ ( code_divmod_abs @ K2 @ L2 ) ) )
            @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
              @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
                @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K2 @ L2 )
                  @ ( produc6916734918728496179nteger
                    @ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L2 ) @ S6 ) ) )
                    @ ( code_divmod_abs @ K2 @ L2 ) ) ) ) ) ) ) ) ) ).

% divmod_integer_code
thf(fact_9462_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_9463_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_9464_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
      = ( minus_minus_nat @ U @ L ) ) ).

% card_atLeastLessThan
thf(fact_9465_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_9466_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L @ U ) )
      = ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).

% card_atLeastAtMost
thf(fact_9467_complex__Re__numeral,axiom,
    ! [V: num] :
      ( ( re @ ( numera6690914467698888265omplex @ V ) )
      = ( numeral_numeral_real @ V ) ) ).

% complex_Re_numeral
thf(fact_9468_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_9469_complex__Re__le__cmod,axiom,
    ! [X3: complex] : ( ord_less_eq_real @ ( re @ X3 ) @ ( real_V1022390504157884413omplex @ X3 ) ) ).

% complex_Re_le_cmod
thf(fact_9470_scaleR__complex_Osimps_I1_J,axiom,
    ! [R: real,X3: complex] :
      ( ( re @ ( real_V2046097035970521341omplex @ R @ X3 ) )
      = ( times_times_real @ R @ ( re @ X3 ) ) ) ).

% scaleR_complex.simps(1)
thf(fact_9471_abs__Re__le__cmod,axiom,
    ! [X3: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X3 ) ) @ ( real_V1022390504157884413omplex @ X3 ) ) ).

% abs_Re_le_cmod
thf(fact_9472_Re__csqrt,axiom,
    ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).

% Re_csqrt
thf(fact_9473_card__less__Suc2,axiom,
    ! [M7: set_nat,I: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat @ ( suc @ K2 ) @ M7 )
                & ( ord_less_nat @ K2 @ I ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat @ K2 @ M7 )
                & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_9474_card__less__Suc,axiom,
    ! [M7: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K2: nat] :
                  ( ( member_nat @ ( suc @ K2 ) @ M7 )
                  & ( ord_less_nat @ K2 @ I ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat @ K2 @ M7 )
                & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_9475_card__less,axiom,
    ! [M7: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat @ K2 @ M7 )
                & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_9476_subset__card__intvl__is__intvl,axiom,
    ! [A4: set_nat,K: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A4 ) ) ) )
     => ( A4
        = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A4 ) ) ) ) ) ).

% subset_card_intvl_is_intvl
thf(fact_9477_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N4: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
     => ( ord_less_eq_nat @ ( finite_card_nat @ N4 ) @ N ) ) ).

% subset_eq_atLeast0_lessThan_card
thf(fact_9478_card__sum__le__nat__sum,axiom,
    ! [S2: set_nat] :
      ( ord_less_eq_nat
      @ ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
      @ ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ S2 ) ) ).

% card_sum_le_nat_sum
thf(fact_9479_card__nth__roots,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( finite_card_complex
            @ ( collect_complex
              @ ^ [Z2: complex] :
                  ( ( power_power_complex @ Z2 @ N )
                  = C ) ) )
          = N ) ) ) ).

% card_nth_roots
thf(fact_9480_card__roots__unity__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( finite_card_complex
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = one_one_complex ) ) )
        = N ) ) ).

% card_roots_unity_eq
thf(fact_9481_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_9482_cos__n__Re__cis__pow__n,axiom,
    ! [N: nat,A: real] :
      ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
      = ( re @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).

% cos_n_Re_cis_pow_n
thf(fact_9483_csqrt_Ocode,axiom,
    ( csqrt
    = ( ^ [Z2: complex] :
          ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          @ ( times_times_real
            @ ( if_real
              @ ( ( im @ Z2 )
                = zero_zero_real )
              @ one_one_real
              @ ( sgn_sgn_real @ ( im @ Z2 ) ) )
            @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% csqrt.code
thf(fact_9484_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_9485_csqrt__of__real__nonpos,axiom,
    ! [X3: complex] :
      ( ( ( im @ X3 )
        = zero_zero_real )
     => ( ( ord_less_eq_real @ ( re @ X3 ) @ zero_zero_real )
       => ( ( csqrt @ X3 )
          = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X3 ) ) ) ) ) ) ) ) ).

% csqrt_of_real_nonpos
thf(fact_9486_complex__Im__numeral,axiom,
    ! [V: num] :
      ( ( im @ ( numera6690914467698888265omplex @ V ) )
      = zero_zero_real ) ).

% complex_Im_numeral
thf(fact_9487_Im__i__times,axiom,
    ! [Z: complex] :
      ( ( im @ ( times_times_complex @ imaginary_unit @ Z ) )
      = ( re @ Z ) ) ).

% Im_i_times
thf(fact_9488_Re__i__times,axiom,
    ! [Z: complex] :
      ( ( re @ ( times_times_complex @ imaginary_unit @ Z ) )
      = ( uminus_uminus_real @ ( im @ Z ) ) ) ).

% Re_i_times
thf(fact_9489_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_9490_csqrt__of__real__nonneg,axiom,
    ! [X3: complex] :
      ( ( ( im @ X3 )
        = zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X3 ) )
       => ( ( csqrt @ X3 )
          = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X3 ) ) ) ) ) ) ).

% csqrt_of_real_nonneg
thf(fact_9491_csqrt__minus,axiom,
    ! [X3: complex] :
      ( ( ( ord_less_real @ ( im @ X3 ) @ zero_zero_real )
        | ( ( ( im @ X3 )
            = zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ ( re @ X3 ) ) ) )
     => ( ( csqrt @ ( uminus1482373934393186551omplex @ X3 ) )
        = ( times_times_complex @ imaginary_unit @ ( csqrt @ X3 ) ) ) ) ).

% csqrt_minus
thf(fact_9492_scaleR__complex_Osimps_I2_J,axiom,
    ! [R: real,X3: complex] :
      ( ( im @ ( real_V2046097035970521341omplex @ R @ X3 ) )
      = ( times_times_real @ R @ ( im @ X3 ) ) ) ).

% scaleR_complex.simps(2)
thf(fact_9493_abs__Im__le__cmod,axiom,
    ! [X3: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X3 ) ) @ ( real_V1022390504157884413omplex @ X3 ) ) ).

% abs_Im_le_cmod
thf(fact_9494_times__complex_Osimps_I2_J,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( im @ ( times_times_complex @ X3 @ Y3 ) )
      = ( plus_plus_real @ ( times_times_real @ ( re @ X3 ) @ ( im @ Y3 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( re @ Y3 ) ) ) ) ).

% times_complex.simps(2)
thf(fact_9495_cmod__Im__le__iff,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( ( re @ X3 )
        = ( re @ Y3 ) )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X3 ) ) @ ( abs_abs_real @ ( im @ Y3 ) ) ) ) ) ).

% cmod_Im_le_iff
thf(fact_9496_cmod__Re__le__iff,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( ( im @ X3 )
        = ( im @ Y3 ) )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X3 ) ) @ ( abs_abs_real @ ( re @ Y3 ) ) ) ) ) ).

% cmod_Re_le_iff
thf(fact_9497_times__complex_Osimps_I1_J,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( re @ ( times_times_complex @ X3 @ Y3 ) )
      = ( minus_minus_real @ ( times_times_real @ ( re @ X3 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( im @ Y3 ) ) ) ) ).

% times_complex.simps(1)
thf(fact_9498_scaleR__complex_Ocode,axiom,
    ( real_V2046097035970521341omplex
    = ( ^ [R5: real,X: complex] : ( complex2 @ ( times_times_real @ R5 @ ( re @ X ) ) @ ( times_times_real @ R5 @ ( im @ X ) ) ) ) ) ).

% scaleR_complex.code
thf(fact_9499_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_9500_cmod__le,axiom,
    ! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).

% cmod_le
thf(fact_9501_sin__n__Im__cis__pow__n,axiom,
    ! [N: nat,A: real] :
      ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
      = ( im @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).

% sin_n_Im_cis_pow_n
thf(fact_9502_Re__exp,axiom,
    ! [Z: complex] :
      ( ( re @ ( exp_complex @ Z ) )
      = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( cos_real @ ( im @ Z ) ) ) ) ).

% Re_exp
thf(fact_9503_Im__exp,axiom,
    ! [Z: complex] :
      ( ( im @ ( exp_complex @ Z ) )
      = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( sin_real @ ( im @ Z ) ) ) ) ).

% Im_exp
thf(fact_9504_complex__eq,axiom,
    ! [A: complex] :
      ( A
      = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( re @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( im @ A ) ) ) ) ) ).

% complex_eq
thf(fact_9505_times__complex_Ocode,axiom,
    ( times_times_complex
    = ( ^ [X: complex,Y: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) ) ) ) ) ).

% times_complex.code
thf(fact_9506_exp__eq__polar,axiom,
    ( exp_complex
    = ( ^ [Z2: complex] : ( times_times_complex @ ( real_V4546457046886955230omplex @ ( exp_real @ ( re @ Z2 ) ) ) @ ( cis @ ( im @ Z2 ) ) ) ) ) ).

% exp_eq_polar
thf(fact_9507_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_9508_Im__power2,axiom,
    ! [X3: complex] :
      ( ( im @ ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X3 ) ) @ ( im @ X3 ) ) ) ).

% Im_power2
thf(fact_9509_Re__power2,axiom,
    ! [X3: complex] :
      ( ( re @ ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( minus_minus_real @ ( power_power_real @ ( re @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% Re_power2
thf(fact_9510_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_9511_norm__complex__def,axiom,
    ( real_V1022390504157884413omplex
    = ( ^ [Z2: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% norm_complex_def
thf(fact_9512_inverse__complex_Osimps_I1_J,axiom,
    ! [X3: complex] :
      ( ( re @ ( invers8013647133539491842omplex @ X3 ) )
      = ( divide_divide_real @ ( re @ X3 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% inverse_complex.simps(1)
thf(fact_9513_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_9514_Re__divide,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( re @ ( divide1717551699836669952omplex @ X3 @ Y3 ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X3 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( im @ Y3 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Re_divide
thf(fact_9515_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_9516_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_9517_inverse__complex_Osimps_I2_J,axiom,
    ! [X3: complex] :
      ( ( im @ ( invers8013647133539491842omplex @ X3 ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X3 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% inverse_complex.simps(2)
thf(fact_9518_Im__divide,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( im @ ( divide1717551699836669952omplex @ X3 @ Y3 ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X3 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( re @ X3 ) @ ( im @ Y3 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Im_divide
thf(fact_9519_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_9520_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_9521_inverse__complex_Ocode,axiom,
    ( invers8013647133539491842omplex
    = ( ^ [X: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% inverse_complex.code
thf(fact_9522_Complex__divide,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [X: complex,Y: complex] : ( complex2 @ ( 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 ) ) ) ) ) @ ( 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 ) ) ) ) ) ) ) ) ).

% Complex_divide
thf(fact_9523_Im__Reals__divide,axiom,
    ! [R: complex,Z: complex] :
      ( ( member_complex @ R @ real_V2521375963428798218omplex )
     => ( ( im @ ( divide1717551699836669952omplex @ R @ Z ) )
        = ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R ) ) @ ( im @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Im_Reals_divide
thf(fact_9524_imaginary__eq__real__iff,axiom,
    ! [Y3: complex,X3: complex] :
      ( ( member_complex @ Y3 @ real_V2521375963428798218omplex )
     => ( ( member_complex @ X3 @ real_V2521375963428798218omplex )
       => ( ( ( times_times_complex @ imaginary_unit @ Y3 )
            = X3 )
          = ( ( X3 = zero_zero_complex )
            & ( Y3 = zero_zero_complex ) ) ) ) ) ).

% imaginary_eq_real_iff
thf(fact_9525_real__eq__imaginary__iff,axiom,
    ! [Y3: complex,X3: complex] :
      ( ( member_complex @ Y3 @ real_V2521375963428798218omplex )
     => ( ( member_complex @ X3 @ real_V2521375963428798218omplex )
       => ( ( X3
            = ( times_times_complex @ imaginary_unit @ Y3 ) )
          = ( ( X3 = zero_zero_complex )
            & ( Y3 = zero_zero_complex ) ) ) ) ) ).

% real_eq_imaginary_iff
thf(fact_9526_Re__Reals__divide,axiom,
    ! [R: complex,Z: complex] :
      ( ( member_complex @ R @ real_V2521375963428798218omplex )
     => ( ( re @ ( divide1717551699836669952omplex @ R @ Z ) )
        = ( divide_divide_real @ ( times_times_real @ ( re @ R ) @ ( re @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Re_Reals_divide
thf(fact_9527_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_9528_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_9529_complex__div__cnj,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A5: complex,B4: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A5 @ ( cnj @ B4 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% complex_div_cnj
thf(fact_9530_complex__cnj__mult,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( cnj @ ( times_times_complex @ X3 @ Y3 ) )
      = ( times_times_complex @ ( cnj @ X3 ) @ ( cnj @ Y3 ) ) ) ).

% complex_cnj_mult
thf(fact_9531_complex__cnj__numeral,axiom,
    ! [W: num] :
      ( ( cnj @ ( numera6690914467698888265omplex @ W ) )
      = ( numera6690914467698888265omplex @ W ) ) ).

% complex_cnj_numeral
thf(fact_9532_complex__cnj__neg__numeral,axiom,
    ! [W: num] :
      ( ( cnj @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).

% complex_cnj_neg_numeral
thf(fact_9533_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_9534_Re__complex__div__eq__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( re @ ( divide1717551699836669952omplex @ A @ B2 ) )
        = zero_zero_real )
      = ( ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) )
        = zero_zero_real ) ) ).

% Re_complex_div_eq_0
thf(fact_9535_Im__complex__div__eq__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( im @ ( divide1717551699836669952omplex @ A @ B2 ) )
        = zero_zero_real )
      = ( ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) )
        = zero_zero_real ) ) ).

% Im_complex_div_eq_0
thf(fact_9536_complex__mod__sqrt__Re__mult__cnj,axiom,
    ( real_V1022390504157884413omplex
    = ( ^ [Z2: complex] : ( sqrt @ ( re @ ( times_times_complex @ Z2 @ ( cnj @ Z2 ) ) ) ) ) ) ).

% complex_mod_sqrt_Re_mult_cnj
thf(fact_9537_Re__complex__div__lt__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) @ zero_zero_real )
      = ( ord_less_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).

% Re_complex_div_lt_0
thf(fact_9538_Re__complex__div__gt__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
      = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ).

% Re_complex_div_gt_0
thf(fact_9539_Re__complex__div__le__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) @ zero_zero_real )
      = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).

% Re_complex_div_le_0
thf(fact_9540_Re__complex__div__ge__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
      = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ).

% Re_complex_div_ge_0
thf(fact_9541_Im__complex__div__lt__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) @ zero_zero_real )
      = ( ord_less_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).

% Im_complex_div_lt_0
thf(fact_9542_Im__complex__div__gt__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
      = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ).

% Im_complex_div_gt_0
thf(fact_9543_Im__complex__div__le__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) @ zero_zero_real )
      = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).

% Im_complex_div_le_0
thf(fact_9544_Im__complex__div__ge__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
      = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ).

% Im_complex_div_ge_0
thf(fact_9545_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_9546_complex__div__gt__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
        = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) )
      & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
        = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ) ).

% complex_div_gt_0
thf(fact_9547_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_9548_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_9549_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_9550_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
      = ( case_nat_o @ $false
        @ ^ [Uu3: nat] : $true
        @ Nat ) ) ).

% nat.disc_eq_case(2)
thf(fact_9551_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
      = ( case_nat_o @ $true
        @ ^ [Uu3: nat] : $false
        @ Nat ) ) ).

% nat.disc_eq_case(1)
thf(fact_9552_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).

% less_eq_nat.simps(2)
thf(fact_9553_max__Suc2,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_max_nat @ M @ ( suc @ N ) )
      = ( case_nat_nat @ ( suc @ N )
        @ ^ [M5: nat] : ( suc @ ( ord_max_nat @ M5 @ N ) )
        @ M ) ) ).

% max_Suc2
thf(fact_9554_max__Suc1,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_max_nat @ ( suc @ N ) @ M )
      = ( case_nat_nat @ ( suc @ N )
        @ ^ [M5: nat] : ( suc @ ( ord_max_nat @ N @ M5 ) )
        @ M ) ) ).

% max_Suc1
thf(fact_9555_diff__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [K2: nat] : K2
        @ ( minus_minus_nat @ M @ N ) ) ) ).

% diff_Suc
thf(fact_9556_pred__def,axiom,
    ( pred
    = ( case_nat_nat @ zero_zero_nat
      @ ^ [X23: nat] : X23 ) ) ).

% pred_def
thf(fact_9557_floor__real__def,axiom,
    ( archim6058952711729229775r_real
    = ( ^ [X: real] :
          ( the_int
          @ ^ [Z2: int] :
              ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X )
              & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).

% floor_real_def
thf(fact_9558_floor__rat__def,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [X: rat] :
          ( the_int
          @ ^ [Z2: int] :
              ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X )
              & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).

% floor_rat_def
thf(fact_9559_bezw__0,axiom,
    ! [X3: nat] :
      ( ( bezw @ X3 @ zero_zero_nat )
      = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).

% bezw_0
thf(fact_9560_prod__decode__aux_Oelims,axiom,
    ! [X3: nat,Xa2: nat,Y3: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X3 @ Xa2 )
        = Y3 )
     => ( ( ( ord_less_eq_nat @ Xa2 @ X3 )
         => ( Y3
            = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X3 @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa2 @ X3 )
         => ( Y3
            = ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X3 ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_9561_less__eq__rat__def,axiom,
    ( ord_less_eq_rat
    = ( ^ [X: rat,Y: rat] :
          ( ( ord_less_rat @ X @ Y )
          | ( X = Y ) ) ) ) ).

% less_eq_rat_def
thf(fact_9562_obtain__pos__sum,axiom,
    ! [R: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ R )
     => ~ ! [S3: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ S3 )
           => ! [T4: rat] :
                ( ( ord_less_rat @ zero_zero_rat @ T4 )
               => ( R
                 != ( plus_plus_rat @ S3 @ T4 ) ) ) ) ) ).

% obtain_pos_sum
thf(fact_9563_sgn__rat__def,axiom,
    ( sgn_sgn_rat
    = ( ^ [A5: rat] : ( if_rat @ ( A5 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A5 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_rat_def
thf(fact_9564_abs__rat__def,axiom,
    ( abs_abs_rat
    = ( ^ [A5: rat] : ( if_rat @ ( ord_less_rat @ A5 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A5 ) @ A5 ) ) ) ).

% abs_rat_def
thf(fact_9565_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K2: nat,M3: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M3 @ K2 ) @ ( product_Pair_nat_nat @ M3 @ ( minus_minus_nat @ K2 @ M3 ) ) @ ( nat_prod_decode_aux @ ( suc @ K2 ) @ ( minus_minus_nat @ M3 @ ( suc @ K2 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_9566_prod__decode__aux_Opelims,axiom,
    ! [X3: nat,Xa2: nat,Y3: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X3 @ Xa2 )
        = Y3 )
     => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) )
       => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X3 )
               => ( Y3
                  = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X3 @ Xa2 ) ) ) )
              & ( ~ ( ord_less_eq_nat @ Xa2 @ X3 )
               => ( Y3
                  = ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X3 ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) ) ) ) ) ).

% prod_decode_aux.pelims
thf(fact_9567_rat__inverse__code,axiom,
    ! [P4: rat] :
      ( ( quotient_of @ ( inverse_inverse_rat @ P4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( A5 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A5 ) @ B4 ) @ ( abs_abs_int @ A5 ) ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_inverse_code
thf(fact_9568_normalize__negative,axiom,
    ! [Q2: int,P4: int] :
      ( ( ord_less_int @ Q2 @ zero_zero_int )
     => ( ( normalize @ ( product_Pair_int_int @ P4 @ Q2 ) )
        = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P4 ) @ ( uminus_uminus_int @ Q2 ) ) ) ) ) ).

% normalize_negative
thf(fact_9569_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_9570_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_9571_rat__one__code,axiom,
    ( ( quotient_of @ one_one_rat )
    = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).

% rat_one_code
thf(fact_9572_rat__zero__code,axiom,
    ( ( quotient_of @ zero_zero_rat )
    = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% rat_zero_code
thf(fact_9573_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_9574_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_9575_divide__rat__def,axiom,
    ( divide_divide_rat
    = ( ^ [Q4: rat,R5: rat] : ( times_times_rat @ Q4 @ ( inverse_inverse_rat @ R5 ) ) ) ) ).

% divide_rat_def
thf(fact_9576_rat__divide__code,axiom,
    ! [P4: rat,Q2: rat] :
      ( ( quotient_of @ ( divide_divide_rat @ P4 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ C3 @ B4 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_divide_code
thf(fact_9577_rat__times__code,axiom,
    ! [P4: rat,Q2: rat] :
      ( ( quotient_of @ ( times_times_rat @ P4 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A5 @ B4 ) @ ( times_times_int @ C3 @ D2 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_times_code
thf(fact_9578_quotient__of__div,axiom,
    ! [R: rat,N: int,D: int] :
      ( ( ( quotient_of @ R )
        = ( product_Pair_int_int @ N @ D ) )
     => ( R
        = ( divide_divide_rat @ ( ring_1_of_int_rat @ N ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).

% quotient_of_div
thf(fact_9579_rat__plus__code,axiom,
    ! [P4: rat,Q2: rat] :
      ( ( quotient_of @ ( plus_plus_rat @ P4 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ B4 @ C3 ) ) @ ( times_times_int @ C3 @ D2 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_plus_code
thf(fact_9580_rat__minus__code,axiom,
    ! [P4: rat,Q2: rat] :
      ( ( quotient_of @ ( minus_minus_rat @ P4 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ B4 @ C3 ) ) @ ( times_times_int @ C3 @ D2 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_minus_code
thf(fact_9581_quotient__of__denom__pos,axiom,
    ! [R: rat,P4: int,Q2: int] :
      ( ( ( quotient_of @ R )
        = ( product_Pair_int_int @ P4 @ Q2 ) )
     => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).

% quotient_of_denom_pos
thf(fact_9582_rat__uminus__code,axiom,
    ! [P4: rat] :
      ( ( quotient_of @ ( uminus_uminus_rat @ P4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A5 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_uminus_code
thf(fact_9583_rat__abs__code,axiom,
    ! [P4: rat] :
      ( ( quotient_of @ ( abs_abs_rat @ P4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int] : ( product_Pair_int_int @ ( abs_abs_int @ A5 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_abs_code
thf(fact_9584_normalize__denom__pos,axiom,
    ! [R: product_prod_int_int,P4: int,Q2: int] :
      ( ( ( normalize @ R )
        = ( product_Pair_int_int @ P4 @ Q2 ) )
     => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).

% normalize_denom_pos
thf(fact_9585_normalize__crossproduct,axiom,
    ! [Q2: int,S: int,P4: int,R: int] :
      ( ( Q2 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( ( normalize @ ( product_Pair_int_int @ P4 @ Q2 ) )
            = ( normalize @ ( product_Pair_int_int @ R @ S ) ) )
         => ( ( times_times_int @ P4 @ S )
            = ( times_times_int @ R @ Q2 ) ) ) ) ) ).

% normalize_crossproduct
thf(fact_9586_rat__less__code,axiom,
    ( ord_less_rat
    = ( ^ [P5: rat,Q4: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A5: int,C3: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B4: int,D2: int] : ( ord_less_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ C3 @ B4 ) )
              @ ( quotient_of @ Q4 ) )
          @ ( quotient_of @ P5 ) ) ) ) ).

% rat_less_code
thf(fact_9587_rat__less__eq__code,axiom,
    ( ord_less_eq_rat
    = ( ^ [P5: rat,Q4: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A5: int,C3: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B4: int,D2: int] : ( ord_less_eq_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ C3 @ B4 ) )
              @ ( quotient_of @ Q4 ) )
          @ ( quotient_of @ P5 ) ) ) ) ).

% rat_less_eq_code
thf(fact_9588_quotient__of__int,axiom,
    ! [A: int] :
      ( ( quotient_of @ ( of_int @ A ) )
      = ( product_Pair_int_int @ A @ one_one_int ) ) ).

% quotient_of_int
thf(fact_9589_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_9590_Suc__0__div__numeral,axiom,
    ! [K: num] :
      ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
      = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).

% Suc_0_div_numeral
thf(fact_9591_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_9592_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_9593_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_9594_fst__divmod__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( product_fst_nat_nat @ ( divmod_nat @ M @ N ) )
      = ( divide_divide_nat @ M @ N ) ) ).

% fst_divmod_nat
thf(fact_9595_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_9596_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_9597_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_9598_drop__bit__push__bit__int,axiom,
    ! [M: nat,N: nat,K: int] :
      ( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N @ K ) )
      = ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N @ M ) @ K ) ) ) ).

% drop_bit_push_bit_int
thf(fact_9599_drop__bit__int__def,axiom,
    ( bit_se8568078237143864401it_int
    = ( ^ [N2: nat,K2: int] : ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% drop_bit_int_def
thf(fact_9600_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_9601_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_9602_Suc__0__mod__numeral,axiom,
    ! [K: num] :
      ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
      = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).

% Suc_0_mod_numeral
thf(fact_9603_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_9604_drop__bit__nat__eq,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se8570568707652914677it_nat @ N @ ( nat2 @ K ) )
      = ( nat2 @ ( bit_se8568078237143864401it_int @ N @ K ) ) ) ).

% drop_bit_nat_eq
thf(fact_9605_drop__bit__nat__def,axiom,
    ( bit_se8570568707652914677it_nat
    = ( ^ [N2: nat,M3: nat] : ( divide_divide_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% drop_bit_nat_def
thf(fact_9606_Frct__code__post_I1_J,axiom,
    ! [A: int] :
      ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
      = zero_zero_rat ) ).

% Frct_code_post(1)
thf(fact_9607_Frct__code__post_I2_J,axiom,
    ! [A: int] :
      ( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
      = zero_zero_rat ) ).

% Frct_code_post(2)
thf(fact_9608_Frct__code__post_I8_J,axiom,
    ! [A: int,B2: int] :
      ( ( frct @ ( product_Pair_int_int @ A @ ( uminus_uminus_int @ B2 ) ) )
      = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B2 ) ) ) ) ).

% Frct_code_post(8)
thf(fact_9609_Frct__code__post_I7_J,axiom,
    ! [A: int,B2: int] :
      ( ( frct @ ( product_Pair_int_int @ ( uminus_uminus_int @ A ) @ B2 ) )
      = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B2 ) ) ) ) ).

% Frct_code_post(7)
thf(fact_9610_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_9611_rat__sgn__code,axiom,
    ! [P4: rat] :
      ( ( quotient_of @ ( sgn_sgn_rat @ P4 ) )
      = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P4 ) ) ) @ one_one_int ) ) ).

% rat_sgn_code
thf(fact_9612_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_9613_quotient__of__denom__pos_H,axiom,
    ! [R: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R ) ) ) ).

% quotient_of_denom_pos'
thf(fact_9614_bezw__non__0,axiom,
    ! [Y3: nat,X3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Y3 )
     => ( ( bezw @ X3 @ Y3 )
        = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Y3 ) ) ) ) ) ) ) ).

% bezw_non_0
thf(fact_9615_bezw_Osimps,axiom,
    ( bezw
    = ( ^ [X: nat,Y: nat] : ( if_Pro3027730157355071871nt_int @ ( Y = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ) ).

% bezw.simps
thf(fact_9616_bezw_Oelims,axiom,
    ! [X3: nat,Xa2: nat,Y3: product_prod_int_int] :
      ( ( ( bezw @ X3 @ Xa2 )
        = Y3 )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y3
            = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y3
            = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Xa2 ) ) ) ) ) ) ) ) ) ).

% bezw.elims
thf(fact_9617_bezw_Opelims,axiom,
    ! [X3: nat,Xa2: nat,Y3: product_prod_int_int] :
      ( ( ( bezw @ X3 @ Xa2 )
        = Y3 )
     => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) )
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y3
                  = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y3
                  = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Xa2 ) ) ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) ) ) ) ) ).

% bezw.pelims
thf(fact_9618_minus__one__mod__numeral,axiom,
    ! [N: num] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
      = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).

% minus_one_mod_numeral
thf(fact_9619_one__mod__minus__numeral,axiom,
    ! [N: num] :
      ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ) ).

% one_mod_minus_numeral
thf(fact_9620_minus__numeral__mod__numeral,axiom,
    ! [M: num,N: num] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).

% minus_numeral_mod_numeral
thf(fact_9621_numeral__mod__minus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ) ).

% numeral_mod_minus_numeral
thf(fact_9622_normalize__def,axiom,
    ( normalize
    = ( ^ [P5: product_prod_int_int] :
          ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P5 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P5 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P5 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) )
          @ ( if_Pro3027730157355071871nt_int
            @ ( ( product_snd_int_int @ P5 )
              = zero_zero_int )
            @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
            @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P5 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P5 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) ) ) ) ) ) ).

% normalize_def
thf(fact_9623_gcd__pos__int,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N ) )
      = ( ( M != zero_zero_int )
        | ( N != zero_zero_int ) ) ) ).

% gcd_pos_int
thf(fact_9624_gcd__neg__numeral__2__int,axiom,
    ! [X3: int,N: num] :
      ( ( gcd_gcd_int @ X3 @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( gcd_gcd_int @ X3 @ ( numeral_numeral_int @ N ) ) ) ).

% gcd_neg_numeral_2_int
thf(fact_9625_gcd__neg__numeral__1__int,axiom,
    ! [N: num,X3: int] :
      ( ( gcd_gcd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ X3 )
      = ( gcd_gcd_int @ ( numeral_numeral_int @ N ) @ X3 ) ) ).

% gcd_neg_numeral_1_int
thf(fact_9626_gcd__ge__0__int,axiom,
    ! [X3: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X3 @ Y3 ) ) ).

% gcd_ge_0_int
thf(fact_9627_bezout__int,axiom,
    ! [X3: int,Y3: int] :
    ? [U3: int,V2: int] :
      ( ( plus_plus_int @ ( times_times_int @ U3 @ X3 ) @ ( times_times_int @ V2 @ Y3 ) )
      = ( gcd_gcd_int @ X3 @ Y3 ) ) ).

% bezout_int
thf(fact_9628_gcd__mult__distrib__int,axiom,
    ! [K: int,M: int,N: int] :
      ( ( times_times_int @ ( abs_abs_int @ K ) @ ( gcd_gcd_int @ M @ N ) )
      = ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) ) ) ).

% gcd_mult_distrib_int
thf(fact_9629_gcd__le2__int,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B2 ) @ B2 ) ) ).

% gcd_le2_int
thf(fact_9630_gcd__le1__int,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B2 ) @ A ) ) ).

% gcd_le1_int
thf(fact_9631_gcd__cases__int,axiom,
    ! [X3: int,Y3: int,P: int > $o] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
         => ( P @ ( gcd_gcd_int @ X3 @ Y3 ) ) ) )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
         => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
           => ( P @ ( gcd_gcd_int @ X3 @ ( uminus_uminus_int @ Y3 ) ) ) ) )
       => ( ( ( ord_less_eq_int @ X3 @ zero_zero_int )
           => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
             => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X3 ) @ Y3 ) ) ) )
         => ( ( ( ord_less_eq_int @ X3 @ zero_zero_int )
             => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
               => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X3 ) @ ( uminus_uminus_int @ Y3 ) ) ) ) )
           => ( P @ ( gcd_gcd_int @ X3 @ Y3 ) ) ) ) ) ) ).

% gcd_cases_int
thf(fact_9632_gcd__unique__int,axiom,
    ! [D: int,A: int,B2: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ D )
        & ( dvd_dvd_int @ D @ A )
        & ( dvd_dvd_int @ D @ B2 )
        & ! [E3: int] :
            ( ( ( dvd_dvd_int @ E3 @ A )
              & ( dvd_dvd_int @ E3 @ B2 ) )
           => ( dvd_dvd_int @ E3 @ D ) ) )
      = ( D
        = ( gcd_gcd_int @ A @ B2 ) ) ) ).

% gcd_unique_int
thf(fact_9633_gcd__non__0__int,axiom,
    ! [Y3: int,X3: int] :
      ( ( ord_less_int @ zero_zero_int @ Y3 )
     => ( ( gcd_gcd_int @ X3 @ Y3 )
        = ( gcd_gcd_int @ Y3 @ ( modulo_modulo_int @ X3 @ Y3 ) ) ) ) ).

% gcd_non_0_int
thf(fact_9634_nat__descend__induct,axiom,
    ! [N: nat,P: nat > $o,M: nat] :
      ( ! [K3: nat] :
          ( ( ord_less_nat @ N @ K3 )
         => ( P @ K3 ) )
     => ( ! [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N )
           => ( ! [I2: nat] :
                  ( ( ord_less_nat @ K3 @ I2 )
                 => ( P @ I2 ) )
             => ( P @ K3 ) ) )
       => ( P @ M ) ) ) ).

% nat_descend_induct
thf(fact_9635_finite__enumerate,axiom,
    ! [S2: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ? [R3: nat > nat] :
          ( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S2 ) ) )
          & ! [N9: nat] :
              ( ( ord_less_nat @ N9 @ ( finite_card_nat @ S2 ) )
             => ( member_nat @ ( R3 @ N9 ) @ S2 ) ) ) ) ).

% finite_enumerate
thf(fact_9636_divmod__integer__eq__cases,axiom,
    ( code_divmod_integer
    = ( ^ [K2: code_integer,L2: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
            @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
              @ ( if_Pro6119634080678213985nteger
                @ ( ( sgn_sgn_Code_integer @ K2 )
                  = ( sgn_sgn_Code_integer @ L2 ) )
                @ ( code_divmod_abs @ K2 @ L2 )
                @ ( produc6916734918728496179nteger
                  @ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L2 ) @ S6 ) ) )
                  @ ( code_divmod_abs @ K2 @ L2 ) ) ) ) ) ) ) ) ).

% divmod_integer_eq_cases
thf(fact_9637_gcd__pos__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
      = ( ( M != zero_zero_nat )
        | ( N != zero_zero_nat ) ) ) ).

% gcd_pos_nat
thf(fact_9638_gcd__mult__distrib__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N ) )
      = ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).

% gcd_mult_distrib_nat
thf(fact_9639_gcd__le2__nat,axiom,
    ! [B2: nat,A: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B2 ) @ B2 ) ) ).

% gcd_le2_nat
thf(fact_9640_gcd__le1__nat,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B2 ) @ A ) ) ).

% gcd_le1_nat
thf(fact_9641_gcd__diff1__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
        = ( gcd_gcd_nat @ M @ N ) ) ) ).

% gcd_diff1_nat
thf(fact_9642_gcd__diff2__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
        = ( gcd_gcd_nat @ M @ N ) ) ) ).

% gcd_diff2_nat
thf(fact_9643_bezout__nat,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ? [X5: nat,Y4: nat] :
          ( ( times_times_nat @ A @ X5 )
          = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y4 ) @ ( gcd_gcd_nat @ A @ B2 ) ) ) ) ).

% bezout_nat
thf(fact_9644_bezout__gcd__nat_H,axiom,
    ! [B2: nat,A: nat] :
    ? [X5: nat,Y4: nat] :
      ( ( ( ord_less_eq_nat @ ( times_times_nat @ B2 @ Y4 ) @ ( times_times_nat @ A @ X5 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ A @ X5 ) @ ( times_times_nat @ B2 @ Y4 ) )
          = ( gcd_gcd_nat @ A @ B2 ) ) )
      | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y4 ) @ ( times_times_nat @ B2 @ X5 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ B2 @ X5 ) @ ( times_times_nat @ A @ Y4 ) )
          = ( gcd_gcd_nat @ A @ B2 ) ) ) ) ).

% bezout_gcd_nat'
thf(fact_9645_bezw__aux,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X3 @ Y3 ) )
      = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X3 @ Y3 ) ) @ ( semiri1314217659103216013at_int @ X3 ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X3 @ Y3 ) ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) ) ).

% bezw_aux
thf(fact_9646_gcd__nat_Opelims,axiom,
    ! [X3: nat,Xa2: nat,Y3: nat] :
      ( ( ( gcd_gcd_nat @ X3 @ Xa2 )
        = Y3 )
     => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) )
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y3 = X3 ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y3
                  = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) ) ) ) ) ).

% gcd_nat.pelims
thf(fact_9647_Code__Numeral_Onegative__def,axiom,
    ( code_negative
    = ( comp_C3531382070062128313er_num @ uminus1351360451143612070nteger @ numera6620942414471956472nteger ) ) ).

% Code_Numeral.negative_def
thf(fact_9648_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_9649_root__powr__inverse,axiom,
    ! [N: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( root @ N @ X3 )
          = ( powr_real @ X3 @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).

% root_powr_inverse
thf(fact_9650_real__root__zero,axiom,
    ! [N: nat] :
      ( ( root @ N @ zero_zero_real )
      = zero_zero_real ) ).

% real_root_zero
thf(fact_9651_real__root__Suc__0,axiom,
    ! [X3: real] :
      ( ( root @ ( suc @ zero_zero_nat ) @ X3 )
      = X3 ) ).

% real_root_Suc_0
thf(fact_9652_root__0,axiom,
    ! [X3: real] :
      ( ( root @ zero_zero_nat @ X3 )
      = zero_zero_real ) ).

% root_0
thf(fact_9653_real__root__eq__iff,axiom,
    ! [N: nat,X3: real,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X3 )
          = ( root @ N @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% real_root_eq_iff
thf(fact_9654_real__root__eq__0__iff,axiom,
    ! [N: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X3 )
          = zero_zero_real )
        = ( X3 = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
thf(fact_9655_real__root__less__iff,axiom,
    ! [N: nat,X3: real,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X3 ) @ ( root @ N @ Y3 ) )
        = ( ord_less_real @ X3 @ Y3 ) ) ) ).

% real_root_less_iff
thf(fact_9656_real__root__le__iff,axiom,
    ! [N: nat,X3: real,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X3 ) @ ( root @ N @ Y3 ) )
        = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ).

% real_root_le_iff
thf(fact_9657_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_9658_real__root__eq__1__iff,axiom,
    ! [N: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X3 )
          = one_one_real )
        = ( X3 = one_one_real ) ) ) ).

% real_root_eq_1_iff
thf(fact_9659_real__root__lt__0__iff,axiom,
    ! [N: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X3 ) @ zero_zero_real )
        = ( ord_less_real @ X3 @ zero_zero_real ) ) ) ).

% real_root_lt_0_iff
thf(fact_9660_real__root__gt__0__iff,axiom,
    ! [N: nat,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y3 ) )
        = ( ord_less_real @ zero_zero_real @ Y3 ) ) ) ).

% real_root_gt_0_iff
thf(fact_9661_real__root__le__0__iff,axiom,
    ! [N: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X3 ) @ zero_zero_real )
        = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ) ).

% real_root_le_0_iff
thf(fact_9662_real__root__ge__0__iff,axiom,
    ! [N: nat,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y3 ) )
        = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ) ).

% real_root_ge_0_iff
thf(fact_9663_real__root__gt__1__iff,axiom,
    ! [N: nat,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y3 ) )
        = ( ord_less_real @ one_one_real @ Y3 ) ) ) ).

% real_root_gt_1_iff
thf(fact_9664_real__root__lt__1__iff,axiom,
    ! [N: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X3 ) @ one_one_real )
        = ( ord_less_real @ X3 @ one_one_real ) ) ) ).

% real_root_lt_1_iff
thf(fact_9665_real__root__le__1__iff,axiom,
    ! [N: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X3 ) @ one_one_real )
        = ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ).

% real_root_le_1_iff
thf(fact_9666_real__root__ge__1__iff,axiom,
    ! [N: nat,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y3 ) )
        = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ) ).

% real_root_ge_1_iff
thf(fact_9667_real__root__pow__pos2,axiom,
    ! [N: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
       => ( ( power_power_real @ ( root @ N @ X3 ) @ N )
          = X3 ) ) ) ).

% real_root_pow_pos2
thf(fact_9668_real__root__inverse,axiom,
    ! [N: nat,X3: real] :
      ( ( root @ N @ ( inverse_inverse_real @ X3 ) )
      = ( inverse_inverse_real @ ( root @ N @ X3 ) ) ) ).

% real_root_inverse
thf(fact_9669_real__root__minus,axiom,
    ! [N: nat,X3: real] :
      ( ( root @ N @ ( uminus_uminus_real @ X3 ) )
      = ( uminus_uminus_real @ ( root @ N @ X3 ) ) ) ).

% real_root_minus
thf(fact_9670_real__root__commute,axiom,
    ! [M: nat,N: nat,X3: real] :
      ( ( root @ M @ ( root @ N @ X3 ) )
      = ( root @ N @ ( root @ M @ X3 ) ) ) ).

% real_root_commute
thf(fact_9671_real__root__mult,axiom,
    ! [N: nat,X3: real,Y3: real] :
      ( ( root @ N @ ( times_times_real @ X3 @ Y3 ) )
      = ( times_times_real @ ( root @ N @ X3 ) @ ( root @ N @ Y3 ) ) ) ).

% real_root_mult
thf(fact_9672_real__root__mult__exp,axiom,
    ! [M: nat,N: nat,X3: real] :
      ( ( root @ ( times_times_nat @ M @ N ) @ X3 )
      = ( root @ M @ ( root @ N @ X3 ) ) ) ).

% real_root_mult_exp
thf(fact_9673_real__root__divide,axiom,
    ! [N: nat,X3: real,Y3: real] :
      ( ( root @ N @ ( divide_divide_real @ X3 @ Y3 ) )
      = ( divide_divide_real @ ( root @ N @ X3 ) @ ( root @ N @ Y3 ) ) ) ).

% real_root_divide
thf(fact_9674_real__root__pos__pos__le,axiom,
    ! [X3: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X3 ) ) ) ).

% real_root_pos_pos_le
thf(fact_9675_real__root__less__mono,axiom,
    ! [N: nat,X3: real,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ X3 @ Y3 )
       => ( ord_less_real @ ( root @ N @ X3 ) @ ( root @ N @ Y3 ) ) ) ) ).

% real_root_less_mono
thf(fact_9676_real__root__le__mono,axiom,
    ! [N: nat,X3: real,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ X3 @ Y3 )
       => ( ord_less_eq_real @ ( root @ N @ X3 ) @ ( root @ N @ Y3 ) ) ) ) ).

% real_root_le_mono
thf(fact_9677_real__root__power,axiom,
    ! [N: nat,X3: real,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( power_power_real @ X3 @ K ) )
        = ( power_power_real @ ( root @ N @ X3 ) @ K ) ) ) ).

% real_root_power
thf(fact_9678_real__root__abs,axiom,
    ! [N: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( abs_abs_real @ X3 ) )
        = ( abs_abs_real @ ( root @ N @ X3 ) ) ) ) ).

% real_root_abs
thf(fact_9679_sgn__root,axiom,
    ! [N: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( sgn_sgn_real @ ( root @ N @ X3 ) )
        = ( sgn_sgn_real @ X3 ) ) ) ).

% sgn_root
thf(fact_9680_real__root__gt__zero,axiom,
    ! [N: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ord_less_real @ zero_zero_real @ ( root @ N @ X3 ) ) ) ) ).

% real_root_gt_zero
thf(fact_9681_real__root__strict__decreasing,axiom,
    ! [N: nat,N4: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ( ord_less_real @ one_one_real @ X3 )
         => ( ord_less_real @ ( root @ N4 @ X3 ) @ ( root @ N @ X3 ) ) ) ) ) ).

% real_root_strict_decreasing
thf(fact_9682_sqrt__def,axiom,
    ( sqrt
    = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% sqrt_def
thf(fact_9683_root__abs__power,axiom,
    ! [N: nat,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y3 @ N ) ) )
        = ( abs_abs_real @ Y3 ) ) ) ).

% root_abs_power
thf(fact_9684_real__root__pos__pos,axiom,
    ! [N: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X3 ) ) ) ) ).

% real_root_pos_pos
thf(fact_9685_real__root__strict__increasing,axiom,
    ! [N: nat,N4: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ( ord_less_real @ zero_zero_real @ X3 )
         => ( ( ord_less_real @ X3 @ one_one_real )
           => ( ord_less_real @ ( root @ N @ X3 ) @ ( root @ N4 @ X3 ) ) ) ) ) ) ).

% real_root_strict_increasing
thf(fact_9686_real__root__decreasing,axiom,
    ! [N: nat,N4: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ( ord_less_eq_real @ one_one_real @ X3 )
         => ( ord_less_eq_real @ ( root @ N4 @ X3 ) @ ( root @ N @ X3 ) ) ) ) ) ).

% real_root_decreasing
thf(fact_9687_real__root__pow__pos,axiom,
    ! [N: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( power_power_real @ ( root @ N @ X3 ) @ N )
          = X3 ) ) ) ).

% real_root_pow_pos
thf(fact_9688_real__root__pos__unique,axiom,
    ! [N: nat,Y3: real,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ( power_power_real @ Y3 @ N )
            = X3 )
         => ( ( root @ N @ X3 )
            = Y3 ) ) ) ) ).

% real_root_pos_unique
thf(fact_9689_real__root__power__cancel,axiom,
    ! [N: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
       => ( ( root @ N @ ( power_power_real @ X3 @ N ) )
          = X3 ) ) ) ).

% real_root_power_cancel
thf(fact_9690_odd__real__root__power__cancel,axiom,
    ! [N: nat,X3: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( root @ N @ ( power_power_real @ X3 @ N ) )
        = X3 ) ) ).

% odd_real_root_power_cancel
thf(fact_9691_odd__real__root__unique,axiom,
    ! [N: nat,Y3: real,X3: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ( power_power_real @ Y3 @ N )
          = X3 )
       => ( ( root @ N @ X3 )
          = Y3 ) ) ) ).

% odd_real_root_unique
thf(fact_9692_odd__real__root__pow,axiom,
    ! [N: nat,X3: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( root @ N @ X3 ) @ N )
        = X3 ) ) ).

% odd_real_root_pow
thf(fact_9693_real__root__increasing,axiom,
    ! [N: nat,N4: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
         => ( ( ord_less_eq_real @ X3 @ one_one_real )
           => ( ord_less_eq_real @ ( root @ N @ X3 ) @ ( root @ N4 @ X3 ) ) ) ) ) ) ).

% real_root_increasing
thf(fact_9694_sgn__power__root,axiom,
    ! [N: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X3 ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X3 ) ) @ N ) )
        = X3 ) ) ).

% sgn_power_root
thf(fact_9695_root__sgn__power,axiom,
    ! [N: nat,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N ) ) )
        = Y3 ) ) ).

% root_sgn_power
thf(fact_9696_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_9697_log__root,axiom,
    ! [N: nat,A: real,B2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( log @ B2 @ ( root @ N @ A ) )
          = ( divide_divide_real @ ( log @ B2 @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_root
thf(fact_9698_log__base__root,axiom,
    ! [N: nat,B2: real,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ( log @ ( root @ N @ B2 ) @ X3 )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ X3 ) ) ) ) ) ).

% log_base_root
thf(fact_9699_split__root,axiom,
    ! [P: real > $o,N: nat,X3: real] :
      ( ( P @ ( root @ N @ X3 ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ zero_zero_real ) )
        & ( ( ord_less_nat @ zero_zero_nat @ N )
         => ! [Y: real] :
              ( ( ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
                = X3 )
             => ( P @ Y ) ) ) ) ) ).

% split_root
thf(fact_9700_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
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = one_one_complex ) )
          @ ( collect_complex
            @ ^ [Z2: complex] :
                ( ( power_power_complex @ Z2 @ N )
                = C ) ) ) ) ) ).

% bij_betw_nth_root_unity
thf(fact_9701_xor__minus__numerals_I2_J,axiom,
    ! [K: int,N: num] :
      ( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N @ one ) ) ) ) ).

% xor_minus_numerals(2)
thf(fact_9702_xor__minus__numerals_I1_J,axiom,
    ! [N: num,K: int] :
      ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ K )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N @ one ) @ K ) ) ) ).

% xor_minus_numerals(1)
thf(fact_9703_sub__BitM__One__eq,axiom,
    ! [N: num] :
      ( ( neg_numeral_sub_int @ ( bitM @ N ) @ one )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N @ one ) ) ) ).

% sub_BitM_One_eq
thf(fact_9704_Suc__funpow,axiom,
    ! [N: nat] :
      ( ( compow_nat_nat @ N @ suc )
      = ( plus_plus_nat @ N ) ) ).

% Suc_funpow
thf(fact_9705_max__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
    @ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
    @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X ) ) ).

% max_nat.semilattice_neutr_order_axioms
thf(fact_9706_times__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
      ( ( times_times_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X: nat,Y: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U2 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U2 ) ) ) )
          @ Xa2
          @ X3 ) ) ) ).

% times_int.abs_eq
thf(fact_9707_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X5: nat,Y4: nat] :
          ( Z
         != ( abs_Integ @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) ) ).

% eq_Abs_Integ
thf(fact_9708_nat_Oabs__eq,axiom,
    ! [X3: product_prod_nat_nat] :
      ( ( nat2 @ ( abs_Integ @ X3 ) )
      = ( produc6842872674320459806at_nat @ minus_minus_nat @ X3 ) ) ).

% nat.abs_eq
thf(fact_9709_zero__int__def,axiom,
    ( zero_zero_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).

% zero_int_def
thf(fact_9710_int__def,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N2: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N2 @ zero_zero_nat ) ) ) ) ).

% int_def
thf(fact_9711_uminus__int_Oabs__eq,axiom,
    ! [X3: product_prod_nat_nat] :
      ( ( uminus_uminus_int @ ( abs_Integ @ X3 ) )
      = ( abs_Integ
        @ ( produc2626176000494625587at_nat
          @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X )
          @ X3 ) ) ) ).

% uminus_int.abs_eq
thf(fact_9712_one__int__def,axiom,
    ( one_one_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).

% one_int_def
thf(fact_9713_less__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
      ( ( ord_less_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X: nat,Y: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) )
        @ Xa2
        @ X3 ) ) ).

% less_int.abs_eq
thf(fact_9714_less__eq__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
      ( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X: nat,Y: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) )
        @ Xa2
        @ X3 ) ) ).

% less_eq_int.abs_eq
thf(fact_9715_plus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
      ( ( plus_plus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X: nat,Y: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) )
          @ Xa2
          @ X3 ) ) ) ).

% plus_int.abs_eq
thf(fact_9716_minus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
      ( ( minus_minus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X: nat,Y: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) )
          @ Xa2
          @ X3 ) ) ) ).

% minus_int.abs_eq
thf(fact_9717_less__eq__int_Orep__eq,axiom,
    ( ord_less_eq_int
    = ( ^ [X: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y: nat,Z2: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U2 @ Z2 ) ) )
          @ ( rep_Integ @ X )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_eq_int.rep_eq
thf(fact_9718_less__int_Orep__eq,axiom,
    ( ord_less_int
    = ( ^ [X: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y: nat,Z2: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U2 @ Z2 ) ) )
          @ ( rep_Integ @ X )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_int.rep_eq
thf(fact_9719_nat_Orep__eq,axiom,
    ( nat2
    = ( ^ [X: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X ) ) ) ) ).

% nat.rep_eq
thf(fact_9720_prod__encode__def,axiom,
    ( nat_prod_encode
    = ( produc6842872674320459806at_nat
      @ ^ [M3: nat,N2: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M3 @ N2 ) ) @ M3 ) ) ) ).

% prod_encode_def
thf(fact_9721_uminus__int__def,axiom,
    ( uminus_uminus_int
    = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
      @ ( produc2626176000494625587at_nat
        @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X ) ) ) ) ).

% uminus_int_def
thf(fact_9722_le__prod__encode__1,axiom,
    ! [A: nat,B2: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B2 ) ) ) ).

% le_prod_encode_1
thf(fact_9723_le__prod__encode__2,axiom,
    ! [B2: nat,A: nat] : ( ord_less_eq_nat @ B2 @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B2 ) ) ) ).

% le_prod_encode_2
thf(fact_9724_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,M: nat] :
      ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
      = ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).

% prod_encode_prod_decode_aux
thf(fact_9725_times__int__def,axiom,
    ( times_times_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X: nat,Y: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U2 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U2 ) ) ) ) ) ) ) ).

% times_int_def
thf(fact_9726_minus__int__def,axiom,
    ( minus_minus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X: nat,Y: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) ) ) ) ) ).

% minus_int_def
thf(fact_9727_plus__int__def,axiom,
    ( plus_plus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X: nat,Y: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) ) ) ) ).

% plus_int_def
thf(fact_9728_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( num_of_nat @ ( suc @ N ) )
          = ( inc @ ( num_of_nat @ N ) ) ) )
      & ( ~ ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( num_of_nat @ ( suc @ N ) )
          = one ) ) ) ).

% num_of_nat.simps(2)
thf(fact_9729_pow_Osimps_I3_J,axiom,
    ! [X3: num,Y3: num] :
      ( ( pow @ X3 @ ( bit1 @ Y3 ) )
      = ( times_times_num @ ( sqr @ ( pow @ X3 @ Y3 ) ) @ X3 ) ) ).

% pow.simps(3)
thf(fact_9730_num__of__nat__numeral__eq,axiom,
    ! [Q2: num] :
      ( ( num_of_nat @ ( numeral_numeral_nat @ Q2 ) )
      = Q2 ) ).

% num_of_nat_numeral_eq
thf(fact_9731_sqr_Osimps_I2_J,axiom,
    ! [N: num] :
      ( ( sqr @ ( bit0 @ N ) )
      = ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).

% sqr.simps(2)
thf(fact_9732_sqr_Osimps_I1_J,axiom,
    ( ( sqr @ one )
    = one ) ).

% sqr.simps(1)
thf(fact_9733_sqr__conv__mult,axiom,
    ( sqr
    = ( ^ [X: num] : ( times_times_num @ X @ X ) ) ) ).

% sqr_conv_mult
thf(fact_9734_num__of__nat_Osimps_I1_J,axiom,
    ( ( num_of_nat @ zero_zero_nat )
    = one ) ).

% num_of_nat.simps(1)
thf(fact_9735_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
        = N ) ) ).

% numeral_num_of_nat
thf(fact_9736_num__of__nat__One,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ one_one_nat )
     => ( ( num_of_nat @ N )
        = one ) ) ).

% num_of_nat_One
thf(fact_9737_pow_Osimps_I2_J,axiom,
    ! [X3: num,Y3: num] :
      ( ( pow @ X3 @ ( bit0 @ Y3 ) )
      = ( sqr @ ( pow @ X3 @ Y3 ) ) ) ).

% pow.simps(2)
thf(fact_9738_num__of__nat__double,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
        = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).

% num_of_nat_double
thf(fact_9739_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
          = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).

% num_of_nat_plus_distrib
thf(fact_9740_sqr_Osimps_I3_J,axiom,
    ! [N: num] :
      ( ( sqr @ ( bit1 @ N ) )
      = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).

% sqr.simps(3)
thf(fact_9741_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_valid @ X3 @ Xa2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X3
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Xa2 = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X3
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( ( Deg2 = Xa2 )
                & ! [X5: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                   => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                & ( case_o184042715313410164at_nat
                  @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
                    & ! [X: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
                  @ ( produc6081775807080527818_nat_o
                    @ ^ [Mi3: nat,Ma3: nat] :
                        ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                        & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                        & ! [I4: nat] :
                            ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                           => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X7 ) )
                              = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                        & ( ( Mi3 = Ma3 )
                         => ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
                        & ( ( Mi3 != Ma3 )
                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                            & ! [X: nat] :
                                ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                 => ( ( ord_less_nat @ Mi3 @ X )
                                    & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                  @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
thf(fact_9742_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X3
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Xa2 != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X3
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
             => ~ ( ( Deg2 = Xa2 )
                  & ! [X6: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                     => ( vEBT_VEBT_valid @ X6 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( case_o184042715313410164at_nat
                    @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
                      & ! [X: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
                    @ ( produc6081775807080527818_nat_o
                      @ ^ [Mi3: nat,Ma3: nat] :
                          ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                          & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                          & ! [I4: nat] :
                              ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                             => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X7 ) )
                                = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                          & ( ( Mi3 = Ma3 )
                           => ! [X: vEBT_VEBT] :
                                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                               => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
                          & ( ( Mi3 != Ma3 )
                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                              & ! [X: nat] :
                                  ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                   => ( ( ord_less_nat @ Mi3 @ X )
                                      & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                    @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
thf(fact_9743_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
      ( ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
        = Y3 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X3
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Y3
            = ( Xa2 != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X3
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( Y3
                = ( ~ ( ( Deg2 = Xa2 )
                      & ! [X: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I4: nat] :
                                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X7 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X: nat] :
                                      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                       => ( ( ord_less_nat @ Mi3 @ X )
                                          & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
thf(fact_9744_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg4 )
      = ( ( Deg = Deg4 )
        & ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
           => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( case_o184042715313410164at_nat
          @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X7 )
            & ! [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
               => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
          @ ( produc6081775807080527818_nat_o
            @ ^ [Mi3: nat,Ma3: nat] :
                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                & ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                   => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X7 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
                & ( ( Mi3 = Ma3 )
                 => ! [X: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
                & ( ( Mi3 != Ma3 )
                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                    & ! [X: nat] :
                        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
                         => ( ( ord_less_nat @ Mi3 @ X )
                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
          @ Mima2 ) ) ) ).

% VEBT_internal.valid'.simps(2)
thf(fact_9745_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_valid @ X3 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X3
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
               => ( Xa2 = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
                 => ( ( Deg2 = Xa2 )
                    & ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                    & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                    & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( case_o184042715313410164at_nat
                      @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
                        & ! [X: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                           => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
                      @ ( produc6081775807080527818_nat_o
                        @ ^ [Mi3: nat,Ma3: nat] :
                            ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                            & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                            & ! [I4: nat] :
                                ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                               => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X7 ) )
                                  = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                            & ( ( Mi3 = Ma3 )
                             => ! [X: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                 => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
                            & ( ( Mi3 != Ma3 )
                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                & ! [X: nat] :
                                    ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                     => ( ( ord_less_nat @ Mi3 @ X )
                                        & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                      @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
thf(fact_9746_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X3
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
               => ( Xa2 != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
                 => ~ ( ( Deg2 = Xa2 )
                      & ! [X6: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_VEBT_valid @ X6 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I4: nat] :
                                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X7 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X: nat] :
                                      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                       => ( ( ord_less_nat @ Mi3 @ X )
                                          & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
thf(fact_9747_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
      ( ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
        = Y3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X3
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Y3
                  = ( Xa2 = one_one_nat ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( Y3
                    = ( ( Deg2 = Xa2 )
                      & ! [X: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I4: nat] :
                                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X7 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X: nat] :
                                      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                       => ( ( ord_less_nat @ Mi3 @ X )
                                          & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
thf(fact_9748_Sup__int__def,axiom,
    ( complete_Sup_Sup_int
    = ( ^ [X7: set_int] :
          ( the_int
          @ ^ [X: int] :
              ( ( member_int @ X @ X7 )
              & ! [Y: int] :
                  ( ( member_int @ Y @ X7 )
                 => ( ord_less_eq_int @ Y @ X ) ) ) ) ) ) ).

% Sup_int_def
thf(fact_9749_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( case_option_int_num @ zero_zero_int
        @ ^ [Q4: num] : ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_int @ Q4 ) ) )
        @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N ) ) ) ).

% take_bit_numeral_minus_numeral_int
thf(fact_9750_and__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).

% and_minus_numerals(3)
thf(fact_9751_take__bit__num__simps_I1_J,axiom,
    ! [M: num] :
      ( ( bit_take_bit_num @ zero_zero_nat @ M )
      = none_num ) ).

% take_bit_num_simps(1)
thf(fact_9752_take__bit__num__simps_I2_J,axiom,
    ! [N: nat] :
      ( ( bit_take_bit_num @ ( suc @ N ) @ one )
      = ( some_num @ one ) ) ).

% take_bit_num_simps(2)
thf(fact_9753_take__bit__num__simps_I5_J,axiom,
    ! [R: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ one )
      = ( some_num @ one ) ) ).

% take_bit_num_simps(5)
thf(fact_9754_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
        @ ( bit_take_bit_num @ N @ M ) ) ) ).

% take_bit_num_simps(3)
thf(fact_9755_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).

% take_bit_num_simps(4)
thf(fact_9756_take__bit__num__simps_I6_J,axiom,
    ! [R: num,M: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
        @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ).

% take_bit_num_simps(6)
thf(fact_9757_take__bit__num__simps_I7_J,axiom,
    ! [R: num,M: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ ( bit1 @ M ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ) ).

% take_bit_num_simps(7)
thf(fact_9758_and__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).

% and_minus_numerals(4)
thf(fact_9759_and__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).

% and_minus_numerals(8)
thf(fact_9760_and__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).

% and_minus_numerals(7)
thf(fact_9761_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [N: nat,M: num] :
      ( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
      = ( case_nat_option_num @ none_num
        @ ^ [N2: nat] :
            ( case_o6005452278849405969um_num @ none_num
            @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
            @ ( bit_take_bit_num @ N2 @ M ) )
        @ N ) ) ).

% Code_Abstract_Nat.take_bit_num_code(2)
thf(fact_9762_and__not__num_Osimps_I1_J,axiom,
    ( ( bit_and_not_num @ one @ one )
    = none_num ) ).

% and_not_num.simps(1)
thf(fact_9763_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [N: nat] :
      ( ( bit_take_bit_num @ N @ one )
      = ( case_nat_option_num @ none_num
        @ ^ [N2: nat] : ( some_num @ one )
        @ N ) ) ).

% Code_Abstract_Nat.take_bit_num_code(1)
thf(fact_9764_and__not__num_Osimps_I4_J,axiom,
    ! [M: num] :
      ( ( bit_and_not_num @ ( bit0 @ M ) @ one )
      = ( some_num @ ( bit0 @ M ) ) ) ).

% and_not_num.simps(4)
thf(fact_9765_and__not__num_Osimps_I2_J,axiom,
    ! [N: num] :
      ( ( bit_and_not_num @ one @ ( bit0 @ N ) )
      = ( some_num @ one ) ) ).

% and_not_num.simps(2)
thf(fact_9766_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_9767_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_9768_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_9769_and__not__num_Osimps_I3_J,axiom,
    ! [N: num] :
      ( ( bit_and_not_num @ one @ ( bit1 @ N ) )
      = none_num ) ).

% and_not_num.simps(3)
thf(fact_9770_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [N: nat,M: num] :
      ( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
      = ( case_nat_option_num @ none_num
        @ ^ [N2: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) )
        @ N ) ) ).

% Code_Abstract_Nat.take_bit_num_code(3)
thf(fact_9771_and__not__num_Osimps_I7_J,axiom,
    ! [M: num] :
      ( ( bit_and_not_num @ ( bit1 @ M ) @ one )
      = ( some_num @ ( bit0 @ M ) ) ) ).

% and_not_num.simps(7)
thf(fact_9772_and__not__num__eq__Some__iff,axiom,
    ! [M: num,N: num,Q2: num] :
      ( ( ( bit_and_not_num @ M @ N )
        = ( some_num @ Q2 ) )
      = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
        = ( numeral_numeral_int @ Q2 ) ) ) ).

% and_not_num_eq_Some_iff
thf(fact_9773_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( case_o6005452278849405969um_num @ ( some_num @ one )
        @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
        @ ( bit_and_not_num @ M @ N ) ) ) ).

% and_not_num.simps(8)
thf(fact_9774_and__not__num__eq__None__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( bit_and_not_num @ M @ N )
        = none_num )
      = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
        = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
thf(fact_9775_int__numeral__not__and__num,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N @ M ) ) ) ).

% int_numeral_not_and_num
thf(fact_9776_int__numeral__and__not__num,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N ) ) ) ).

% int_numeral_and_not_num
thf(fact_9777_take__bit__num__def,axiom,
    ( bit_take_bit_num
    = ( ^ [N2: nat,M3: num] :
          ( if_option_num
          @ ( ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M3 ) )
            = zero_zero_nat )
          @ none_num
          @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M3 ) ) ) ) ) ) ) ).

% take_bit_num_def
thf(fact_9778_Bit__Operations_Otake__bit__num__code,axiom,
    ( bit_take_bit_num
    = ( ^ [N2: nat,M3: num] :
          ( produc478579273971653890on_num
          @ ^ [A5: nat,X: num] :
              ( case_nat_option_num @ none_num
              @ ^ [O: nat] :
                  ( case_num_option_num @ ( some_num @ one )
                  @ ^ [P5: num] :
                      ( case_o6005452278849405969um_num @ none_num
                      @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
                      @ ( bit_take_bit_num @ O @ P5 ) )
                  @ ^ [P5: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P5 ) ) )
                  @ X )
              @ A5 )
          @ ( product_Pair_nat_num @ N2 @ M3 ) ) ) ) ).

% Bit_Operations.take_bit_num_code
thf(fact_9779_integer__of__num__triv_I2_J,axiom,
    ( ( code_integer_of_num @ ( bit0 @ one ) )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% integer_of_num_triv(2)
thf(fact_9780_integer__of__num__def,axiom,
    code_integer_of_num = numera6620942414471956472nteger ).

% integer_of_num_def
thf(fact_9781_integer__of__num__triv_I1_J,axiom,
    ( ( code_integer_of_num @ one )
    = one_one_Code_integer ) ).

% integer_of_num_triv(1)
thf(fact_9782_integer__of__num_I2_J,axiom,
    ! [N: num] :
      ( ( code_integer_of_num @ ( bit0 @ N ) )
      = ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).

% integer_of_num(2)
thf(fact_9783_and__not__num_Oelims,axiom,
    ! [X3: num,Xa2: num,Y3: option_num] :
      ( ( ( bit_and_not_num @ X3 @ Xa2 )
        = Y3 )
     => ( ( ( X3 = one )
         => ( ( Xa2 = one )
           => ( Y3 != none_num ) ) )
       => ( ( ( X3 = one )
           => ( ? [N3: num] :
                  ( Xa2
                  = ( bit0 @ N3 ) )
             => ( Y3
               != ( some_num @ one ) ) ) )
         => ( ( ( X3 = one )
             => ( ? [N3: num] :
                    ( Xa2
                    = ( bit1 @ N3 ) )
               => ( Y3 != none_num ) ) )
           => ( ! [M4: num] :
                  ( ( X3
                    = ( bit0 @ M4 ) )
                 => ( ( Xa2 = one )
                   => ( Y3
                     != ( some_num @ ( bit0 @ M4 ) ) ) ) )
             => ( ! [M4: num] :
                    ( ( X3
                      = ( bit0 @ M4 ) )
                   => ! [N3: num] :
                        ( ( Xa2
                          = ( bit0 @ N3 ) )
                       => ( Y3
                         != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) ) ) )
               => ( ! [M4: num] :
                      ( ( X3
                        = ( bit0 @ M4 ) )
                     => ! [N3: num] :
                          ( ( Xa2
                            = ( bit1 @ N3 ) )
                         => ( Y3
                           != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) ) ) )
                 => ( ! [M4: num] :
                        ( ( X3
                          = ( bit1 @ M4 ) )
                       => ( ( Xa2 = one )
                         => ( Y3
                           != ( some_num @ ( bit0 @ M4 ) ) ) ) )
                   => ( ! [M4: num] :
                          ( ( X3
                            = ( bit1 @ M4 ) )
                         => ! [N3: num] :
                              ( ( Xa2
                                = ( bit0 @ N3 ) )
                             => ( Y3
                               != ( case_o6005452278849405969um_num @ ( some_num @ one )
                                  @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
                                  @ ( bit_and_not_num @ M4 @ N3 ) ) ) ) )
                     => ~ ! [M4: num] :
                            ( ( X3
                              = ( bit1 @ M4 ) )
                           => ! [N3: num] :
                                ( ( Xa2
                                  = ( bit1 @ N3 ) )
                               => ( Y3
                                 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
thf(fact_9784_xor__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).

% xor_num.simps(6)
thf(fact_9785_xor__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).

% xor_num.simps(8)
thf(fact_9786_xor__num_Osimps_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).

% xor_num.simps(5)
thf(fact_9787_xor__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).

% xor_num.simps(9)
thf(fact_9788_and__not__num_Osimps_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).

% and_not_num.simps(5)
thf(fact_9789_and__not__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).

% and_not_num.simps(6)
thf(fact_9790_and__not__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).

% and_not_num.simps(9)
thf(fact_9791_xor__num_Osimps_I1_J,axiom,
    ( ( bit_un2480387367778600638or_num @ one @ one )
    = none_num ) ).

% xor_num.simps(1)
thf(fact_9792_xor__num_Oelims,axiom,
    ! [X3: num,Xa2: num,Y3: option_num] :
      ( ( ( bit_un2480387367778600638or_num @ X3 @ Xa2 )
        = Y3 )
     => ( ( ( X3 = one )
         => ( ( Xa2 = one )
           => ( Y3 != none_num ) ) )
       => ( ( ( X3 = one )
           => ! [N3: num] :
                ( ( Xa2
                  = ( bit0 @ N3 ) )
               => ( Y3
                 != ( some_num @ ( bit1 @ N3 ) ) ) ) )
         => ( ( ( X3 = one )
             => ! [N3: num] :
                  ( ( Xa2
                    = ( bit1 @ N3 ) )
                 => ( Y3
                   != ( some_num @ ( bit0 @ N3 ) ) ) ) )
           => ( ! [M4: num] :
                  ( ( X3
                    = ( bit0 @ M4 ) )
                 => ( ( Xa2 = one )
                   => ( Y3
                     != ( some_num @ ( bit1 @ M4 ) ) ) ) )
             => ( ! [M4: num] :
                    ( ( X3
                      = ( bit0 @ M4 ) )
                   => ! [N3: num] :
                        ( ( Xa2
                          = ( bit0 @ N3 ) )
                       => ( Y3
                         != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) ) ) )
               => ( ! [M4: num] :
                      ( ( X3
                        = ( bit0 @ M4 ) )
                     => ! [N3: num] :
                          ( ( Xa2
                            = ( bit1 @ N3 ) )
                         => ( Y3
                           != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) ) ) ) )
                 => ( ! [M4: num] :
                        ( ( X3
                          = ( bit1 @ M4 ) )
                       => ( ( Xa2 = one )
                         => ( Y3
                           != ( some_num @ ( bit0 @ M4 ) ) ) ) )
                   => ( ! [M4: num] :
                          ( ( X3
                            = ( bit1 @ M4 ) )
                         => ! [N3: num] :
                              ( ( Xa2
                                = ( bit0 @ N3 ) )
                             => ( Y3
                               != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) ) ) ) )
                     => ~ ! [M4: num] :
                            ( ( X3
                              = ( bit1 @ M4 ) )
                           => ! [N3: num] :
                                ( ( Xa2
                                  = ( bit1 @ N3 ) )
                               => ( Y3
                                 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
thf(fact_9793_xor__num_Osimps_I7_J,axiom,
    ! [M: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
      = ( some_num @ ( bit0 @ M ) ) ) ).

% xor_num.simps(7)
thf(fact_9794_xor__num_Osimps_I4_J,axiom,
    ! [M: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
      = ( some_num @ ( bit1 @ M ) ) ) ).

% xor_num.simps(4)
thf(fact_9795_xor__num_Osimps_I3_J,axiom,
    ! [N: num] :
      ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N ) )
      = ( some_num @ ( bit0 @ N ) ) ) ).

% xor_num.simps(3)
thf(fact_9796_xor__num_Osimps_I2_J,axiom,
    ! [N: num] :
      ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N ) )
      = ( some_num @ ( bit1 @ N ) ) ) ).

% xor_num.simps(2)
thf(fact_9797_and__not__num_Opelims,axiom,
    ! [X3: num,Xa2: num,Y3: option_num] :
      ( ( ( bit_and_not_num @ X3 @ Xa2 )
        = Y3 )
     => ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X3 @ Xa2 ) )
       => ( ( ( X3 = one )
           => ( ( Xa2 = one )
             => ( ( Y3 = none_num )
               => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X3 = one )
             => ! [N3: num] :
                  ( ( Xa2
                    = ( bit0 @ N3 ) )
                 => ( ( Y3
                      = ( some_num @ one ) )
                   => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
           => ( ( ( X3 = one )
               => ! [N3: num] :
                    ( ( Xa2
                      = ( bit1 @ N3 ) )
                   => ( ( Y3 = none_num )
                     => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
             => ( ! [M4: num] :
                    ( ( X3
                      = ( bit0 @ M4 ) )
                   => ( ( Xa2 = one )
                     => ( ( Y3
                          = ( some_num @ ( bit0 @ M4 ) ) )
                       => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
               => ( ! [M4: num] :
                      ( ( X3
                        = ( bit0 @ M4 ) )
                     => ! [N3: num] :
                          ( ( Xa2
                            = ( bit0 @ N3 ) )
                         => ( ( Y3
                              = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N3 ) ) ) ) ) )
                 => ( ! [M4: num] :
                        ( ( X3
                          = ( bit0 @ M4 ) )
                       => ! [N3: num] :
                            ( ( Xa2
                              = ( bit1 @ N3 ) )
                           => ( ( Y3
                                = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) )
                   => ( ! [M4: num] :
                          ( ( X3
                            = ( bit1 @ M4 ) )
                         => ( ( Xa2 = one )
                           => ( ( Y3
                                = ( some_num @ ( bit0 @ M4 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
                     => ( ! [M4: num] :
                            ( ( X3
                              = ( bit1 @ M4 ) )
                           => ! [N3: num] :
                                ( ( Xa2
                                  = ( bit0 @ N3 ) )
                               => ( ( Y3
                                    = ( case_o6005452278849405969um_num @ ( some_num @ one )
                                      @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
                                      @ ( bit_and_not_num @ M4 @ N3 ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N3 ) ) ) ) ) )
                       => ~ ! [M4: num] :
                              ( ( X3
                                = ( bit1 @ M4 ) )
                             => ! [N3: num] :
                                  ( ( Xa2
                                    = ( bit1 @ N3 ) )
                                 => ( ( Y3
                                      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.pelims
thf(fact_9798_xor__num_Opelims,axiom,
    ! [X3: num,Xa2: num,Y3: option_num] :
      ( ( ( bit_un2480387367778600638or_num @ X3 @ Xa2 )
        = Y3 )
     => ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X3 @ Xa2 ) )
       => ( ( ( X3 = one )
           => ( ( Xa2 = one )
             => ( ( Y3 = none_num )
               => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X3 = one )
             => ! [N3: num] :
                  ( ( Xa2
                    = ( bit0 @ N3 ) )
                 => ( ( Y3
                      = ( some_num @ ( bit1 @ N3 ) ) )
                   => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
           => ( ( ( X3 = one )
               => ! [N3: num] :
                    ( ( Xa2
                      = ( bit1 @ N3 ) )
                   => ( ( Y3
                        = ( some_num @ ( bit0 @ N3 ) ) )
                     => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
             => ( ! [M4: num] :
                    ( ( X3
                      = ( bit0 @ M4 ) )
                   => ( ( Xa2 = one )
                     => ( ( Y3
                          = ( some_num @ ( bit1 @ M4 ) ) )
                       => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
               => ( ! [M4: num] :
                      ( ( X3
                        = ( bit0 @ M4 ) )
                     => ! [N3: num] :
                          ( ( Xa2
                            = ( bit0 @ N3 ) )
                         => ( ( Y3
                              = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N3 ) ) ) ) ) )
                 => ( ! [M4: num] :
                        ( ( X3
                          = ( bit0 @ M4 ) )
                       => ! [N3: num] :
                            ( ( Xa2
                              = ( bit1 @ N3 ) )
                           => ( ( Y3
                                = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) )
                   => ( ! [M4: num] :
                          ( ( X3
                            = ( bit1 @ M4 ) )
                         => ( ( Xa2 = one )
                           => ( ( Y3
                                = ( some_num @ ( bit0 @ M4 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
                     => ( ! [M4: num] :
                            ( ( X3
                              = ( bit1 @ M4 ) )
                           => ! [N3: num] :
                                ( ( Xa2
                                  = ( bit0 @ N3 ) )
                               => ( ( Y3
                                    = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N3 ) ) ) ) ) )
                       => ~ ! [M4: num] :
                              ( ( X3
                                = ( bit1 @ M4 ) )
                             => ! [N3: num] :
                                  ( ( Xa2
                                    = ( bit1 @ N3 ) )
                                 => ( ( Y3
                                      = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.pelims
thf(fact_9799_and__num_Oelims,axiom,
    ! [X3: num,Xa2: num,Y3: option_num] :
      ( ( ( bit_un7362597486090784418nd_num @ X3 @ Xa2 )
        = Y3 )
     => ( ( ( X3 = one )
         => ( ( Xa2 = one )
           => ( Y3
             != ( some_num @ one ) ) ) )
       => ( ( ( X3 = one )
           => ( ? [N3: num] :
                  ( Xa2
                  = ( bit0 @ N3 ) )
             => ( Y3 != none_num ) ) )
         => ( ( ( X3 = one )
             => ( ? [N3: num] :
                    ( Xa2
                    = ( bit1 @ N3 ) )
               => ( Y3
                 != ( some_num @ one ) ) ) )
           => ( ( ? [M4: num] :
                    ( X3
                    = ( bit0 @ M4 ) )
               => ( ( Xa2 = one )
                 => ( Y3 != none_num ) ) )
             => ( ! [M4: num] :
                    ( ( X3
                      = ( bit0 @ M4 ) )
                   => ! [N3: num] :
                        ( ( Xa2
                          = ( bit0 @ N3 ) )
                       => ( Y3
                         != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) ) ) )
               => ( ! [M4: num] :
                      ( ( X3
                        = ( bit0 @ M4 ) )
                     => ! [N3: num] :
                          ( ( Xa2
                            = ( bit1 @ N3 ) )
                         => ( Y3
                           != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) ) ) )
                 => ( ( ? [M4: num] :
                          ( X3
                          = ( bit1 @ M4 ) )
                     => ( ( Xa2 = one )
                       => ( Y3
                         != ( some_num @ one ) ) ) )
                   => ( ! [M4: num] :
                          ( ( X3
                            = ( bit1 @ M4 ) )
                         => ! [N3: num] :
                              ( ( Xa2
                                = ( bit0 @ N3 ) )
                             => ( Y3
                               != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) ) ) )
                     => ~ ! [M4: num] :
                            ( ( X3
                              = ( bit1 @ M4 ) )
                           => ! [N3: num] :
                                ( ( Xa2
                                  = ( bit1 @ N3 ) )
                               => ( Y3
                                 != ( case_o6005452278849405969um_num @ ( some_num @ one )
                                    @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
                                    @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
thf(fact_9800_xor__num__rel__dict,axiom,
    bit_un2901131394128224187um_rel = bit_un3595099601533988841um_rel ).

% xor_num_rel_dict
thf(fact_9801_xor__num__dict,axiom,
    bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).

% xor_num_dict
thf(fact_9802_and__num_Osimps_I1_J,axiom,
    ( ( bit_un7362597486090784418nd_num @ one @ one )
    = ( some_num @ one ) ) ).

% and_num.simps(1)
thf(fact_9803_and__num_Osimps_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).

% and_num.simps(5)
thf(fact_9804_and__num_Osimps_I3_J,axiom,
    ! [N: num] :
      ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N ) )
      = ( some_num @ one ) ) ).

% and_num.simps(3)
thf(fact_9805_and__num_Osimps_I7_J,axiom,
    ! [M: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
      = ( some_num @ one ) ) ).

% and_num.simps(7)
thf(fact_9806_and__num_Osimps_I4_J,axiom,
    ! [M: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
      = none_num ) ).

% and_num.simps(4)
thf(fact_9807_and__num_Osimps_I2_J,axiom,
    ! [N: num] :
      ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N ) )
      = none_num ) ).

% and_num.simps(2)
thf(fact_9808_and__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).

% and_num.simps(6)
thf(fact_9809_and__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).

% and_num.simps(8)
thf(fact_9810_and__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( case_o6005452278849405969um_num @ ( some_num @ one )
        @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
        @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).

% and_num.simps(9)
thf(fact_9811_and__num_Opelims,axiom,
    ! [X3: num,Xa2: num,Y3: option_num] :
      ( ( ( bit_un7362597486090784418nd_num @ X3 @ Xa2 )
        = Y3 )
     => ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X3 @ Xa2 ) )
       => ( ( ( X3 = one )
           => ( ( Xa2 = one )
             => ( ( Y3
                  = ( some_num @ one ) )
               => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X3 = one )
             => ! [N3: num] :
                  ( ( Xa2
                    = ( bit0 @ N3 ) )
                 => ( ( Y3 = none_num )
                   => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
           => ( ( ( X3 = one )
               => ! [N3: num] :
                    ( ( Xa2
                      = ( bit1 @ N3 ) )
                   => ( ( Y3
                        = ( some_num @ one ) )
                     => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
             => ( ! [M4: num] :
                    ( ( X3
                      = ( bit0 @ M4 ) )
                   => ( ( Xa2 = one )
                     => ( ( Y3 = none_num )
                       => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
               => ( ! [M4: num] :
                      ( ( X3
                        = ( bit0 @ M4 ) )
                     => ! [N3: num] :
                          ( ( Xa2
                            = ( bit0 @ N3 ) )
                         => ( ( Y3
                              = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N3 ) ) ) ) ) )
                 => ( ! [M4: num] :
                        ( ( X3
                          = ( bit0 @ M4 ) )
                       => ! [N3: num] :
                            ( ( Xa2
                              = ( bit1 @ N3 ) )
                           => ( ( Y3
                                = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) )
                   => ( ! [M4: num] :
                          ( ( X3
                            = ( bit1 @ M4 ) )
                         => ( ( Xa2 = one )
                           => ( ( Y3
                                = ( some_num @ one ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
                     => ( ! [M4: num] :
                            ( ( X3
                              = ( bit1 @ M4 ) )
                           => ! [N3: num] :
                                ( ( Xa2
                                  = ( bit0 @ N3 ) )
                               => ( ( Y3
                                    = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N3 ) ) ) ) ) )
                       => ~ ! [M4: num] :
                              ( ( X3
                                = ( bit1 @ M4 ) )
                             => ! [N3: num] :
                                  ( ( Xa2
                                    = ( bit1 @ N3 ) )
                                 => ( ( Y3
                                      = ( case_o6005452278849405969um_num @ ( some_num @ one )
                                        @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
                                        @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
thf(fact_9812_and__num__dict,axiom,
    bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).

% and_num_dict
thf(fact_9813_and__num__rel__dict,axiom,
    bit_un4731106466462545111um_rel = bit_un5425074673868309765um_rel ).

% and_num_rel_dict
thf(fact_9814_Rats__eq__int__div__nat,axiom,
    ( field_5140801741446780682s_real
    = ( collect_real
      @ ^ [Uu3: real] :
        ? [I4: int,N2: nat] :
          ( ( Uu3
            = ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
          & ( N2 != zero_zero_nat ) ) ) ) ).

% Rats_eq_int_div_nat
thf(fact_9815_rat__floor__lemma,axiom,
    ! [A: int,B2: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B2 ) ) @ ( fract @ A @ B2 ) )
      & ( ord_less_rat @ ( fract @ A @ B2 ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ) ).

% rat_floor_lemma
thf(fact_9816_Rats__abs__iff,axiom,
    ! [X3: real] :
      ( ( member_real @ ( abs_abs_real @ X3 ) @ field_5140801741446780682s_real )
      = ( member_real @ X3 @ field_5140801741446780682s_real ) ) ).

% Rats_abs_iff
thf(fact_9817_mult__rat,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( times_times_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
      = ( fract @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ).

% mult_rat
thf(fact_9818_divide__rat,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( divide_divide_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
      = ( fract @ ( times_times_int @ A @ D ) @ ( times_times_int @ B2 @ C ) ) ) ).

% divide_rat
thf(fact_9819_less__rat,axiom,
    ! [B2: int,D: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ord_less_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
          = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B2 @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B2 ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% less_rat
thf(fact_9820_add__rat,axiom,
    ! [B2: int,D: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( plus_plus_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
          = ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B2 ) ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ).

% add_rat
thf(fact_9821_le__rat,axiom,
    ! [B2: int,D: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ord_less_eq_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
          = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B2 @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B2 ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).

% le_rat
thf(fact_9822_diff__rat,axiom,
    ! [B2: int,D: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( minus_minus_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
          = ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B2 ) ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ).

% diff_rat
thf(fact_9823_sgn__rat,axiom,
    ! [A: int,B2: int] :
      ( ( sgn_sgn_rat @ ( fract @ A @ B2 ) )
      = ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B2 ) ) ) ) ).

% sgn_rat
thf(fact_9824_Rats__no__top__le,axiom,
    ! [X3: real] :
    ? [X5: real] :
      ( ( member_real @ X5 @ field_5140801741446780682s_real )
      & ( ord_less_eq_real @ X3 @ X5 ) ) ).

% Rats_no_top_le
thf(fact_9825_Rats__no__bot__less,axiom,
    ! [X3: real] :
    ? [X5: real] :
      ( ( member_real @ X5 @ field_5140801741446780682s_real )
      & ( ord_less_real @ X5 @ X3 ) ) ).

% Rats_no_bot_less
thf(fact_9826_Rats__dense__in__real,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ? [X5: real] :
          ( ( member_real @ X5 @ field_5140801741446780682s_real )
          & ( ord_less_real @ X3 @ X5 )
          & ( ord_less_real @ X5 @ Y3 ) ) ) ).

% Rats_dense_in_real
thf(fact_9827_Rat__induct__pos,axiom,
    ! [P: rat > $o,Q2: rat] :
      ( ! [A3: int,B3: int] :
          ( ( ord_less_int @ zero_zero_int @ B3 )
         => ( P @ ( fract @ A3 @ B3 ) ) )
     => ( P @ Q2 ) ) ).

% Rat_induct_pos
thf(fact_9828_eq__rat_I1_J,axiom,
    ! [B2: int,D: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ( fract @ A @ B2 )
            = ( fract @ C @ D ) )
          = ( ( times_times_int @ A @ D )
            = ( times_times_int @ C @ B2 ) ) ) ) ) ).

% eq_rat(1)
thf(fact_9829_mult__rat__cancel,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( ( fract @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
        = ( fract @ A @ B2 ) ) ) ).

% mult_rat_cancel
thf(fact_9830_quotient__of__eq,axiom,
    ! [A: int,B2: int,P4: int,Q2: int] :
      ( ( ( quotient_of @ ( fract @ A @ B2 ) )
        = ( product_Pair_int_int @ P4 @ Q2 ) )
     => ( ( fract @ P4 @ Q2 )
        = ( fract @ A @ B2 ) ) ) ).

% quotient_of_eq
thf(fact_9831_normalize__eq,axiom,
    ! [A: int,B2: int,P4: int,Q2: int] :
      ( ( ( normalize @ ( product_Pair_int_int @ A @ B2 ) )
        = ( product_Pair_int_int @ P4 @ Q2 ) )
     => ( ( fract @ P4 @ Q2 )
        = ( fract @ A @ B2 ) ) ) ).

% normalize_eq
thf(fact_9832_rat__number__expand_I3_J,axiom,
    ( numeral_numeral_rat
    = ( ^ [K2: num] : ( fract @ ( numeral_numeral_int @ K2 ) @ one_one_int ) ) ) ).

% rat_number_expand(3)
thf(fact_9833_rat__number__collapse_I3_J,axiom,
    ! [W: num] :
      ( ( fract @ ( numeral_numeral_int @ W ) @ one_one_int )
      = ( numeral_numeral_rat @ W ) ) ).

% rat_number_collapse(3)
thf(fact_9834_quotient__of__Fract,axiom,
    ! [A: int,B2: int] :
      ( ( quotient_of @ ( fract @ A @ B2 ) )
      = ( normalize @ ( product_Pair_int_int @ A @ B2 ) ) ) ).

% quotient_of_Fract
thf(fact_9835_zero__less__Fract__iff,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B2 ) )
        = ( ord_less_int @ zero_zero_int @ A ) ) ) ).

% zero_less_Fract_iff
thf(fact_9836_Fract__less__zero__iff,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_rat @ ( fract @ A @ B2 ) @ zero_zero_rat )
        = ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% Fract_less_zero_iff
thf(fact_9837_one__less__Fract__iff,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B2 ) )
        = ( ord_less_int @ B2 @ A ) ) ) ).

% one_less_Fract_iff
thf(fact_9838_Fract__less__one__iff,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_rat @ ( fract @ A @ B2 ) @ one_one_rat )
        = ( ord_less_int @ A @ B2 ) ) ) ).

% Fract_less_one_iff
thf(fact_9839_Rats__eq__int__div__int,axiom,
    ( field_5140801741446780682s_real
    = ( collect_real
      @ ^ [Uu3: real] :
        ? [I4: int,J3: int] :
          ( ( Uu3
            = ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( ring_1_of_int_real @ J3 ) ) )
          & ( J3 != zero_zero_int ) ) ) ) ).

% Rats_eq_int_div_int
thf(fact_9840_Fract__le__zero__iff,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_eq_rat @ ( fract @ A @ B2 ) @ zero_zero_rat )
        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).

% Fract_le_zero_iff
thf(fact_9841_zero__le__Fract__iff,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B2 ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% zero_le_Fract_iff
thf(fact_9842_Fract__le__one__iff,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_eq_rat @ ( fract @ A @ B2 ) @ one_one_rat )
        = ( ord_less_eq_int @ A @ B2 ) ) ) ).

% Fract_le_one_iff
thf(fact_9843_one__le__Fract__iff,axiom,
    ! [B2: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B2 ) )
        = ( ord_less_eq_int @ B2 @ A ) ) ) ).

% one_le_Fract_iff
thf(fact_9844_rat__number__expand_I5_J,axiom,
    ! [K: num] :
      ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) )
      = ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).

% rat_number_expand(5)
thf(fact_9845_rat__number__collapse_I4_J,axiom,
    ! [W: num] :
      ( ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ one_one_int )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ).

% rat_number_collapse(4)
thf(fact_9846_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y3: nat,X3: nat] :
      ( ( ( ord_less_nat @ C @ Y3 )
       => ( ( image_nat_nat
            @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
            @ ( set_or4665077453230672383an_nat @ X3 @ Y3 ) )
          = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X3 @ C ) @ ( minus_minus_nat @ Y3 @ C ) ) ) )
      & ( ~ ( ord_less_nat @ C @ Y3 )
       => ( ( ( ord_less_nat @ X3 @ Y3 )
           => ( ( image_nat_nat
                @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
                @ ( set_or4665077453230672383an_nat @ X3 @ Y3 ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X3 @ Y3 )
           => ( ( image_nat_nat
                @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
                @ ( set_or4665077453230672383an_nat @ X3 @ Y3 ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_9847_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
thf(fact_9848_bij__betw__Suc,axiom,
    ! [M7: set_nat,N4: set_nat] :
      ( ( bij_betw_nat_nat @ suc @ M7 @ N4 )
      = ( ( image_nat_nat @ suc @ M7 )
        = N4 ) ) ).

% bij_betw_Suc
thf(fact_9849_less__int__def,axiom,
    ( ord_less_int
    = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
      @ ( produc8739625826339149834_nat_o
        @ ^ [X: nat,Y: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ) ).

% less_int_def
thf(fact_9850_less__eq__int__def,axiom,
    ( ord_less_eq_int
    = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
      @ ( produc8739625826339149834_nat_o
        @ ^ [X: nat,Y: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ) ).

% less_eq_int_def
thf(fact_9851_nat__def,axiom,
    ( nat2
    = ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).

% nat_def
thf(fact_9852_positive__rat,axiom,
    ! [A: int,B2: int] :
      ( ( positive @ ( fract @ A @ B2 ) )
      = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ).

% positive_rat
thf(fact_9853_Inf__real__def,axiom,
    ( comple4887499456419720421f_real
    = ( ^ [X7: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X7 ) ) ) ) ) ).

% Inf_real_def
thf(fact_9854_Rat_Opositive__mult,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( positive @ X3 )
     => ( ( positive @ Y3 )
       => ( positive @ ( times_times_rat @ X3 @ Y3 ) ) ) ) ).

% Rat.positive_mult
thf(fact_9855_finite__int__iff__bounded__le,axiom,
    ( finite_finite_int
    = ( ^ [S4: set_int] :
        ? [K2: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S4 ) @ ( set_ord_atMost_int @ K2 ) ) ) ) ).

% finite_int_iff_bounded_le
thf(fact_9856_finite__int__iff__bounded,axiom,
    ( finite_finite_int
    = ( ^ [S4: set_int] :
        ? [K2: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S4 ) @ ( set_ord_lessThan_int @ K2 ) ) ) ) ).

% finite_int_iff_bounded
thf(fact_9857_suminf__eq__SUP__real,axiom,
    ! [X9: nat > real] :
      ( ( summable_real @ X9 )
     => ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X9 @ I3 ) )
       => ( ( suminf_real @ X9 )
          = ( comple1385675409528146559p_real
            @ ( image_nat_real
              @ ^ [I4: nat] : ( groups6591440286371151544t_real @ X9 @ ( set_ord_lessThan_nat @ I4 ) )
              @ top_top_set_nat ) ) ) ) ) ).

% suminf_eq_SUP_real
thf(fact_9858_less__rat__def,axiom,
    ( ord_less_rat
    = ( ^ [X: rat,Y: rat] : ( positive @ ( minus_minus_rat @ Y @ X ) ) ) ) ).

% less_rat_def
thf(fact_9859_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ U )
     => ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
        = ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).

% image_atLeastZeroLessThan_int
thf(fact_9860_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_9861_Rat_Opositive_Orep__eq,axiom,
    ( positive
    = ( ^ [X: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X ) ) @ ( product_snd_int_int @ ( rep_Rat @ X ) ) ) ) ) ) ).

% Rat.positive.rep_eq
thf(fact_9862_card__UNIV__bool,axiom,
    ( ( finite_card_o @ top_top_set_o )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% card_UNIV_bool
thf(fact_9863_range__mult,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
          = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
      & ( ( A != zero_zero_real )
       => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
          = top_top_set_real ) ) ) ).

% range_mult
thf(fact_9864_Rat_Opositive__def,axiom,
    ( positive
    = ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
      @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).

% Rat.positive_def
thf(fact_9865_root__def,axiom,
    ( root
    = ( ^ [N2: nat,X: real] :
          ( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
          @ ( the_in5290026491893676941l_real @ top_top_set_real
            @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) )
            @ X ) ) ) ) ).

% root_def
thf(fact_9866_card__UNIV__char,axiom,
    ( ( finite_card_char @ top_top_set_char )
    = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% card_UNIV_char
thf(fact_9867_UNIV__char__of__nat,axiom,
    ( top_top_set_char
    = ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% UNIV_char_of_nat
thf(fact_9868_nat__of__char__less__256,axiom,
    ! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% nat_of_char_less_256
thf(fact_9869_range__nat__of__char,axiom,
    ( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
    = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% range_nat_of_char
thf(fact_9870_integer__of__char__code,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
      ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ B72 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B62 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B52 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B42 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B32 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B22 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B1 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B0 ) ) ) ).

% integer_of_char_code
thf(fact_9871_of__rat__dense,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ? [Q3: rat] :
          ( ( ord_less_real @ X3 @ ( field_7254667332652039916t_real @ Q3 ) )
          & ( ord_less_real @ ( field_7254667332652039916t_real @ Q3 ) @ Y3 ) ) ) ).

% of_rat_dense
thf(fact_9872_plus__rat__def,axiom,
    ( plus_plus_rat
    = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
      @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ) ).

% plus_rat_def
thf(fact_9873_inverse__rat__def,axiom,
    ( inverse_inverse_rat
    = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
      @ ^ [X: product_prod_int_int] :
          ( if_Pro3027730157355071871nt_int
          @ ( ( product_fst_int_int @ X )
            = zero_zero_int )
          @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
          @ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) ) ) ) ).

% inverse_rat_def
thf(fact_9874_String_Ochar__of__ascii__of,axiom,
    ! [C: char] :
      ( ( comm_s629917340098488124ar_nat @ ( ascii_of @ C ) )
      = ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( comm_s629917340098488124ar_nat @ C ) ) ) ).

% String.char_of_ascii_of
thf(fact_9875_one__rat__def,axiom,
    ( one_one_rat
    = ( abs_Rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ) ).

% one_rat_def
thf(fact_9876_Fract_Oabs__eq,axiom,
    ( fract
    = ( ^ [Xa4: int,X: int] : ( abs_Rat @ ( if_Pro3027730157355071871nt_int @ ( X = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ Xa4 @ X ) ) ) ) ) ).

% Fract.abs_eq
thf(fact_9877_zero__rat__def,axiom,
    ( zero_zero_rat
    = ( abs_Rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ) ).

% zero_rat_def
thf(fact_9878_uminus__rat__def,axiom,
    ( uminus_uminus_rat
    = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
      @ ^ [X: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X ) ) @ ( product_snd_int_int @ X ) ) ) ) ).

% uminus_rat_def
thf(fact_9879_times__rat__def,axiom,
    ( times_times_rat
    = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
      @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ) ).

% times_rat_def
thf(fact_9880_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_9881_upto_Opelims,axiom,
    ! [X3: int,Xa2: int,Y3: list_int] :
      ( ( ( upto @ X3 @ Xa2 )
        = Y3 )
     => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) )
       => ~ ( ( ( ( ord_less_eq_int @ X3 @ Xa2 )
               => ( Y3
                  = ( cons_int @ X3 @ ( upto @ ( plus_plus_int @ X3 @ one_one_int ) @ Xa2 ) ) ) )
              & ( ~ ( ord_less_eq_int @ X3 @ Xa2 )
               => ( Y3 = nil_int ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) ) ) ) ) ).

% upto.pelims
thf(fact_9882_upto__empty,axiom,
    ! [J: int,I: int] :
      ( ( ord_less_int @ J @ I )
     => ( ( upto @ I @ J )
        = nil_int ) ) ).

% upto_empty
thf(fact_9883_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int
        = ( upto @ I @ J ) )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil2
thf(fact_9884_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( ( upto @ I @ J )
        = nil_int )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil
thf(fact_9885_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
     => ( ( nth_int @ ( upto @ I @ J ) @ K )
        = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).

% nth_upto
thf(fact_9886_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
          = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(1)
thf(fact_9887_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(4)
thf(fact_9888_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(3)
thf(fact_9889_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(2)
thf(fact_9890_atLeastAtMost__upto,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ I4 @ J3 ) ) ) ) ).

% atLeastAtMost_upto
thf(fact_9891_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I @ K )
          = ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).

% upto_split2
thf(fact_9892_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I @ K )
          = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).

% upto_split1
thf(fact_9893_atLeastLessThan__upto,axiom,
    ( set_or4662586982721622107an_int
    = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).

% atLeastLessThan_upto
thf(fact_9894_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( upto @ I @ J )
        = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).

% upto_rec1
thf(fact_9895_upto_Osimps,axiom,
    ( upto
    = ( ^ [I4: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I4 @ J3 ) @ ( cons_int @ I4 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).

% upto.simps
thf(fact_9896_upto_Oelims,axiom,
    ! [X3: int,Xa2: int,Y3: list_int] :
      ( ( ( upto @ X3 @ Xa2 )
        = Y3 )
     => ( ( ( ord_less_eq_int @ X3 @ Xa2 )
         => ( Y3
            = ( cons_int @ X3 @ ( upto @ ( plus_plus_int @ X3 @ one_one_int ) @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_int @ X3 @ Xa2 )
         => ( Y3 = nil_int ) ) ) ) ).

% upto.elims
thf(fact_9897_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( upto @ I @ J )
        = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).

% upto_rec2
thf(fact_9898_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I @ K )
          = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).

% upto_split3
thf(fact_9899_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
     => ( ( ( ord_less_eq_int @ I @ J )
         => ( ( upto @ I @ J )
            = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
        & ( ~ ( ord_less_eq_int @ I @ J )
         => ( ( upto @ I @ J )
            = nil_int ) ) ) ) ).

% upto.psimps
thf(fact_9900_plus__rat_Oabs__eq,axiom,
    ! [Xa2: product_prod_int_int,X3: product_prod_int_int] :
      ( ( ratrel @ Xa2 @ Xa2 )
     => ( ( ratrel @ X3 @ X3 )
       => ( ( plus_plus_rat @ ( abs_Rat @ Xa2 ) @ ( abs_Rat @ X3 ) )
          = ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa2 ) @ ( product_snd_int_int @ X3 ) ) @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa2 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ) ) ).

% plus_rat.abs_eq
thf(fact_9901_ratrel__iff,axiom,
    ( ratrel
    = ( ^ [X: product_prod_int_int,Y: product_prod_int_int] :
          ( ( ( product_snd_int_int @ X )
           != zero_zero_int )
          & ( ( product_snd_int_int @ Y )
           != zero_zero_int )
          & ( ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) )
            = ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ).

% ratrel_iff
thf(fact_9902_one__rat_Orsp,axiom,
    ratrel @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ).

% one_rat.rsp
thf(fact_9903_zero__rat_Orsp,axiom,
    ratrel @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ).

% zero_rat.rsp
thf(fact_9904_ratrel__def,axiom,
    ( ratrel
    = ( ^ [X: product_prod_int_int,Y: product_prod_int_int] :
          ( ( ( product_snd_int_int @ X )
           != zero_zero_int )
          & ( ( product_snd_int_int @ Y )
           != zero_zero_int )
          & ( ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) )
            = ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ).

% ratrel_def
thf(fact_9905_uminus__rat_Oabs__eq,axiom,
    ! [X3: product_prod_int_int] :
      ( ( ratrel @ X3 @ X3 )
     => ( ( uminus_uminus_rat @ ( abs_Rat @ X3 ) )
        = ( abs_Rat @ ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X3 ) ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ).

% uminus_rat.abs_eq
thf(fact_9906_times__rat_Oabs__eq,axiom,
    ! [Xa2: product_prod_int_int,X3: product_prod_int_int] :
      ( ( ratrel @ Xa2 @ Xa2 )
     => ( ( ratrel @ X3 @ X3 )
       => ( ( times_times_rat @ ( abs_Rat @ Xa2 ) @ ( abs_Rat @ X3 ) )
          = ( abs_Rat @ ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ Xa2 ) @ ( product_fst_int_int @ X3 ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa2 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ) ) ).

% times_rat.abs_eq
thf(fact_9907_Rat_Opositive_Oabs__eq,axiom,
    ! [X3: product_prod_int_int] :
      ( ( ratrel @ X3 @ X3 )
     => ( ( positive @ ( abs_Rat @ X3 ) )
        = ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ).

% Rat.positive.abs_eq
thf(fact_9908_inverse__rat_Oabs__eq,axiom,
    ! [X3: product_prod_int_int] :
      ( ( ratrel @ X3 @ X3 )
     => ( ( inverse_inverse_rat @ ( abs_Rat @ X3 ) )
        = ( abs_Rat
          @ ( if_Pro3027730157355071871nt_int
            @ ( ( product_fst_int_int @ X3 )
              = zero_zero_int )
            @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
            @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) ) ) ) ) ).

% inverse_rat.abs_eq
thf(fact_9909_DERIV__even__real__root,axiom,
    ! [N: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( ord_less_real @ X3 @ 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 @ X3 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ).

% DERIV_even_real_root
thf(fact_9910_DERIV__real__root__generic,axiom,
    ! [N: nat,X3: real,D4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( X3 != zero_zero_real )
       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
           => ( ( ord_less_real @ zero_zero_real @ X3 )
             => ( D4
                = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X3 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
             => ( ( ord_less_real @ X3 @ zero_zero_real )
               => ( D4
                  = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X3 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
           => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
               => ( D4
                  = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X3 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
             => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ) ) ).

% DERIV_real_root_generic
thf(fact_9911_DERIV__neg__imp__decreasing,axiom,
    ! [A: real,B2: real,F: real > real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ! [X5: real] :
            ( ( ord_less_eq_real @ A @ X5 )
           => ( ( ord_less_eq_real @ X5 @ B2 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
                  & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
       => ( ord_less_real @ ( F @ B2 ) @ ( F @ A ) ) ) ) ).

% DERIV_neg_imp_decreasing
thf(fact_9912_DERIV__pos__imp__increasing,axiom,
    ! [A: real,B2: real,F: real > real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ! [X5: real] :
            ( ( ord_less_eq_real @ A @ X5 )
           => ( ( ord_less_eq_real @ X5 @ B2 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
       => ( ord_less_real @ ( F @ A ) @ ( F @ B2 ) ) ) ) ).

% DERIV_pos_imp_increasing
thf(fact_9913_deriv__nonneg__imp__mono,axiom,
    ! [A: real,B2: real,G: real > real,G2: real > real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ ( set_or1222579329274155063t_real @ A @ B2 ) )
         => ( has_fi5821293074295781190e_real @ G @ ( G2 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ ( set_or1222579329274155063t_real @ A @ B2 ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X5 ) ) )
       => ( ( ord_less_eq_real @ A @ B2 )
         => ( ord_less_eq_real @ ( G @ A ) @ ( G @ B2 ) ) ) ) ) ).

% deriv_nonneg_imp_mono
thf(fact_9914_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A: real,B2: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ! [X5: real] :
            ( ( ord_less_eq_real @ A @ X5 )
           => ( ( ord_less_eq_real @ X5 @ B2 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ zero_zero_real @ Y5 ) ) ) )
       => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B2 ) ) ) ) ).

% DERIV_nonneg_imp_nondecreasing
thf(fact_9915_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A: real,B2: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ! [X5: real] :
            ( ( ord_less_eq_real @ A @ X5 )
           => ( ( ord_less_eq_real @ X5 @ B2 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ Y5 @ zero_zero_real ) ) ) )
       => ( ord_less_eq_real @ ( F @ B2 ) @ ( F @ A ) ) ) ) ).

% DERIV_nonpos_imp_nonincreasing
thf(fact_9916_DERIV__const__ratio__const,axiom,
    ! [A: real,B2: real,F: real > real,K: real] :
      ( ( A != B2 )
     => ( ! [X5: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
       => ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) )
          = ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ K ) ) ) ) ).

% DERIV_const_ratio_const
thf(fact_9917_has__real__derivative__pos__inc__left,axiom,
    ! [F: real > real,L: real,X3: real,S2: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S2 ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( minus_minus_real @ X3 @ H4 ) @ S2 )
                 => ( ( ord_less_real @ H4 @ D3 )
                   => ( ord_less_real @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
thf(fact_9918_has__real__derivative__neg__dec__left,axiom,
    ! [F: real > real,L: real,X3: real,S2: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S2 ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( minus_minus_real @ X3 @ H4 ) @ S2 )
                 => ( ( ord_less_real @ H4 @ D3 )
                   => ( ord_less_real @ ( F @ X3 ) @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
thf(fact_9919_has__real__derivative__pos__inc__right,axiom,
    ! [F: real > real,L: real,X3: real,S2: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S2 ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( plus_plus_real @ X3 @ H4 ) @ S2 )
                 => ( ( ord_less_real @ H4 @ D3 )
                   => ( ord_less_real @ ( F @ X3 ) @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
thf(fact_9920_has__real__derivative__neg__dec__right,axiom,
    ! [F: real > real,L: real,X3: real,S2: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S2 ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( plus_plus_real @ X3 @ H4 ) @ S2 )
                 => ( ( ord_less_real @ H4 @ D3 )
                   => ( ord_less_real @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
thf(fact_9921_DERIV__pos__inc__right,axiom,
    ! [F: real > real,L: real,X3: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D3 )
                 => ( ord_less_real @ ( F @ X3 ) @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ).

% DERIV_pos_inc_right
thf(fact_9922_DERIV__neg__dec__right,axiom,
    ! [F: real > real,L: real,X3: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D3 )
                 => ( ord_less_real @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ).

% DERIV_neg_dec_right
thf(fact_9923_DERIV__neg__dec__left,axiom,
    ! [F: real > real,L: real,X3: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D3 )
                 => ( ord_less_real @ ( F @ X3 ) @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ).

% DERIV_neg_dec_left
thf(fact_9924_DERIV__pos__inc__left,axiom,
    ! [F: real > real,L: real,X3: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D3 )
                 => ( ord_less_real @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ).

% DERIV_pos_inc_left
thf(fact_9925_MVT2,axiom,
    ! [A: real,B2: real,F: real > real,F4: real > real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ! [X5: real] :
            ( ( ord_less_eq_real @ A @ X5 )
           => ( ( ord_less_eq_real @ X5 @ B2 )
             => ( has_fi5821293074295781190e_real @ F @ ( F4 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
       => ? [Z4: real] :
            ( ( ord_less_real @ A @ Z4 )
            & ( ord_less_real @ Z4 @ B2 )
            & ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) )
              = ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ ( F4 @ Z4 ) ) ) ) ) ) ).

% MVT2
thf(fact_9926_DERIV__local__const,axiom,
    ! [F: real > real,L: real,X3: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y4: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y4 ) ) @ D )
             => ( ( F @ X3 )
                = ( F @ Y4 ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
thf(fact_9927_DERIV__ln,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).

% DERIV_ln
thf(fact_9928_DERIV__const__average,axiom,
    ! [A: real,B2: real,V: real > real,K: real] :
      ( ( A != B2 )
     => ( ! [X5: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
       => ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% DERIV_const_average
thf(fact_9929_DERIV__local__max,axiom,
    ! [F: real > real,L: real,X3: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y4: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y4 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X3 ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
thf(fact_9930_DERIV__local__min,axiom,
    ! [F: real > real,L: real,X3: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y4: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y4 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
thf(fact_9931_DERIV__ln__divide,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).

% DERIV_ln_divide
thf(fact_9932_DERIV__pow,axiom,
    ! [N: nat,X3: real,S: set_real] :
      ( has_fi5821293074295781190e_real
      @ ^ [X: real] : ( power_power_real @ X @ N )
      @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X3 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
      @ ( topolo2177554685111907308n_real @ X3 @ S ) ) ).

% DERIV_pow
thf(fact_9933_DERIV__fun__pow,axiom,
    ! [G: real > real,M: real,X3: real,N: nat] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( has_fi5821293074295781190e_real
        @ ^ [X: real] : ( power_power_real @ ( G @ X ) @ N )
        @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G @ X3 ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M )
        @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).

% DERIV_fun_pow
thf(fact_9934_has__real__derivative__powr,axiom,
    ! [Z: real,R: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( has_fi5821293074295781190e_real
        @ ^ [Z2: real] : ( powr_real @ Z2 @ R )
        @ ( times_times_real @ R @ ( powr_real @ Z @ ( minus_minus_real @ R @ one_one_real ) ) )
        @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).

% has_real_derivative_powr
thf(fact_9935_DERIV__log,axiom,
    ! [X3: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( has_fi5821293074295781190e_real @ ( log @ B2 ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B2 ) @ X3 ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).

% DERIV_log
thf(fact_9936_DERIV__fun__powr,axiom,
    ! [G: real > real,M: real,X3: real,R: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X3 ) )
       => ( has_fi5821293074295781190e_real
          @ ^ [X: real] : ( powr_real @ ( G @ X ) @ R )
          @ ( times_times_real @ ( times_times_real @ R @ ( powr_real @ ( G @ X3 ) @ ( minus_minus_real @ R @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
          @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).

% DERIV_fun_powr
thf(fact_9937_DERIV__powr,axiom,
    ! [G: real > real,M: real,X3: real,F: real > real,R: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X3 ) )
       => ( ( has_fi5821293074295781190e_real @ F @ R @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
         => ( has_fi5821293074295781190e_real
            @ ^ [X: real] : ( powr_real @ ( G @ X ) @ ( F @ X ) )
            @ ( times_times_real @ ( powr_real @ ( G @ X3 ) @ ( F @ X3 ) ) @ ( plus_plus_real @ ( times_times_real @ R @ ( ln_ln_real @ ( G @ X3 ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X3 ) ) @ ( G @ X3 ) ) ) )
            @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ).

% DERIV_powr
thf(fact_9938_DERIV__real__sqrt,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).

% DERIV_real_sqrt
thf(fact_9939_DERIV__arctan,axiom,
    ! [X3: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ).

% DERIV_arctan
thf(fact_9940_arsinh__real__has__field__derivative,axiom,
    ! [X3: real,A4: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ A4 ) ) ).

% arsinh_real_has_field_derivative
thf(fact_9941_DERIV__real__sqrt__generic,axiom,
    ! [X3: real,D4: real] :
      ( ( X3 != zero_zero_real )
     => ( ( ( ord_less_real @ zero_zero_real @ X3 )
         => ( D4
            = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( ( ord_less_real @ X3 @ zero_zero_real )
           => ( D4
              = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
         => ( has_fi5821293074295781190e_real @ sqrt @ D4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ).

% DERIV_real_sqrt_generic
thf(fact_9942_arcosh__real__has__field__derivative,axiom,
    ! [X3: real,A4: set_real] :
      ( ( ord_less_real @ one_one_real @ X3 )
     => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ A4 ) ) ) ).

% arcosh_real_has_field_derivative
thf(fact_9943_artanh__real__has__field__derivative,axiom,
    ! [X3: real,A4: set_real] :
      ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ A4 ) ) ) ).

% artanh_real_has_field_derivative
thf(fact_9944_DERIV__real__root,axiom,
    ! [N: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X3 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).

% DERIV_real_root
thf(fact_9945_DERIV__arccos,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_real @ X3 @ one_one_real )
       => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).

% DERIV_arccos
thf(fact_9946_DERIV__arcsin,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_real @ X3 @ one_one_real )
       => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).

% DERIV_arcsin
thf(fact_9947_Maclaurin__all__le,axiom,
    ! [Diff: nat > real > real,F: real > real,X3: real,N: nat] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M4: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
       => ? [T4: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X3 ) )
            & ( ( F @ X3 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ) ).

% Maclaurin_all_le
thf(fact_9948_Maclaurin__all__le__objl,axiom,
    ! [Diff: nat > real > real,F: real > real,X3: real,N: nat] :
      ( ( ( ( Diff @ zero_zero_nat )
          = F )
        & ! [M4: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
     => ? [T4: real] :
          ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X3 ) )
          & ( ( F @ X3 )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
                @ ( set_ord_lessThan_nat @ N ) )
              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ).

% Maclaurin_all_le_objl
thf(fact_9949_DERIV__odd__real__root,axiom,
    ! [N: nat,X3: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( X3 != zero_zero_real )
       => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X3 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).

% DERIV_odd_real_root
thf(fact_9950_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,T4: real] :
                ( ( ( ord_less_nat @ M4 @ N )
                  & ( ord_less_eq_real @ zero_zero_real @ T4 )
                  & ( ord_less_eq_real @ T4 @ H2 ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
           => ? [T4: real] :
                ( ( ord_less_real @ zero_zero_real @ T4 )
                & ( ord_less_real @ T4 @ 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 @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin
thf(fact_9951_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,T4: real] :
              ( ( ( ord_less_nat @ M4 @ N )
                & ( ord_less_eq_real @ zero_zero_real @ T4 )
                & ( ord_less_eq_real @ T4 @ H2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
         => ? [T4: real] :
              ( ( ord_less_real @ zero_zero_real @ T4 )
              & ( ord_less_eq_real @ T4 @ 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 @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ).

% Maclaurin2
thf(fact_9952_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,T4: real] :
                ( ( ( ord_less_nat @ M4 @ N )
                  & ( ord_less_eq_real @ H2 @ T4 )
                  & ( ord_less_eq_real @ T4 @ zero_zero_real ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
           => ? [T4: real] :
                ( ( ord_less_real @ H2 @ T4 )
                & ( ord_less_real @ T4 @ 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 @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin_minus
thf(fact_9953_Maclaurin__all__lt,axiom,
    ! [Diff: nat > real > real,F: real > real,N: nat,X3: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( X3 != zero_zero_real )
         => ( ! [M4: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
           => ? [T4: real] :
                ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T4 ) )
                & ( ord_less_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X3 ) )
                & ( ( F @ X3 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin_all_lt
thf(fact_9954_Maclaurin__bi__le,axiom,
    ! [Diff: nat > real > real,F: real > real,N: nat,X3: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M4: nat,T4: real] :
            ( ( ( ord_less_nat @ M4 @ N )
              & ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X3 ) ) )
           => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
       => ? [T4: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X3 ) )
            & ( ( F @ X3 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ) ).

% Maclaurin_bi_le
thf(fact_9955_Taylor,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B2: real,C: real,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T4: real] :
              ( ( ( ord_less_nat @ M4 @ N )
                & ( ord_less_eq_real @ A @ T4 )
                & ( ord_less_eq_real @ T4 @ B2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B2 )
             => ( ( ord_less_eq_real @ A @ X3 )
               => ( ( ord_less_eq_real @ X3 @ B2 )
                 => ( ( X3 != C )
                   => ? [T4: real] :
                        ( ( ( ord_less_real @ X3 @ C )
                         => ( ( ord_less_real @ X3 @ T4 )
                            & ( ord_less_real @ T4 @ C ) ) )
                        & ( ~ ( ord_less_real @ X3 @ C )
                         => ( ( ord_less_real @ C @ T4 )
                            & ( ord_less_real @ T4 @ X3 ) ) )
                        & ( ( F @ X3 )
                          = ( 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 @ X3 @ C ) @ M3 ) )
                              @ ( set_ord_lessThan_nat @ N ) )
                            @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X3 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
thf(fact_9956_Taylor__up,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B2: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T4: real] :
              ( ( ( ord_less_nat @ M4 @ N )
                & ( ord_less_eq_real @ A @ T4 )
                & ( ord_less_eq_real @ T4 @ B2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_real @ C @ B2 )
             => ? [T4: real] :
                  ( ( ord_less_real @ C @ T4 )
                  & ( ord_less_real @ T4 @ 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 @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B2 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).

% Taylor_up
thf(fact_9957_Taylor__down,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B2: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T4: real] :
              ( ( ( ord_less_nat @ M4 @ N )
                & ( ord_less_eq_real @ A @ T4 )
                & ( ord_less_eq_real @ T4 @ B2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
         => ( ( ord_less_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B2 )
             => ? [T4: real] :
                  ( ( ord_less_real @ A @ T4 )
                  & ( ord_less_real @ T4 @ C )
                  & ( ( F @ A )
                    = ( 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 @ A @ C ) @ M3 ) )
                        @ ( set_ord_lessThan_nat @ N ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).

% Taylor_down
thf(fact_9958_Maclaurin__lemma2,axiom,
    ! [N: nat,H2: real,Diff: nat > real > real,K: nat,B5: real] :
      ( ! [M4: nat,T4: real] :
          ( ( ( ord_less_nat @ M4 @ N )
            & ( ord_less_eq_real @ zero_zero_real @ T4 )
            & ( ord_less_eq_real @ T4 @ H2 ) )
         => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
     => ( ( N
          = ( suc @ K ) )
       => ! [M2: nat,T5: real] :
            ( ( ( ord_less_nat @ M2 @ N )
              & ( ord_less_eq_real @ zero_zero_real @ T5 )
              & ( ord_less_eq_real @ T5 @ H2 ) )
           => ( has_fi5821293074295781190e_real
              @ ^ [U2: real] :
                  ( minus_minus_real @ ( Diff @ M2 @ U2 )
                  @ ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M2 @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ U2 @ P5 ) )
                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M2 ) ) )
                    @ ( times_times_real @ B5 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N @ M2 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) )
              @ ( minus_minus_real @ ( Diff @ ( suc @ M2 ) @ T5 )
                @ ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M2 ) @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ T5 @ P5 ) )
                    @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) )
                  @ ( times_times_real @ B5 @ ( divide_divide_real @ ( power_power_real @ T5 @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) ) ) ) )
              @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) ) ) ) ).

% Maclaurin_lemma2
thf(fact_9959_DERIV__arctan__series,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( has_fi5821293074295781190e_real
        @ ^ [X10: real] :
            ( suminf_real
            @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X10 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
        @ ( suminf_real
          @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( power_power_real @ X3 @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).

% DERIV_arctan_series
thf(fact_9960_DERIV__power__series_H,axiom,
    ! [R2: real,F: nat > real,X0: real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R2 ) @ R2 ) )
         => ( summable_real
            @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X5 @ N2 ) ) ) )
     => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R2 ) @ R2 ) )
       => ( ( ord_less_real @ zero_zero_real @ R2 )
         => ( has_fi5821293074295781190e_real
            @ ^ [X: real] :
                ( suminf_real
                @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X @ ( suc @ N2 ) ) ) )
            @ ( suminf_real
              @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X0 @ N2 ) ) )
            @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).

% DERIV_power_series'
thf(fact_9961_DERIV__isconst3,axiom,
    ! [A: real,B2: real,X3: real,Y3: real,F: real > real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
       => ( ( member_real @ Y3 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
           => ( ( F @ X3 )
              = ( F @ Y3 ) ) ) ) ) ) ).

% DERIV_isconst3
thf(fact_9962_DERIV__series_H,axiom,
    ! [F: real > nat > real,F4: real > nat > real,X0: real,A: real,B2: real,L5: nat > real] :
      ( ! [N3: nat] :
          ( has_fi5821293074295781190e_real
          @ ^ [X: real] : ( F @ X @ N3 )
          @ ( F4 @ X0 @ N3 )
          @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
           => ( summable_real @ ( F @ X5 ) ) )
       => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
         => ( ( summable_real @ ( F4 @ X0 ) )
           => ( ( summable_real @ L5 )
             => ( ! [N3: nat,X5: real,Y4: real] :
                    ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
                   => ( ( member_real @ Y4 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
                     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X5 @ N3 ) @ ( F @ Y4 @ N3 ) ) ) @ ( times_times_real @ ( L5 @ N3 ) @ ( abs_abs_real @ ( minus_minus_real @ X5 @ Y4 ) ) ) ) ) )
               => ( has_fi5821293074295781190e_real
                  @ ^ [X: real] : ( suminf_real @ ( F @ X ) )
                  @ ( suminf_real @ ( F4 @ X0 ) )
                  @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_series'
thf(fact_9963_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L @ U ) )
      = ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).

% card_greaterThanLessThan
thf(fact_9964_isCont__Lb__Ub,axiom,
    ! [A: real,B2: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ! [X5: real] :
            ( ( ( ord_less_eq_real @ A @ X5 )
              & ( ord_less_eq_real @ X5 @ B2 ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ F ) )
       => ? [L6: real,M8: real] :
            ( ! [X6: real] :
                ( ( ( ord_less_eq_real @ A @ X6 )
                  & ( ord_less_eq_real @ X6 @ B2 ) )
               => ( ( ord_less_eq_real @ L6 @ ( F @ X6 ) )
                  & ( ord_less_eq_real @ ( F @ X6 ) @ M8 ) ) )
            & ! [Y5: real] :
                ( ( ( ord_less_eq_real @ L6 @ Y5 )
                  & ( ord_less_eq_real @ Y5 @ M8 ) )
               => ? [X5: real] :
                    ( ( ord_less_eq_real @ A @ X5 )
                    & ( ord_less_eq_real @ X5 @ B2 )
                    & ( ( F @ X5 )
                      = Y5 ) ) ) ) ) ) ).

% isCont_Lb_Ub
thf(fact_9965_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 )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ! [X6: real] :
                ( ( ( X6 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X6 ) ) @ R3 ) )
               => ( ord_less_real @ ( F @ X6 ) @ zero_zero_real ) ) ) ) ) ).

% LIM_fun_less_zero
thf(fact_9966_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 )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ! [X6: real] :
                ( ( ( X6 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X6 ) ) @ R3 ) )
               => ( ( F @ X6 )
                 != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
thf(fact_9967_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 )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ! [X6: real] :
                ( ( ( X6 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X6 ) ) @ R3 ) )
               => ( ord_less_real @ zero_zero_real @ ( F @ X6 ) ) ) ) ) ) ).

% LIM_fun_gt_zero
thf(fact_9968_isCont__real__sqrt,axiom,
    ! [X3: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ sqrt ) ).

% isCont_real_sqrt
thf(fact_9969_isCont__real__root,axiom,
    ! [X3: real,N: nat] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ ( root @ N ) ) ).

% isCont_real_root
thf(fact_9970_isCont__inverse__function2,axiom,
    ! [A: real,X3: real,B2: real,G: real > real,F: real > real] :
      ( ( ord_less_real @ A @ X3 )
     => ( ( ord_less_real @ X3 @ B2 )
       => ( ! [Z4: real] :
              ( ( ord_less_eq_real @ A @ Z4 )
             => ( ( ord_less_eq_real @ Z4 @ B2 )
               => ( ( G @ ( F @ Z4 ) )
                  = Z4 ) ) )
         => ( ! [Z4: real] :
                ( ( ord_less_eq_real @ A @ Z4 )
               => ( ( ord_less_eq_real @ Z4 @ B2 )
                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X3 ) @ top_top_set_real ) @ G ) ) ) ) ) ).

% isCont_inverse_function2
thf(fact_9971_isCont__arcosh,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ one_one_real @ X3 )
     => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ arcosh_real ) ) ).

% isCont_arcosh
thf(fact_9972_LIM__cos__div__sin,axiom,
    ( filterlim_real_real
    @ ^ [X: real] : ( divide_divide_real @ ( cos_real @ X ) @ ( sin_real @ X ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).

% LIM_cos_div_sin
thf(fact_9973_DERIV__inverse__function,axiom,
    ! [F: real > real,D4: real,G: real > real,X3: real,A: real,B2: real] :
      ( ( has_fi5821293074295781190e_real @ F @ D4 @ ( topolo2177554685111907308n_real @ ( G @ X3 ) @ top_top_set_real ) )
     => ( ( D4 != zero_zero_real )
       => ( ( ord_less_real @ A @ X3 )
         => ( ( ord_less_real @ X3 @ B2 )
           => ( ! [Y4: real] :
                  ( ( ord_less_real @ A @ Y4 )
                 => ( ( ord_less_real @ Y4 @ B2 )
                   => ( ( F @ ( G @ Y4 ) )
                      = Y4 ) ) )
             => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ G )
               => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D4 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_inverse_function
thf(fact_9974_isCont__arccos,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_real @ X3 @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ arccos ) ) ) ).

% isCont_arccos
thf(fact_9975_isCont__arcsin,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_real @ X3 @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ arcsin ) ) ) ).

% isCont_arcsin
thf(fact_9976_LIM__less__bound,axiom,
    ! [B2: real,X3: real,F: real > real] :
      ( ( ord_less_real @ B2 @ X3 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ B2 @ X3 ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) ) ) ) ).

% LIM_less_bound
thf(fact_9977_isCont__artanh,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_real @ X3 @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ artanh_real ) ) ) ).

% isCont_artanh
thf(fact_9978_greaterThanLessThan__upto,axiom,
    ( set_or5832277885323065728an_int
    = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).

% greaterThanLessThan_upto
thf(fact_9979_isCont__inverse__function,axiom,
    ! [D: real,X3: real,G: real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ D )
     => ( ! [Z4: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z4 @ X3 ) ) @ D )
           => ( ( G @ ( F @ Z4 ) )
              = Z4 ) )
       => ( ! [Z4: real] :
              ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z4 @ X3 ) ) @ D )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) )
         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X3 ) @ top_top_set_real ) @ G ) ) ) ) ).

% isCont_inverse_function
thf(fact_9980_GMVT_H,axiom,
    ! [A: real,B2: real,F: real > real,G: real > real,G2: real > real,F4: real > real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ! [Z4: real] :
            ( ( ord_less_eq_real @ A @ Z4 )
           => ( ( ord_less_eq_real @ Z4 @ B2 )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) ) )
       => ( ! [Z4: real] :
              ( ( ord_less_eq_real @ A @ Z4 )
             => ( ( ord_less_eq_real @ Z4 @ B2 )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ G ) ) )
         => ( ! [Z4: real] :
                ( ( ord_less_real @ A @ Z4 )
               => ( ( ord_less_real @ Z4 @ B2 )
                 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z4 ) @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) )
           => ( ! [Z4: real] :
                  ( ( ord_less_real @ A @ Z4 )
                 => ( ( ord_less_real @ Z4 @ B2 )
                   => ( has_fi5821293074295781190e_real @ F @ ( F4 @ Z4 ) @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) )
             => ? [C2: real] :
                  ( ( ord_less_real @ A @ C2 )
                  & ( ord_less_real @ C2 @ B2 )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) ) @ ( G2 @ C2 ) )
                    = ( times_times_real @ ( minus_minus_real @ ( G @ B2 ) @ ( G @ A ) ) @ ( F4 @ C2 ) ) ) ) ) ) ) ) ) ).

% GMVT'
thf(fact_9981_summable__Leibniz_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
         => ! [N9: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) ) ) ) ) ) ) ).

% summable_Leibniz(3)
thf(fact_9982_summable__Leibniz_I2_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
         => ! [N9: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% summable_Leibniz(2)
thf(fact_9983_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_9984_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( filterlim_nat_nat
        @ ^ [X: nat] : ( times_times_nat @ X @ C )
        @ at_top_nat
        @ at_top_nat ) ) ).

% mult_nat_right_at_top
thf(fact_9985_monoseq__convergent,axiom,
    ! [X9: nat > real,B5: real] :
      ( ( topolo6980174941875973593q_real @ X9 )
     => ( ! [I3: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X9 @ I3 ) ) @ B5 )
       => ~ ! [L6: real] :
              ~ ( filterlim_nat_real @ X9 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat ) ) ) ).

% monoseq_convergent
thf(fact_9986_LIMSEQ__root,axiom,
    ( filterlim_nat_real
    @ ^ [N2: nat] : ( root @ N2 @ ( semiri5074537144036343181t_real @ N2 ) )
    @ ( topolo2815343760600316023s_real @ one_one_real )
    @ at_top_nat ) ).

% LIMSEQ_root
thf(fact_9987_nested__sequence__unique,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ! [N3: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N3 ) ) @ ( G @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
         => ( ( filterlim_nat_real
              @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
              @ ( topolo2815343760600316023s_real @ zero_zero_real )
              @ at_top_nat )
           => ? [L4: real] :
                ( ! [N9: nat] : ( ord_less_eq_real @ ( F @ N9 ) @ L4 )
                & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
                & ! [N9: nat] : ( ord_less_eq_real @ L4 @ ( G @ N9 ) )
                & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).

% nested_sequence_unique
thf(fact_9988_LIMSEQ__inverse__zero,axiom,
    ! [X9: nat > real] :
      ( ! [R3: real] :
        ? [N7: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N7 @ N3 )
         => ( ord_less_real @ R3 @ ( X9 @ N3 ) ) )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( inverse_inverse_real @ ( X9 @ N2 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_zero
thf(fact_9989_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( root @ N2 @ C )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat ) ) ).

% LIMSEQ_root_const
thf(fact_9990_increasing__LIMSEQ,axiom,
    ! [F: nat > real,L: real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ L )
       => ( ! [E2: real] :
              ( ( ord_less_real @ zero_zero_real @ E2 )
             => ? [N9: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N9 ) @ E2 ) ) )
         => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).

% increasing_LIMSEQ
thf(fact_9991_LIMSEQ__realpow__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ one_one_real )
       => ( filterlim_nat_real @ ( power_power_real @ X3 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).

% LIMSEQ_realpow_zero
thf(fact_9992_LIMSEQ__divide__realpow__zero,axiom,
    ! [X3: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X3 )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( divide_divide_real @ A @ ( power_power_real @ X3 @ N2 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_divide_realpow_zero
thf(fact_9993_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_9994_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_9995_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ one_one_real @ X3 )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( inverse_inverse_real @ ( power_power_real @ X3 @ N2 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_realpow_zero
thf(fact_9996_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R: real] :
      ( filterlim_nat_real
      @ ^ [N2: nat] : ( times_times_real @ R @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_9997_summable__Leibniz_I1_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( summable_real
          @ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A @ N2 ) ) ) ) ) ).

% summable_Leibniz(1)
thf(fact_9998_summable,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( summable_real
            @ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A @ N2 ) ) ) ) ) ) ).

% summable
thf(fact_9999_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 )
     => ~ ! [K3: nat > int] :
            ~ ( filterlim_nat_real
              @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
              @ ( topolo2815343760600316023s_real @ Theta2 )
              @ at_top_nat ) ) ).

% cos_diff_limit_1
thf(fact_10000_cos__limit__1,axiom,
    ! [Theta: nat > real] :
      ( ( filterlim_nat_real
        @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ? [K3: nat > int] :
          ( filterlim_nat_real
          @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
          @ ( topolo2815343760600316023s_real @ zero_zero_real )
          @ at_top_nat ) ) ).

% cos_limit_1
thf(fact_10001_summable__Leibniz_I4_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( filterlim_nat_real
          @ ^ [N2: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(4)
thf(fact_10002_zeroseq__arctan__series,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% zeroseq_arctan_series
thf(fact_10003_summable__Leibniz_H_I2_J,axiom,
    ! [A: nat > real,N: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( ord_less_eq_real
            @ ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ 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 ) @ ( A @ I4 ) ) ) ) ) ) ) ).

% summable_Leibniz'(2)
thf(fact_10004_summable__Leibniz_H_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( filterlim_nat_real
            @ ^ [N2: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(3)
thf(fact_10005_sums__alternating__upper__lower,axiom,
    ! [A: nat > real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
         => ? [L4: real] :
              ( ! [N9: nat] :
                  ( ord_less_eq_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) )
                  @ L4 )
              & ( filterlim_nat_real
                @ ^ [N2: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat )
              & ! [N9: nat] :
                  ( ord_less_eq_real @ L4
                  @ ( groups6591440286371151544t_real
                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) ) )
              & ( filterlim_nat_real
                @ ^ [N2: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat ) ) ) ) ) ).

% sums_alternating_upper_lower
thf(fact_10006_summable__Leibniz_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( filterlim_nat_real
          @ ^ [N2: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(5)
thf(fact_10007_summable__Leibniz_H_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( filterlim_nat_real
            @ ^ [N2: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(5)
thf(fact_10008_summable__Leibniz_H_I4_J,axiom,
    ! [A: nat > real,N: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( ord_less_eq_real
            @ ( suminf_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
            @ ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ 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_10009_eventually__sequentially__seg,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat
        @ ^ [N2: nat] : ( P @ ( plus_plus_nat @ N2 @ K ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_seg
thf(fact_10010_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_10011_eventually__sequentially,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat @ P @ at_top_nat )
      = ( ? [N6: nat] :
          ! [N2: nat] :
            ( ( ord_less_eq_nat @ N6 @ N2 )
           => ( P @ N2 ) ) ) ) ).

% eventually_sequentially
thf(fact_10012_eventually__sequentiallyI,axiom,
    ! [C: nat,P: nat > $o] :
      ( ! [X5: nat] :
          ( ( ord_less_eq_nat @ C @ X5 )
         => ( P @ X5 ) )
     => ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentiallyI
thf(fact_10013_le__sequentially,axiom,
    ! [F5: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ F5 @ at_top_nat )
      = ( ! [N6: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N6 ) @ F5 ) ) ) ).

% le_sequentially
thf(fact_10014_real__bounded__linear,axiom,
    ( real_V5970128139526366754l_real
    = ( ^ [F3: real > real] :
        ? [C3: real] :
          ( F3
          = ( ^ [X: real] : ( times_times_real @ X @ C3 ) ) ) ) ) ).

% real_bounded_linear
thf(fact_10015_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L @ U ) )
      = ( minus_minus_nat @ U @ L ) ) ).

% card_greaterThanAtMost
thf(fact_10016_sqrt__at__top,axiom,
    filterlim_real_real @ sqrt @ at_top_real @ at_top_real ).

% sqrt_at_top
thf(fact_10017_eventually__at__left__real,axiom,
    ! [B2: real,A: real] :
      ( ( ord_less_real @ B2 @ A )
     => ( eventually_real
        @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ B2 @ A ) )
        @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ).

% eventually_at_left_real
thf(fact_10018_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_10019_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B2: real,F: real > real,Flim: real] :
      ( ! [X5: real] :
          ( ( ord_less_eq_real @ B2 @ X5 )
         => ? [Y5: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
              & ( ord_less_real @ Y5 @ zero_zero_real ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
       => ( ord_less_real @ Flim @ ( F @ B2 ) ) ) ) ).

% DERIV_neg_imp_decreasing_at_top
thf(fact_10020_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_10021_at__top__le__at__infinity,axiom,
    ord_le4104064031414453916r_real @ at_top_real @ at_infinity_real ).

% at_top_le_at_infinity
thf(fact_10022_greaterThanAtMost__upto,axiom,
    ( set_or6656581121297822940st_int
    = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ).

% greaterThanAtMost_upto
thf(fact_10023_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F: real > real,F5: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
         => ( filterlim_real_real
            @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N )
            @ at_top_real
            @ F5 ) ) ) ) ).

% filterlim_pow_at_bot_even
thf(fact_10024_at__bot__le__at__infinity,axiom,
    ord_le4104064031414453916r_real @ at_bot_real @ at_infinity_real ).

% at_bot_le_at_infinity
thf(fact_10025_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B2: real,F: real > real,Flim: real] :
      ( ! [X5: real] :
          ( ( ord_less_eq_real @ X5 @ B2 )
         => ? [Y5: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
              & ( ord_less_real @ zero_zero_real @ Y5 ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
       => ( ord_less_real @ Flim @ ( F @ B2 ) ) ) ) ).

% DERIV_pos_imp_increasing_at_bot
thf(fact_10026_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F: real > real,F5: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
       => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
         => ( filterlim_real_real
            @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N )
            @ at_bot_real
            @ F5 ) ) ) ) ).

% filterlim_pow_at_bot_odd
thf(fact_10027_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_10028_Bseq__eq__bounded,axiom,
    ! [F: nat > real,A: real,B2: real] :
      ( ( ord_less_eq_set_real @ ( image_nat_real @ F @ top_top_set_nat ) @ ( set_or1222579329274155063t_real @ A @ B2 ) )
     => ( bfun_nat_real @ F @ at_top_nat ) ) ).

% Bseq_eq_bounded
thf(fact_10029_Bseq__realpow,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ X3 @ one_one_real )
       => ( bfun_nat_real @ ( power_power_real @ X3 ) @ at_top_nat ) ) ) ).

% Bseq_realpow
thf(fact_10030_tendsto__exp__limit__at__right,axiom,
    ! [X3: real] :
      ( filterlim_real_real
      @ ^ [Y: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X3 @ Y ) ) @ ( divide_divide_real @ one_one_real @ Y ) )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X3 ) )
      @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).

% tendsto_exp_limit_at_right
thf(fact_10031_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_10032_eventually__at__right__real,axiom,
    ! [A: real,B2: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( eventually_real
        @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B2 ) )
        @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ).

% eventually_at_right_real
thf(fact_10033_decseq__bounded,axiom,
    ! [X9: nat > real,B5: real] :
      ( ( order_9091379641038594480t_real @ X9 )
     => ( ! [I3: nat] : ( ord_less_eq_real @ B5 @ ( X9 @ I3 ) )
       => ( bfun_nat_real @ X9 @ at_top_nat ) ) ) ).

% decseq_bounded
thf(fact_10034_decseq__convergent,axiom,
    ! [X9: nat > real,B5: real] :
      ( ( order_9091379641038594480t_real @ X9 )
     => ( ! [I3: nat] : ( ord_less_eq_real @ B5 @ ( X9 @ I3 ) )
       => ~ ! [L6: real] :
              ( ( filterlim_nat_real @ X9 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
             => ~ ! [I2: nat] : ( ord_less_eq_real @ L6 @ ( X9 @ I2 ) ) ) ) ) ).

% decseq_convergent
thf(fact_10035_GMVT,axiom,
    ! [A: real,B2: real,F: real > real,G: real > real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ! [X5: real] :
            ( ( ( ord_less_eq_real @ A @ X5 )
              & ( ord_less_eq_real @ X5 @ B2 ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ F ) )
       => ( ! [X5: real] :
              ( ( ( ord_less_real @ A @ X5 )
                & ( ord_less_real @ X5 @ B2 ) )
             => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
         => ( ! [X5: real] :
                ( ( ( ord_less_eq_real @ A @ X5 )
                  & ( ord_less_eq_real @ X5 @ B2 ) )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ G ) )
           => ( ! [X5: real] :
                  ( ( ( ord_less_real @ A @ X5 )
                    & ( ord_less_real @ X5 @ B2 ) )
                 => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
             => ? [G_c: real,F_c: real,C2: real] :
                  ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
                  & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
                  & ( ord_less_real @ A @ C2 )
                  & ( ord_less_real @ C2 @ B2 )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) ) @ G_c )
                    = ( times_times_real @ ( minus_minus_real @ ( G @ B2 ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).

% GMVT
thf(fact_10036_MVT,axiom,
    ! [A: real,B2: real,F: real > real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
       => ( ! [X5: real] :
              ( ( ord_less_real @ A @ X5 )
             => ( ( ord_less_real @ X5 @ B2 )
               => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
         => ? [L4: real,Z4: real] :
              ( ( ord_less_real @ A @ Z4 )
              & ( ord_less_real @ Z4 @ B2 )
              & ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) )
              & ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) )
                = ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ L4 ) ) ) ) ) ) ).

% MVT
thf(fact_10037_continuous__on__arcosh_H,axiom,
    ! [A4: set_real,F: real > real] :
      ( ( topolo5044208981011980120l_real @ A4 @ F )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ A4 )
           => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
       => ( topolo5044208981011980120l_real @ A4
          @ ^ [X: real] : ( arcosh_real @ ( F @ X ) ) ) ) ) ).

% continuous_on_arcosh'
thf(fact_10038_continuous__image__closed__interval,axiom,
    ! [A: real,B2: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B2 )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
       => ? [C2: real,D3: real] :
            ( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B2 ) )
              = ( set_or1222579329274155063t_real @ C2 @ D3 ) )
            & ( ord_less_eq_real @ C2 @ D3 ) ) ) ) ).

% continuous_image_closed_interval
thf(fact_10039_continuous__on__arcosh,axiom,
    ! [A4: set_real] :
      ( ( ord_less_eq_set_real @ A4 @ ( set_ord_atLeast_real @ one_one_real ) )
     => ( topolo5044208981011980120l_real @ A4 @ arcosh_real ) ) ).

% continuous_on_arcosh
thf(fact_10040_Rolle__deriv,axiom,
    ! [A: real,B2: real,F: real > real,F4: real > real > real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ( F @ A )
          = ( F @ B2 ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
         => ( ! [X5: real] :
                ( ( ord_less_real @ A @ X5 )
               => ( ( ord_less_real @ X5 @ B2 )
                 => ( has_de1759254742604945161l_real @ F @ ( F4 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
           => ? [Z4: real] :
                ( ( ord_less_real @ A @ Z4 )
                & ( ord_less_real @ Z4 @ B2 )
                & ( ( F4 @ Z4 )
                  = ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).

% Rolle_deriv
thf(fact_10041_mvt,axiom,
    ! [A: real,B2: real,F: real > real,F4: real > real > real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
       => ( ! [X5: real] :
              ( ( ord_less_real @ A @ X5 )
             => ( ( ord_less_real @ X5 @ B2 )
               => ( has_de1759254742604945161l_real @ F @ ( F4 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
         => ~ ! [Xi: real] :
                ( ( ord_less_real @ A @ Xi )
               => ( ( ord_less_real @ Xi @ B2 )
                 => ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) )
                   != ( F4 @ Xi @ ( minus_minus_real @ B2 @ A ) ) ) ) ) ) ) ) ).

% mvt
thf(fact_10042_DERIV__pos__imp__increasing__open,axiom,
    ! [A: real,B2: real,F: real > real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ! [X5: real] :
            ( ( ord_less_real @ A @ X5 )
           => ( ( ord_less_real @ X5 @ B2 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
         => ( ord_less_real @ ( F @ A ) @ ( F @ B2 ) ) ) ) ) ).

% DERIV_pos_imp_increasing_open
thf(fact_10043_DERIV__neg__imp__decreasing__open,axiom,
    ! [A: real,B2: real,F: real > real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ! [X5: real] :
            ( ( ord_less_real @ A @ X5 )
           => ( ( ord_less_real @ X5 @ B2 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
                  & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
         => ( ord_less_real @ ( F @ B2 ) @ ( F @ A ) ) ) ) ) ).

% DERIV_neg_imp_decreasing_open
thf(fact_10044_DERIV__isconst__end,axiom,
    ! [A: real,B2: real,F: real > real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
       => ( ! [X5: real] :
              ( ( ord_less_real @ A @ X5 )
             => ( ( ord_less_real @ X5 @ B2 )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
         => ( ( F @ B2 )
            = ( F @ A ) ) ) ) ) ).

% DERIV_isconst_end
thf(fact_10045_continuous__on__artanh,axiom,
    ! [A4: set_real] :
      ( ( ord_less_eq_set_real @ A4 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
     => ( topolo5044208981011980120l_real @ A4 @ artanh_real ) ) ).

% continuous_on_artanh
thf(fact_10046_DERIV__isconst2,axiom,
    ! [A: real,B2: real,F: real > real,X3: real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
       => ( ! [X5: real] :
              ( ( ord_less_real @ A @ X5 )
             => ( ( ord_less_real @ X5 @ B2 )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
         => ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B2 )
             => ( ( F @ X3 )
                = ( F @ A ) ) ) ) ) ) ) ).

% DERIV_isconst2
thf(fact_10047_Rolle,axiom,
    ! [A: real,B2: real,F: real > real] :
      ( ( ord_less_real @ A @ B2 )
     => ( ( ( F @ A )
          = ( F @ B2 ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
         => ( ! [X5: real] :
                ( ( ord_less_real @ A @ X5 )
               => ( ( ord_less_real @ X5 @ B2 )
                 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
           => ? [Z4: real] :
                ( ( ord_less_real @ A @ Z4 )
                & ( ord_less_real @ Z4 @ B2 )
                & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) ) ) ) ) ).

% Rolle
thf(fact_10048_mono__Suc,axiom,
    order_mono_nat_nat @ suc ).

% mono_Suc
thf(fact_10049_mono__times__nat,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).

% mono_times_nat
thf(fact_10050_incseq__bounded,axiom,
    ! [X9: nat > real,B5: real] :
      ( ( order_mono_nat_real @ X9 )
     => ( ! [I3: nat] : ( ord_less_eq_real @ ( X9 @ I3 ) @ B5 )
       => ( bfun_nat_real @ X9 @ at_top_nat ) ) ) ).

% incseq_bounded
thf(fact_10051_incseq__convergent,axiom,
    ! [X9: nat > real,B5: real] :
      ( ( order_mono_nat_real @ X9 )
     => ( ! [I3: nat] : ( ord_less_eq_real @ ( X9 @ I3 ) @ B5 )
       => ~ ! [L6: real] :
              ( ( filterlim_nat_real @ X9 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
             => ~ ! [I2: nat] : ( ord_less_eq_real @ ( X9 @ I2 ) @ L6 ) ) ) ) ).

% incseq_convergent
thf(fact_10052_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( order_mono_nat_nat
        @ ^ [M3: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M3 ) @ M3 ) ) ) ).

% mono_ge2_power_minus_self
thf(fact_10053_nonneg__incseq__Bseq__subseq__iff,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
     => ( ( order_mono_nat_real @ F )
       => ( ( order_5726023648592871131at_nat @ G )
         => ( ( bfun_nat_real
              @ ^ [X: nat] : ( F @ ( G @ X ) )
              @ at_top_nat )
            = ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).

% nonneg_incseq_Bseq_subseq_iff
thf(fact_10054_strict__mono__imp__increasing,axiom,
    ! [F: nat > nat,N: nat] :
      ( ( order_5726023648592871131at_nat @ F )
     => ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).

% strict_mono_imp_increasing
thf(fact_10055_inj__sgn__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( inj_on_real_real
        @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
        @ top_top_set_real ) ) ).

% inj_sgn_power
thf(fact_10056_list__encode_Oelims,axiom,
    ! [X3: list_nat,Y3: nat] :
      ( ( ( nat_list_encode @ X3 )
        = Y3 )
     => ( ( ( X3 = nil_nat )
         => ( Y3 != zero_zero_nat ) )
       => ~ ! [X5: nat,Xs3: list_nat] :
              ( ( X3
                = ( cons_nat @ X5 @ Xs3 ) )
             => ( Y3
               != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X5 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).

% list_encode.elims
thf(fact_10057_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_10058_list__encode_Osimps_I2_J,axiom,
    ! [X3: nat,Xs2: list_nat] :
      ( ( nat_list_encode @ ( cons_nat @ X3 @ Xs2 ) )
      = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ).

% list_encode.simps(2)
thf(fact_10059_pos__deriv__imp__strict__mono,axiom,
    ! [F: real > real,F4: real > real] :
      ( ! [X5: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
     => ( ! [X5: real] : ( ord_less_real @ zero_zero_real @ ( F4 @ X5 ) )
       => ( order_7092887310737990675l_real @ F ) ) ) ).

% pos_deriv_imp_strict_mono
thf(fact_10060_list__encode_Opelims,axiom,
    ! [X3: list_nat,Y3: nat] :
      ( ( ( nat_list_encode @ X3 )
        = Y3 )
     => ( ( accp_list_nat @ nat_list_encode_rel @ X3 )
       => ( ( ( X3 = nil_nat )
           => ( ( Y3 = zero_zero_nat )
             => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
         => ~ ! [X5: nat,Xs3: list_nat] :
                ( ( X3
                  = ( cons_nat @ X5 @ Xs3 ) )
               => ( ( Y3
                    = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X5 @ ( nat_list_encode @ Xs3 ) ) ) ) )
                 => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X5 @ Xs3 ) ) ) ) ) ) ) ).

% list_encode.pelims
thf(fact_10061_inj__on__diff__nat,axiom,
    ! [N4: set_nat,K: nat] :
      ( ! [N3: nat] :
          ( ( member_nat @ N3 @ N4 )
         => ( ord_less_eq_nat @ K @ N3 ) )
     => ( inj_on_nat_nat
        @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ K )
        @ N4 ) ) ).

% inj_on_diff_nat
thf(fact_10062_inj__Suc,axiom,
    ! [N4: set_nat] : ( inj_on_nat_nat @ suc @ N4 ) ).

% inj_Suc
thf(fact_10063_summable__reindex,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
       => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
         => ( summable_real @ ( comp_nat_real_nat @ F @ G ) ) ) ) ) ).

% summable_reindex
thf(fact_10064_suminf__reindex__mono,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
       => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
         => ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G ) ) @ ( suminf_real @ F ) ) ) ) ) ).

% suminf_reindex_mono
thf(fact_10065_inj__on__char__of__nat,axiom,
    inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% inj_on_char_of_nat
thf(fact_10066_suminf__reindex,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
       => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
         => ( ! [X5: nat] :
                ( ~ ( member_nat @ X5 @ ( image_nat_nat @ G @ top_top_set_nat ) )
               => ( ( F @ X5 )
                  = zero_zero_real ) )
           => ( ( suminf_real @ ( comp_nat_real_nat @ F @ G ) )
              = ( suminf_real @ F ) ) ) ) ) ) ).

% suminf_reindex
thf(fact_10067_powr__real__of__int_H,axiom,
    ! [X3: real,N: int] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ( X3 != zero_zero_real )
          | ( ord_less_int @ zero_zero_int @ N ) )
       => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ N ) )
          = ( power_int_real @ X3 @ N ) ) ) ) ).

% powr_real_of_int'
thf(fact_10068_uniformity__real__def,axiom,
    ( topolo1511823702728130853y_real
    = ( comple2936214249959783750l_real
      @ ( image_2178119161166701260l_real
        @ ^ [E3: real] :
            ( princi6114159922880469582l_real
            @ ( collec3799799289383736868l_real
              @ ( produc5414030515140494994real_o
                @ ^ [X: real,Y: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X @ Y ) @ E3 ) ) ) )
        @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% uniformity_real_def
thf(fact_10069_uniformity__complex__def,axiom,
    ( topolo896644834953643431omplex
    = ( comple8358262395181532106omplex
      @ ( image_5971271580939081552omplex
        @ ^ [E3: real] :
            ( princi3496590319149328850omplex
            @ ( collec8663557070575231912omplex
              @ ( produc6771430404735790350plex_o
                @ ^ [X: complex,Y: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X @ Y ) @ E3 ) ) ) )
        @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% uniformity_complex_def
thf(fact_10070_eventually__prod__sequentially,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
      = ( ? [N6: nat] :
          ! [M3: nat] :
            ( ( ord_less_eq_nat @ N6 @ M3 )
           => ! [N2: nat] :
                ( ( ord_less_eq_nat @ N6 @ N2 )
               => ( P @ ( product_Pair_nat_nat @ N2 @ M3 ) ) ) ) ) ) ).

% eventually_prod_sequentially
thf(fact_10071_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_10072_min__0L,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% min_0L
thf(fact_10073_min__0R,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ N @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_10074_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_10075_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_10076_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
thf(fact_10077_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_10078_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q2 ) )
      = ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).

% nat_mult_min_right
thf(fact_10079_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q2 )
      = ( ord_min_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).

% nat_mult_min_left
thf(fact_10080_concat__bit__assoc__sym,axiom,
    ! [M: nat,N: nat,K: int,L: int,R: int] :
      ( ( bit_concat_bit @ M @ ( bit_concat_bit @ N @ K @ L ) @ R )
      = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_concat_bit @ ( minus_minus_nat @ M @ N ) @ L @ R ) ) ) ).

% concat_bit_assoc_sym
thf(fact_10081_take__bit__concat__bit__eq,axiom,
    ! [M: nat,N: nat,K: int,L: int] :
      ( ( bit_se2923211474154528505it_int @ M @ ( bit_concat_bit @ N @ K @ L ) )
      = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ L ) ) ) ).

% take_bit_concat_bit_eq
thf(fact_10082_min__Suc1,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_min_nat @ ( suc @ N ) @ M )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M5: nat] : ( suc @ ( ord_min_nat @ N @ M5 ) )
        @ M ) ) ).

% min_Suc1
thf(fact_10083_min__Suc2,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_min_nat @ M @ ( suc @ N ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M5: nat] : ( suc @ ( ord_min_nat @ M5 @ N ) )
        @ M ) ) ).

% min_Suc2
thf(fact_10084_min__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_mi8085742599997312461d_enat @ zero_z5237406670263579293d_enat @ Q2 )
      = zero_z5237406670263579293d_enat ) ).

% min_enat_simps(3)
thf(fact_10085_min__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_mi8085742599997312461d_enat @ Q2 @ zero_z5237406670263579293d_enat )
      = zero_z5237406670263579293d_enat ) ).

% min_enat_simps(2)
thf(fact_10086_inf__enat__def,axiom,
    inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).

% inf_enat_def
thf(fact_10087_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( hd_nat @ ( upt @ I @ J ) )
        = I ) ) ).

% hd_upt
thf(fact_10088_drop__upt,axiom,
    ! [M: nat,I: nat,J: nat] :
      ( ( drop_nat @ M @ ( upt @ I @ J ) )
      = ( upt @ ( plus_plus_nat @ I @ M ) @ J ) ) ).

% drop_upt
thf(fact_10089_length__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( size_size_list_nat @ ( upt @ I @ J ) )
      = ( minus_minus_nat @ J @ I ) ) ).

% length_upt
thf(fact_10090_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
     => ( ( nth_nat @ ( upt @ I @ J ) @ K )
        = ( plus_plus_nat @ I @ K ) ) ) ).

% nth_upt
thf(fact_10091_take__upt,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N )
     => ( ( take_nat @ M @ ( upt @ I @ N ) )
        = ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).

% take_upt
thf(fact_10092_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( upt @ I @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_10093_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( ( upt @ I @ J )
        = nil_nat )
      = ( ( J = zero_zero_nat )
        | ( ord_less_eq_nat @ J @ I ) ) ) ).

% upt_eq_Nil_conv
thf(fact_10094_upt__rec__numeral,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_10095_map__add__upt,axiom,
    ! [N: nat,M: nat] :
      ( ( map_nat_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N )
        @ ( upt @ zero_zero_nat @ M ) )
      = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).

% map_add_upt
thf(fact_10096_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
        = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_10097_greaterThanAtMost__upt,axiom,
    ( set_or6659071591806873216st_nat
    = ( ^ [N2: nat,M3: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ ( suc @ M3 ) ) ) ) ) ).

% greaterThanAtMost_upt
thf(fact_10098_greaterThanLessThan__upt,axiom,
    ( set_or5834768355832116004an_nat
    = ( ^ [N2: nat,M3: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ M3 ) ) ) ) ).

% greaterThanLessThan_upt
thf(fact_10099_atLeastAtMost__upt,axiom,
    ( set_or1269000886237332187st_nat
    = ( ^ [N2: nat,M3: nat] : ( set_nat2 @ ( upt @ N2 @ ( suc @ M3 ) ) ) ) ) ).

% atLeastAtMost_upt
thf(fact_10100_atLeastLessThan__upt,axiom,
    ( set_or4665077453230672383an_nat
    = ( ^ [I4: nat,J3: nat] : ( set_nat2 @ ( upt @ I4 @ J3 ) ) ) ) ).

% atLeastLessThan_upt
thf(fact_10101_atLeast__upt,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).

% atLeast_upt
thf(fact_10102_atMost__upto,axiom,
    ( set_ord_atMost_nat
    = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N2 ) ) ) ) ) ).

% atMost_upto
thf(fact_10103_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( upt @ I @ J )
        = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).

% upt_conv_Cons
thf(fact_10104_map__decr__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( map_nat_nat
        @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
      = ( upt @ M @ N ) ) ).

% map_decr_upt
thf(fact_10105_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X3: nat,Xs2: list_nat] :
      ( ( ( upt @ I @ J )
        = ( cons_nat @ X3 @ Xs2 ) )
      = ( ( ord_less_nat @ I @ J )
        & ( I = X3 )
        & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
          = Xs2 ) ) ) ).

% upt_eq_Cons_conv
thf(fact_10106_upt__rec,axiom,
    ( upt
    = ( ^ [I4: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I4 @ J3 ) @ ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_10107_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( upt @ I @ ( suc @ J ) )
        = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).

% upt_Suc_append
thf(fact_10108_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
       => ( ( upt @ I @ ( suc @ J ) )
          = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
      & ( ~ ( ord_less_eq_nat @ I @ J )
       => ( ( upt @ I @ ( suc @ J ) )
          = nil_nat ) ) ) ).

% upt_Suc
thf(fact_10109_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_10110_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_10111_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
        = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanAtMost
thf(fact_10112_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
        = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanLessThan
thf(fact_10113_sum__list__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X: nat] : X
          @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ).

% sum_list_upt
thf(fact_10114_card__length__sum__list__rec,axiom,
    ! [M: nat,N4: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M )
     => ( ( finite_card_list_nat
          @ ( collect_list_nat
            @ ^ [L2: list_nat] :
                ( ( ( size_size_list_nat @ L2 )
                  = M )
                & ( ( groups4561878855575611511st_nat @ L2 )
                  = N4 ) ) ) )
        = ( plus_plus_nat
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L2: list_nat] :
                  ( ( ( size_size_list_nat @ L2 )
                    = ( minus_minus_nat @ M @ one_one_nat ) )
                  & ( ( groups4561878855575611511st_nat @ L2 )
                    = N4 ) ) ) )
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L2: list_nat] :
                  ( ( ( size_size_list_nat @ L2 )
                    = M )
                  & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L2 ) @ one_one_nat )
                    = N4 ) ) ) ) ) ) ) ).

% card_length_sum_list_rec
thf(fact_10115_card__length__sum__list,axiom,
    ! [M: nat,N4: nat] :
      ( ( finite_card_list_nat
        @ ( collect_list_nat
          @ ^ [L2: list_nat] :
              ( ( ( size_size_list_nat @ L2 )
                = M )
              & ( ( groups4561878855575611511st_nat @ L2 )
                = N4 ) ) ) )
      = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N4 @ M ) @ one_one_nat ) @ N4 ) ) ).

% card_length_sum_list
thf(fact_10116_sorted__upt,axiom,
    ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).

% sorted_upt
thf(fact_10117_sorted__wrt__upt,axiom,
    ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).

% sorted_wrt_upt
thf(fact_10118_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I: nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
       => ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).

% sorted_wrt_less_idx
thf(fact_10119_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).

% sorted_wrt_upto
thf(fact_10120_sorted__upto,axiom,
    ! [M: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N ) ) ).

% sorted_upto
thf(fact_10121_VEBT_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
      ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).

% VEBT.size(3)
thf(fact_10122_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
      ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).

% VEBT.size_gen(1)
thf(fact_10123_card__le__Suc__Max,axiom,
    ! [S2: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ord_less_eq_nat @ ( finite_card_nat @ S2 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S2 ) ) ) ) ).

% card_le_Suc_Max
thf(fact_10124_divide__nat__def,axiom,
    ( divide_divide_nat
    = ( ^ [M3: nat,N2: nat] :
          ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
          @ ( lattic8265883725875713057ax_nat
            @ ( collect_nat
              @ ^ [K2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K2 @ N2 ) @ M3 ) ) ) ) ) ) ).

% divide_nat_def
thf(fact_10125_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( gcd_gcd_nat @ M @ N )
        = ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D2: nat] :
                ( ( dvd_dvd_nat @ D2 @ M )
                & ( dvd_dvd_nat @ D2 @ N ) ) ) ) ) ) ).

% gcd_is_Max_divisors_nat
thf(fact_10126_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_10127_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
thf(fact_10128_sup__enat__def,axiom,
    sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).

% sup_enat_def
thf(fact_10129_quotient__of__def,axiom,
    ( quotient_of
    = ( ^ [X: rat] :
          ( the_Pr4378521158711661632nt_int
          @ ^ [Pair: product_prod_int_int] :
              ( ( X
                = ( fract @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) )
              & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Pair ) )
              & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) ) ) ) ) ).

% quotient_of_def
thf(fact_10130_coprime__abs__left__iff,axiom,
    ! [K: int,L: int] :
      ( ( algebr932160517623751201me_int @ ( abs_abs_int @ K ) @ L )
      = ( algebr932160517623751201me_int @ K @ L ) ) ).

% coprime_abs_left_iff
thf(fact_10131_coprime__abs__right__iff,axiom,
    ! [K: int,L: int] :
      ( ( algebr932160517623751201me_int @ K @ ( abs_abs_int @ L ) )
      = ( algebr932160517623751201me_int @ K @ L ) ) ).

% coprime_abs_right_iff
thf(fact_10132_normalize__stable,axiom,
    ! [Q2: int,P4: int] :
      ( ( ord_less_int @ zero_zero_int @ Q2 )
     => ( ( algebr932160517623751201me_int @ P4 @ Q2 )
       => ( ( normalize @ ( product_Pair_int_int @ P4 @ Q2 ) )
          = ( product_Pair_int_int @ P4 @ Q2 ) ) ) ) ).

% normalize_stable
thf(fact_10133_Rat__cases,axiom,
    ! [Q2: rat] :
      ~ ! [A3: int,B3: int] :
          ( ( Q2
            = ( fract @ A3 @ B3 ) )
         => ( ( ord_less_int @ zero_zero_int @ B3 )
           => ~ ( algebr932160517623751201me_int @ A3 @ B3 ) ) ) ).

% Rat_cases
thf(fact_10134_Rat__induct,axiom,
    ! [P: rat > $o,Q2: rat] :
      ( ! [A3: int,B3: int] :
          ( ( ord_less_int @ zero_zero_int @ B3 )
         => ( ( algebr932160517623751201me_int @ A3 @ B3 )
           => ( P @ ( fract @ A3 @ B3 ) ) ) )
     => ( P @ Q2 ) ) ).

% Rat_induct
thf(fact_10135_quotient__of__coprime,axiom,
    ! [R: rat,P4: int,Q2: int] :
      ( ( ( quotient_of @ R )
        = ( product_Pair_int_int @ P4 @ Q2 ) )
     => ( algebr932160517623751201me_int @ P4 @ Q2 ) ) ).

% quotient_of_coprime
thf(fact_10136_normalize__coprime,axiom,
    ! [R: product_prod_int_int,P4: int,Q2: int] :
      ( ( ( normalize @ R )
        = ( product_Pair_int_int @ P4 @ Q2 ) )
     => ( algebr932160517623751201me_int @ P4 @ Q2 ) ) ).

% normalize_coprime
thf(fact_10137_coprime__crossproduct__int,axiom,
    ! [A: int,D: int,B2: int,C: int] :
      ( ( algebr932160517623751201me_int @ A @ D )
     => ( ( algebr932160517623751201me_int @ B2 @ C )
       => ( ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ C ) )
            = ( times_times_int @ ( abs_abs_int @ B2 ) @ ( abs_abs_int @ D ) ) )
          = ( ( ( abs_abs_int @ A )
              = ( abs_abs_int @ B2 ) )
            & ( ( abs_abs_int @ C )
              = ( abs_abs_int @ D ) ) ) ) ) ) ).

% coprime_crossproduct_int
thf(fact_10138_coprime__common__divisor__int,axiom,
    ! [A: int,B2: int,X3: int] :
      ( ( algebr932160517623751201me_int @ A @ B2 )
     => ( ( dvd_dvd_int @ X3 @ A )
       => ( ( dvd_dvd_int @ X3 @ B2 )
         => ( ( abs_abs_int @ X3 )
            = one_one_int ) ) ) ) ).

% coprime_common_divisor_int
thf(fact_10139_Rat__cases__nonzero,axiom,
    ! [Q2: rat] :
      ( ! [A3: int,B3: int] :
          ( ( Q2
            = ( fract @ A3 @ B3 ) )
         => ( ( ord_less_int @ zero_zero_int @ B3 )
           => ( ( A3 != zero_zero_int )
             => ~ ( algebr932160517623751201me_int @ A3 @ B3 ) ) ) )
     => ( Q2 = zero_zero_rat ) ) ).

% Rat_cases_nonzero
thf(fact_10140_quotient__of__unique,axiom,
    ! [R: rat] :
    ? [X5: product_prod_int_int] :
      ( ( R
        = ( fract @ ( product_fst_int_int @ X5 ) @ ( product_snd_int_int @ X5 ) ) )
      & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ X5 ) )
      & ( algebr932160517623751201me_int @ ( product_fst_int_int @ X5 ) @ ( product_snd_int_int @ X5 ) )
      & ! [Y5: product_prod_int_int] :
          ( ( ( R
              = ( fract @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
            & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Y5 ) )
            & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
         => ( Y5 = X5 ) ) ) ).

% quotient_of_unique
thf(fact_10141_coprime__int__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( algebr934650988132801477me_nat @ M @ N ) ) ).

% coprime_int_iff
thf(fact_10142_coprime__nat__abs__right__iff,axiom,
    ! [N: nat,K: int] :
      ( ( algebr934650988132801477me_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
      = ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ).

% coprime_nat_abs_right_iff
thf(fact_10143_coprime__nat__abs__left__iff,axiom,
    ! [K: int,N: nat] :
      ( ( algebr934650988132801477me_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
      = ( algebr932160517623751201me_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% coprime_nat_abs_left_iff
thf(fact_10144_coprime__Suc__0__left,axiom,
    ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N ) ).

% coprime_Suc_0_left
thf(fact_10145_coprime__Suc__0__right,axiom,
    ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ zero_zero_nat ) ) ).

% coprime_Suc_0_right
thf(fact_10146_coprime__crossproduct__nat,axiom,
    ! [A: nat,D: nat,B2: nat,C: nat] :
      ( ( algebr934650988132801477me_nat @ A @ D )
     => ( ( algebr934650988132801477me_nat @ B2 @ C )
       => ( ( ( times_times_nat @ A @ C )
            = ( times_times_nat @ B2 @ D ) )
          = ( ( A = B2 )
            & ( C = D ) ) ) ) ) ).

% coprime_crossproduct_nat
thf(fact_10147_coprime__common__divisor__nat,axiom,
    ! [A: nat,B2: nat,X3: nat] :
      ( ( algebr934650988132801477me_nat @ A @ B2 )
     => ( ( dvd_dvd_nat @ X3 @ A )
       => ( ( dvd_dvd_nat @ X3 @ B2 )
         => ( X3 = one_one_nat ) ) ) ) ).

% coprime_common_divisor_nat
thf(fact_10148_Gcd__int__set__eq__fold,axiom,
    ! [Xs2: list_int] :
      ( ( gcd_Gcd_int @ ( set_int2 @ Xs2 ) )
      = ( fold_int_int @ gcd_gcd_int @ Xs2 @ zero_zero_int ) ) ).

% Gcd_int_set_eq_fold
thf(fact_10149_Gcd__nat__set__eq__fold,axiom,
    ! [Xs2: list_nat] :
      ( ( gcd_Gcd_nat @ ( set_nat2 @ Xs2 ) )
      = ( fold_nat_nat @ gcd_gcd_nat @ Xs2 @ zero_zero_nat ) ) ).

% Gcd_nat_set_eq_fold
thf(fact_10150_Gcd__int__greater__eq__0,axiom,
    ! [K5: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K5 ) ) ).

% Gcd_int_greater_eq_0
thf(fact_10151_coprime__diff__one__right__nat,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( algebr934650988132801477me_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).

% coprime_diff_one_right_nat
thf(fact_10152_coprime__diff__one__left__nat,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).

% coprime_diff_one_left_nat
thf(fact_10153_Rats__abs__nat__div__natE,axiom,
    ! [X3: real] :
      ( ( member_real @ X3 @ field_5140801741446780682s_real )
     => ~ ! [M4: nat,N3: nat] :
            ( ( N3 != zero_zero_nat )
           => ( ( ( abs_abs_real @ X3 )
                = ( divide_divide_real @ ( semiri5074537144036343181t_real @ M4 ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
             => ~ ( algebr934650988132801477me_nat @ M4 @ N3 ) ) ) ) ).

% Rats_abs_nat_div_natE
thf(fact_10154_Rat_Opositive_Orsp,axiom,
    ( bNF_re8699439704749558557nt_o_o @ ratrel
    @ ^ [Y6: $o,Z5: $o] : Y6 = Z5
    @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) )
    @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ).

% Rat.positive.rsp
thf(fact_10155_less__eq__integer_Orsp,axiom,
    ( bNF_re3403563459893282935_int_o
    @ ^ [Y6: int,Z5: int] : Y6 = Z5
    @ ( bNF_re5089333283451836215nt_o_o
      @ ^ [Y6: int,Z5: int] : Y6 = Z5
      @ ^ [Y6: $o,Z5: $o] : Y6 = Z5 )
    @ ord_less_eq_int
    @ ord_less_eq_int ) ).

% less_eq_integer.rsp
thf(fact_10156_less__eq__natural_Orsp,axiom,
    ( bNF_re578469030762574527_nat_o
    @ ^ [Y6: nat,Z5: nat] : Y6 = Z5
    @ ( bNF_re4705727531993890431at_o_o
      @ ^ [Y6: nat,Z5: nat] : Y6 = Z5
      @ ^ [Y6: $o,Z5: $o] : Y6 = Z5 )
    @ ord_less_eq_nat
    @ ord_less_eq_nat ) ).

% less_eq_natural.rsp
thf(fact_10157_plus__rat_Orsp,axiom,
    ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
    @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) )
    @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ).

% plus_rat.rsp
thf(fact_10158_times__rat_Orsp,axiom,
    ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
    @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) )
    @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ).

% times_rat.rsp
thf(fact_10159_Fract_Orsp,axiom,
    ( bNF_re157797125943740599nt_int
    @ ^ [Y6: int,Z5: int] : Y6 = Z5
    @ ( bNF_re6250860962936578807nt_int
      @ ^ [Y6: int,Z5: int] : Y6 = Z5
      @ ratrel )
    @ ^ [A5: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( B4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A5 @ B4 ) )
    @ ^ [A5: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( B4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A5 @ B4 ) ) ) ).

% Fract.rsp
thf(fact_10160_plus__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y6: nat,Z5: nat] : Y6 = Z5
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y6: nat,Z5: nat] : Y6 = Z5
      @ ^ [Y6: nat,Z5: nat] : Y6 = Z5 )
    @ plus_plus_nat
    @ plus_plus_nat ) ).

% plus_natural.rsp
thf(fact_10161_minus__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y6: nat,Z5: nat] : Y6 = Z5
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y6: nat,Z5: nat] : Y6 = Z5
      @ ^ [Y6: nat,Z5: nat] : Y6 = Z5 )
    @ minus_minus_nat
    @ minus_minus_nat ) ).

% minus_natural.rsp
thf(fact_10162_times__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y6: nat,Z5: nat] : Y6 = Z5
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y6: nat,Z5: nat] : Y6 = Z5
      @ ^ [Y6: nat,Z5: nat] : Y6 = Z5 )
    @ times_times_nat
    @ times_times_nat ) ).

% times_natural.rsp
thf(fact_10163_times__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y6: int,Z5: int] : Y6 = Z5
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y6: int,Z5: int] : Y6 = Z5
      @ ^ [Y6: int,Z5: int] : Y6 = Z5 )
    @ times_times_int
    @ times_times_int ) ).

% times_integer.rsp
thf(fact_10164_divide__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y6: nat,Z5: nat] : Y6 = Z5
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y6: nat,Z5: nat] : Y6 = Z5
      @ ^ [Y6: nat,Z5: nat] : Y6 = Z5 )
    @ divide_divide_nat
    @ divide_divide_nat ) ).

% divide_natural.rsp
thf(fact_10165_sub_Orsp,axiom,
    ( bNF_re8402795839162346335um_int
    @ ^ [Y6: num,Z5: num] : Y6 = Z5
    @ ( bNF_re1822329894187522285nt_int
      @ ^ [Y6: num,Z5: num] : Y6 = Z5
      @ ^ [Y6: int,Z5: int] : Y6 = Z5 )
    @ ^ [M3: num,N2: num] : ( minus_minus_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) )
    @ ^ [M3: num,N2: num] : ( minus_minus_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) ) ).

% sub.rsp
thf(fact_10166_less__natural_Orsp,axiom,
    ( bNF_re578469030762574527_nat_o
    @ ^ [Y6: nat,Z5: nat] : Y6 = Z5
    @ ( bNF_re4705727531993890431at_o_o
      @ ^ [Y6: nat,Z5: nat] : Y6 = Z5
      @ ^ [Y6: $o,Z5: $o] : Y6 = Z5 )
    @ ord_less_nat
    @ ord_less_nat ) ).

% less_natural.rsp
thf(fact_10167_less__integer_Orsp,axiom,
    ( bNF_re3403563459893282935_int_o
    @ ^ [Y6: int,Z5: int] : Y6 = Z5
    @ ( bNF_re5089333283451836215nt_o_o
      @ ^ [Y6: int,Z5: int] : Y6 = Z5
      @ ^ [Y6: $o,Z5: $o] : Y6 = Z5 )
    @ ord_less_int
    @ ord_less_int ) ).

% less_integer.rsp
thf(fact_10168_uminus__rat_Orsp,axiom,
    ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
    @ ^ [X: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X ) ) @ ( product_snd_int_int @ X ) )
    @ ^ [X: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X ) ) @ ( product_snd_int_int @ X ) ) ) ).

% uminus_rat.rsp
thf(fact_10169_inverse__rat_Orsp,axiom,
    ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
    @ ^ [X: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) )
    @ ^ [X: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) ) ) ).

% inverse_rat.rsp
thf(fact_10170_plus__rat_Otransfer,axiom,
    ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
    @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) )
    @ plus_plus_rat ) ).

% plus_rat.transfer
thf(fact_10171_one__rat_Otransfer,axiom,
    pcr_rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ one_one_rat ).

% one_rat.transfer
thf(fact_10172_zero__rat_Otransfer,axiom,
    pcr_rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ zero_zero_rat ).

% zero_rat.transfer
thf(fact_10173_Fract_Otransfer,axiom,
    ( bNF_re3461391660133120880nt_rat
    @ ^ [Y6: int,Z5: int] : Y6 = Z5
    @ ( bNF_re2214769303045360666nt_rat
      @ ^ [Y6: int,Z5: int] : Y6 = Z5
      @ pcr_rat )
    @ ^ [A5: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( B4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A5 @ B4 ) )
    @ fract ) ).

% Fract.transfer
thf(fact_10174_uminus__rat_Otransfer,axiom,
    ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
    @ ^ [X: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X ) ) @ ( product_snd_int_int @ X ) )
    @ uminus_uminus_rat ) ).

% uminus_rat.transfer
thf(fact_10175_times__rat_Otransfer,axiom,
    ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
    @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) )
    @ times_times_rat ) ).

% times_rat.transfer
thf(fact_10176_Rat_Opositive_Otransfer,axiom,
    ( bNF_re1494630372529172596at_o_o @ pcr_rat
    @ ^ [Y6: $o,Z5: $o] : Y6 = Z5
    @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) )
    @ positive ) ).

% Rat.positive.transfer
thf(fact_10177_inverse__rat_Otransfer,axiom,
    ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
    @ ^ [X: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) )
    @ inverse_inverse_rat ) ).

% inverse_rat.transfer
thf(fact_10178_times__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X: nat,Y: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U2 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U2 ) ) ) ) )
    @ times_times_int ) ).

% times_int.transfer
thf(fact_10179_zero__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).

% zero_int.transfer
thf(fact_10180_int__transfer,axiom,
    ( bNF_re6830278522597306478at_int
    @ ^ [Y6: nat,Z5: nat] : Y6 = Z5
    @ pcr_int
    @ ^ [N2: nat] : ( product_Pair_nat_nat @ N2 @ zero_zero_nat )
    @ semiri1314217659103216013at_int ) ).

% int_transfer
thf(fact_10181_uminus__int_Otransfer,axiom,
    ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int
    @ ( produc2626176000494625587at_nat
      @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X ) )
    @ uminus_uminus_int ) ).

% uminus_int.transfer
thf(fact_10182_nat_Otransfer,axiom,
    ( bNF_re4555766996558763186at_nat @ pcr_int
    @ ^ [Y6: nat,Z5: nat] : Y6 = Z5
    @ ( produc6842872674320459806at_nat @ minus_minus_nat )
    @ nat2 ) ).

% nat.transfer
thf(fact_10183_one__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).

% one_int.transfer
thf(fact_10184_less__int_Otransfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y6: $o,Z5: $o] : Y6 = Z5 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X: nat,Y: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
    @ ord_less_int ) ).

% less_int.transfer
thf(fact_10185_less__eq__int_Otransfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y6: $o,Z5: $o] : Y6 = Z5 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X: nat,Y: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
    @ ord_less_eq_int ) ).

% less_eq_int.transfer
thf(fact_10186_plus__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X: nat,Y: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) )
    @ plus_plus_int ) ).

% plus_int.transfer
thf(fact_10187_minus__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X: nat,Y: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) ) )
    @ minus_minus_int ) ).

% minus_int.transfer
thf(fact_10188_times__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X: nat,Y: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U2 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U2 ) ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X: nat,Y: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U2 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U2 ) ) ) ) ) ) ).

% times_int.rsp
thf(fact_10189_intrel__iff,axiom,
    ! [X3: nat,Y3: nat,U: nat,V: nat] :
      ( ( intrel @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( product_Pair_nat_nat @ U @ V ) )
      = ( ( plus_plus_nat @ X3 @ V )
        = ( plus_plus_nat @ U @ Y3 ) ) ) ).

% intrel_iff
thf(fact_10190_nat_Orsp,axiom,
    ( bNF_re8246922863344978751at_nat @ intrel
    @ ^ [Y6: nat,Z5: nat] : Y6 = Z5
    @ ( produc6842872674320459806at_nat @ minus_minus_nat )
    @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ).

% nat.rsp
thf(fact_10191_uminus__int_Orsp,axiom,
    ( bNF_re2241393799969408733at_nat @ intrel @ intrel
    @ ( produc2626176000494625587at_nat
      @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X ) )
    @ ( produc2626176000494625587at_nat
      @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X ) ) ) ).

% uminus_int.rsp
thf(fact_10192_one__int_Orsp,axiom,
    intrel @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ).

% one_int.rsp
thf(fact_10193_zero__int_Orsp,axiom,
    intrel @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ).

% zero_int.rsp
thf(fact_10194_intrel__def,axiom,
    ( intrel
    = ( produc8739625826339149834_nat_o
      @ ^ [X: nat,Y: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] :
              ( ( plus_plus_nat @ X @ V4 )
              = ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ).

% intrel_def
thf(fact_10195_less__int_Orsp,axiom,
    ( bNF_re4202695980764964119_nat_o @ intrel
    @ ( bNF_re3666534408544137501at_o_o @ intrel
      @ ^ [Y6: $o,Z5: $o] : Y6 = Z5 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X: nat,Y: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X: nat,Y: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ).

% less_int.rsp
thf(fact_10196_less__eq__int_Orsp,axiom,
    ( bNF_re4202695980764964119_nat_o @ intrel
    @ ( bNF_re3666534408544137501at_o_o @ intrel
      @ ^ [Y6: $o,Z5: $o] : Y6 = Z5 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X: nat,Y: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X: nat,Y: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ).

% less_eq_int.rsp
thf(fact_10197_minus__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X: nat,Y: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X: nat,Y: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) ) ) ) ).

% minus_int.rsp
thf(fact_10198_plus__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X: nat,Y: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X: nat,Y: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) ) ) ).

% plus_int.rsp
thf(fact_10199_vanishes__mult__bounded,axiom,
    ! [X9: nat > rat,Y7: nat > rat] :
      ( ? [A8: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ A8 )
          & ! [N3: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N3 ) ) @ A8 ) )
     => ( ( vanishes @ Y7 )
       => ( vanishes
          @ ^ [N2: nat] : ( times_times_rat @ ( X9 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).

% vanishes_mult_bounded
thf(fact_10200_vanishes__const,axiom,
    ! [C: rat] :
      ( ( vanishes
        @ ^ [N2: nat] : C )
      = ( C = zero_zero_rat ) ) ).

% vanishes_const
thf(fact_10201_vanishes__minus,axiom,
    ! [X9: nat > rat] :
      ( ( vanishes @ X9 )
     => ( vanishes
        @ ^ [N2: nat] : ( uminus_uminus_rat @ ( X9 @ N2 ) ) ) ) ).

% vanishes_minus
thf(fact_10202_vanishes__add,axiom,
    ! [X9: nat > rat,Y7: nat > rat] :
      ( ( vanishes @ X9 )
     => ( ( vanishes @ Y7 )
       => ( vanishes
          @ ^ [N2: nat] : ( plus_plus_rat @ ( X9 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).

% vanishes_add
thf(fact_10203_vanishes__diff,axiom,
    ! [X9: nat > rat,Y7: nat > rat] :
      ( ( vanishes @ X9 )
     => ( ( vanishes @ Y7 )
       => ( vanishes
          @ ^ [N2: nat] : ( minus_minus_rat @ ( X9 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).

% vanishes_diff
thf(fact_10204_vanishes__def,axiom,
    ( vanishes
    = ( ^ [X7: nat > rat] :
        ! [R5: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R5 )
         => ? [K2: nat] :
            ! [N2: nat] :
              ( ( ord_less_eq_nat @ K2 @ N2 )
             => ( ord_less_rat @ ( abs_abs_rat @ ( X7 @ N2 ) ) @ R5 ) ) ) ) ) ).

% vanishes_def
thf(fact_10205_vanishesI,axiom,
    ! [X9: nat > rat] :
      ( ! [R3: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R3 )
         => ? [K4: nat] :
            ! [N3: nat] :
              ( ( ord_less_eq_nat @ K4 @ N3 )
             => ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N3 ) ) @ R3 ) ) )
     => ( vanishes @ X9 ) ) ).

% vanishesI
thf(fact_10206_vanishesD,axiom,
    ! [X9: nat > rat,R: rat] :
      ( ( vanishes @ X9 )
     => ( ( ord_less_rat @ zero_zero_rat @ R )
       => ? [K3: nat] :
          ! [N9: nat] :
            ( ( ord_less_eq_nat @ K3 @ N9 )
           => ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N9 ) ) @ R ) ) ) ) ).

% vanishesD
thf(fact_10207_Field__natLeq__on,axiom,
    ! [N: nat] :
      ( ( field_nat
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X: nat,Y: nat] :
                ( ( ord_less_nat @ X @ N )
                & ( ord_less_nat @ Y @ N )
                & ( ord_less_eq_nat @ X @ Y ) ) ) ) )
      = ( collect_nat
        @ ^ [X: nat] : ( ord_less_nat @ X @ N ) ) ) ).

% Field_natLeq_on
thf(fact_10208_natLess__def,axiom,
    ( bNF_Ca8459412986667044542atLess
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).

% natLess_def

% Helper facts (38)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X3: int,Y3: int] :
      ( ( if_int @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X3: int,Y3: int] :
      ( ( if_int @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( if_nat @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( if_nat @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
    ! [X3: num,Y3: num] :
      ( ( if_num @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
    ! [X3: num,Y3: num] :
      ( ( if_num @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( if_rat @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( if_rat @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X3: real,Y3: real] :
      ( ( if_real @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X3: real,Y3: real] :
      ( ( if_real @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: real > $o] :
      ( ( P @ ( fChoice_real @ P ) )
      = ( ? [X7: real] : ( P @ X7 ) ) ) ).

thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( if_complex @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( if_complex @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X3: extended_enat,Y3: extended_enat] :
      ( ( if_Extended_enat @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X3: extended_enat,Y3: extended_enat] :
      ( ( if_Extended_enat @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X3: code_integer,Y3: code_integer] :
      ( ( if_Code_integer @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X3: code_integer,Y3: code_integer] :
      ( ( if_Code_integer @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X3: set_int,Y3: set_int] :
      ( ( if_set_int @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X3: set_int,Y3: set_int] :
      ( ( if_set_int @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X3: vEBT_VEBT,Y3: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X3: vEBT_VEBT,Y3: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X3: list_int,Y3: list_int] :
      ( ( if_list_int @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X3: list_int,Y3: list_int] :
      ( ( if_list_int @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X3: list_nat,Y3: list_nat] :
      ( ( if_list_nat @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X3: list_nat,Y3: list_nat] :
      ( ( if_list_nat @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X3: option_nat,Y3: option_nat] :
      ( ( if_option_nat @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X3: option_nat,Y3: option_nat] :
      ( ( if_option_nat @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X3: option_num,Y3: option_num] :
      ( ( if_option_num @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X3: option_num,Y3: option_num] :
      ( ( if_option_num @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X3: product_prod_int_int,Y3: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X3: product_prod_int_int,Y3: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X3: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X3: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X3: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X3: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $true @ X3 @ Y3 )
      = X3 ) ).

% Conjectures (2)
thf(conj_0,hypothesis,
    ! [Minilow: nat] :
      ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        = ( some_nat @ Minilow ) )
     => thesis ) ).

thf(conj_1,conjecture,
    thesis ).

%------------------------------------------------------------------------------